+ All Categories
Home > Documents > Calculus Video Book

Calculus Video Book

Date post: 14-Apr-2018
Category:
Upload: harpreet-bedi
View: 227 times
Download: 0 times
Share this document with a friend

of 35

Transcript
  • 7/27/2019 Calculus Video Book

    1/35

    Calculus

    H a r p r e e t S i n g h B e d i

    h 2 b e d i @ g m a i l . c o m

    5 7 0 4 2 8 5 4 0 4

    O c t o b e r 2 7 , 2 0 1 3

    mailto:[email protected]
  • 7/27/2019 Calculus Video Book

    2/35

    2

  • 7/27/2019 Calculus Video Book

    3/35

    C o n t e n t s

    1 S e t s , R e l a t i o n s a n d F u n c t i o n s 7

    2 L i m i t s a n d D e r i v a t i v e s 1 1

    3 C o n t i n u i t y a n d D i e r e n t i a b i l i t y 1 5

    4 A p p l i c a t i o n s o f D i e r e n t i a t i o n 1 9

    5 I n t e g r a t i o n 2 5

    6 A p p l i c a t i o n o f I n t e g r a t i o n 3 1

    7 D i e r e n t i a l E q u a t i o n s 3 3

    3

  • 7/27/2019 Calculus Video Book

    4/35

    4 C o n t e n t s

  • 7/27/2019 Calculus Video Book

    5/35

    P r e f a c e

    T h i s i s a v i d e o b o o k f o r w o r k i n g p r o f e s s i o n a l s

    w h o w a n t t o l e a r n c a l c u l u s f r o m t h e b e g i n n i n g

    o r b r u s h u p t h e i r c u r r e n t k n o w l e d g e . T h e l i n k s

    t o v i d e o s a r e g i v e n i n e a c h c h a p t e r i n c o l o r , a n d

    t h e a c c o m p a n y i n g t e x t g i v e s a s m a l l s u m m a r y o f

    t h e l e c t u r e .

    E a c h v i d e o i s a p p r o x i m a t e l y t e n m i n u t e s , t h u s

    c a n b e p r o t a b l y s e e n a t l u n c h b r e a k .

    5

  • 7/27/2019 Calculus Video Book

    6/35

    6 C o n t e n t s

  • 7/27/2019 Calculus Video Book

    7/35

    1 . S e t s , R e l a t i o n s

    a n d F u n c t i o n s

    G a l i l e o G a l i l e i

    I f I w e r e a g a i n b e g i n n i n g m y s t u d i e s , I w o u l d

    f o l l o w t h e a d v i c e o f P l a t o a n d s t a r t w i t h m a t h -

    e m a t i c s .

    S e t s f o r m t h e f u n d a m e n t a l b u i l d i n g b l o c k o f m a t h -

    e m a t i c s . W e s t a r t b y e x p l a i n i n g t h e c o n c e p t o f

    a s e t a n d t h e n p r o c e e d o n t o r e l a t i o n s a n d f u n c -

    7

  • 7/27/2019 Calculus Video Book

    8/35

    8 S e t s , R e l a t i o n s a n d F u n c t i o n s

    t i o n s .

    1 . Define sets in tabular form, set builder formW e s t a r t b y d e n i n g s e t s , a n d h o w t o w r i t e

    t h e m i n t a b u l a r f o r m , o r s e t b u i l d e r f o r m .

    W e t h e n m o v e o n t o d e n i n g e m p t y s e t s ,

    n i t e s e t s a n d i n n i t e s e t s .

    2 . Define Subsets and Venn diagrams I n t h i s l e c t u r e w e d e n e s u b s e t s o f s e t s . A s a p a r -

    t i c u l a r e x a m p l e w e s h o w i n t e r v a l s a s s u b -

    s e t s o f r e a l l i n e a n d d r a w u n i v e r s a l s e t s a n d

    v e n n d i a g r a m s .

    3 .

    Operations on sets, unions, intersectionsand their properties.W e c a r r y o u t b a s i c

    o p e r a t i o n s o f u n i o n s a n d i n t e r s e c t i o n s o n

    s e t s .

    4 . Difference and Complements of Sets W e e x - p l a i n t h e d i e r e n c e o f t w o s e t s a n d c o m -

    p l e m e n t o f s e t s .

    5 .Cartesian product of sets and Relations

    W e

    d e n e t h e c a r t e s i a n p r o d u c t o f s e t s , w e

    t h e n c o n t i n u e t o d e n e r e l a t i o n a s a s u b -

    s e t o f c a r t e s i a n p r o d u c t o f s e t s . W e a l s o

    http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f266794a6f6351e000003?auth=1697328653-0-kbi5crzy-a1d3eaf494091ee306eecb1cbee4621ehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f266794a6f6351e000003?auth=1697328653-0-kbi5crzy-a1d3eaf494091ee306eecb1cbee4621ehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f266794a6f6351e000003?auth=1697328653-0-kbi5crzy-a1d3eaf494091ee306eecb1cbee4621ehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f266794a6f6351e000003?auth=1697328653-0-kbi5crzy-a1d3eaf494091ee306eecb1cbee4621ehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f266794a6f6351e000003?auth=1697328653-0-kbi5crzy-a1d3eaf494091ee306eecb1cbee4621ehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f266794a6f6351e000003?auth=1697328653-0-kbi5crzy-a1d3eaf494091ee306eecb1cbee4621ehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f26f406361d4ead000008?auth=1697328795-0-na8ezvx7-35f3388e2ed0a393f4eba8eb82aaa320http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f26f406361d4ead000008?auth=1697328795-0-na8ezvx7-35f3388e2ed0a393f4eba8eb82aaa320http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f26f406361d4ead000008?auth=1697328795-0-na8ezvx7-35f3388e2ed0a393f4eba8eb82aaa320http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f26f406361d4ead000008?auth=1697328795-0-na8ezvx7-35f3388e2ed0a393f4eba8eb82aaa320http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f26f406361d4ead000008?auth=1697328795-0-na8ezvx7-35f3388e2ed0a393f4eba8eb82aaa320http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f26f406361d4ead000008?auth=1697328795-0-na8ezvx7-35f3388e2ed0a393f4eba8eb82aaa320http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f26f406361d4ead000008?auth=1697328795-0-na8ezvx7-35f3388e2ed0a393f4eba8eb82aaa320http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f276306361d4e9a000005?auth=1697328978-0-1yc7mt3i-81482cacffe6043720240ba92cf760f7http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f276306361d4e9a000005?auth=1697328978-0-1yc7mt3i-81482cacffe6043720240ba92cf760f7http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f276306361d4e9a000005?auth=1697328978-0-1yc7mt3i-81482cacffe6043720240ba92cf760f7http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f276306361d4e9a000005?auth=1697328978-0-1yc7mt3i-81482cacffe6043720240ba92cf760f7http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f276306361d4e9a000005?auth=1697328978-0-1yc7mt3i-81482cacffe6043720240ba92cf760f7http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f276306361d4e9a000005?auth=1697328978-0-1yc7mt3i-81482cacffe6043720240ba92cf760f7http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f27df06361d4ef8000001?auth=1697329053-0-1butyrgw-b3823789f47f5627092d2e2451c5c10bhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f27df06361d4ef8000001?auth=1697329053-0-1butyrgw-b3823789f47f5627092d2e2451c5c10bhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f27df06361d4ef8000001?auth=1697329053-0-1butyrgw-b3823789f47f5627092d2e2451c5c10bhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f27df06361d4ef8000001?auth=1697329053-0-1butyrgw-b3823789f47f5627092d2e2451c5c10bhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f27df06361d4ef8000001?auth=1697329053-0-1butyrgw-b3823789f47f5627092d2e2451c5c10bhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f288906361d4ef8000002?auth=1697329134-0-w0xdj39t-fe161d0801386c36e8f626373ef4bfe9http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f288906361d4ef8000002?auth=1697329134-0-w0xdj39t-fe161d0801386c36e8f626373ef4bfe9http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f288906361d4ef8000002?auth=1697329134-0-w0xdj39t-fe161d0801386c36e8f626373ef4bfe9http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f288906361d4ef8000002?auth=1697329134-0-w0xdj39t-fe161d0801386c36e8f626373ef4bfe9http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f288906361d4ef8000002?auth=1697329134-0-w0xdj39t-fe161d0801386c36e8f626373ef4bfe9http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f288906361d4ef8000002?auth=1697329134-0-w0xdj39t-fe161d0801386c36e8f626373ef4bfe9http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f27df06361d4ef8000001?auth=1697329053-0-1butyrgw-b3823789f47f5627092d2e2451c5c10bhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f276306361d4e9a000005?auth=1697328978-0-1yc7mt3i-81482cacffe6043720240ba92cf760f7http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f276306361d4e9a000005?auth=1697328978-0-1yc7mt3i-81482cacffe6043720240ba92cf760f7http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f26f406361d4ead000008?auth=1697328795-0-na8ezvx7-35f3388e2ed0a393f4eba8eb82aaa320http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f266794a6f6351e000003?auth=1697328653-0-kbi5crzy-a1d3eaf494091ee306eecb1cbee4621e
  • 7/27/2019 Calculus Video Book

    9/35

    9

    d e n e d o m a i n , c o d o m a i n a n d r a n g e o f r e -

    l a t i o n s .

    6 . Real Valued Functions H e r e w e d e n e f u n c - t i o n s w h i c h a r e a s p e c i a l c a s e r e l a t i o n s .

    W e t h e n g o o n t o d e n e r e a l v a l u e d f u n c -

    t i o n s .

    7 .Algebra of functions

    W e a d d , s u b t r a c t ,

    m u l t i p l y a n d d i v i d e t w o f u n c t i o n s . A f t e r

    t h i s w e d e n e i n j e c t i v e a n d s u r j e c t i v e f u n c -

    t i o n s .

    8 .Composition of functions

    W e d e n e c o m -

    p o s i t i o n o f f u n c t i o n s a n d a l s o i n v e r t i b l e

    f u n c t i o n s .

    9 .Graphing Functions I

    W e g r a p h t h e f u n c -

    t i o n s x

    2

    ; x

    4

    ; x

    3

    ; x

    6

    ; x

    5

    :

    1 0 .

    Graphing Functions III n t h i s l e c t u r e w e

    g r a p h f u n c t i o n s j x j ; 1 = x :

    http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f291d94a6f635a5000000?auth=1697329251-0-z4mod2j0-3d0e0c6e78c08076a64f7a9052b17fd1http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f291d94a6f635a5000000?auth=1697329251-0-z4mod2j0-3d0e0c6e78c08076a64f7a9052b17fd1http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f291d94a6f635a5000000?auth=1697329251-0-z4mod2j0-3d0e0c6e78c08076a64f7a9052b17fd1http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f291d94a6f635a5000000?auth=1697329251-0-z4mod2j0-3d0e0c6e78c08076a64f7a9052b17fd1http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f291d94a6f635a5000000?auth=1697329251-0-z4mod2j0-3d0e0c6e78c08076a64f7a9052b17fd1http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f291d94a6f635a5000000?auth=1697329251-0-z4mod2j0-3d0e0c6e78c08076a64f7a9052b17fd1http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f29a194a6f634a6000005?auth=1697329368-0-7u4c40v1-90f601882ffe8e7e2b1041f701c62105http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f29a194a6f634a6000005?auth=1697329368-0-7u4c40v1-90f601882ffe8e7e2b1041f701c62105http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f29a194a6f634a6000005?auth=1697329368-0-7u4c40v1-90f601882ffe8e7e2b1041f701c62105http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f29a194a6f634a6000005?auth=1697329368-0-7u4c40v1-90f601882ffe8e7e2b1041f701c62105http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f29a194a6f634a6000005?auth=1697329368-0-7u4c40v1-90f601882ffe8e7e2b1041f701c62105http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f29a194a6f634a6000005?auth=1697329368-0-7u4c40v1-90f601882ffe8e7e2b1041f701c62105http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f2a4b06361d4f61000001?auth=1697329514-0-91vlus2u-9bc5e0b053138bf3b3ae952169631da7http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f2a4b06361d4f61000001?auth=1697329514-0-91vlus2u-9bc5e0b053138bf3b3ae952169631da7http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f2a4b06361d4f61000001?auth=1697329514-0-91vlus2u-9bc5e0b053138bf3b3ae952169631da7http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f2a4b06361d4f61000001?auth=1697329514-0-91vlus2u-9bc5e0b053138bf3b3ae952169631da7http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f2a4b06361d4f61000001?auth=1697329514-0-91vlus2u-9bc5e0b053138bf3b3ae952169631da7http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f2a8a06361d4e9a000006?auth=1697329576-0-p02gv1se-7176422fc7a4a857b9c101a73124e2b6http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f2a8a06361d4e9a000006?auth=1697329576-0-p02gv1se-7176422fc7a4a857b9c101a73124e2b6http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f2a8a06361d4e9a000006?auth=1697329576-0-p02gv1se-7176422fc7a4a857b9c101a73124e2b6http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f2a8a06361d4e9a000006?auth=1697329576-0-p02gv1se-7176422fc7a4a857b9c101a73124e2b6http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f2a8a06361d4e9a000006?auth=1697329576-0-p02gv1se-7176422fc7a4a857b9c101a73124e2b6http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f2a8a06361d4e9a000006?auth=1697329576-0-p02gv1se-7176422fc7a4a857b9c101a73124e2b6http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f2a8a06361d4e9a000006?auth=1697329576-0-p02gv1se-7176422fc7a4a857b9c101a73124e2b6http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f2a8a06361d4e9a000006?auth=1697329576-0-p02gv1se-7176422fc7a4a857b9c101a73124e2b6http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f2a8a06361d4e9a000006?auth=1697329576-0-p02gv1se-7176422fc7a4a857b9c101a73124e2b6http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f2a8a06361d4e9a000006?auth=1697329576-0-p02gv1se-7176422fc7a4a857b9c101a73124e2b6http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f2a8a06361d4e9a000006?auth=1697329576-0-p02gv1se-7176422fc7a4a857b9c101a73124e2b6http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f2a8a06361d4e9a000006?auth=1697329576-0-p02gv1se-7176422fc7a4a857b9c101a73124e2b6http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f2a8a06361d4e9a000006?auth=1697329576-0-p02gv1se-7176422fc7a4a857b9c101a73124e2b6http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f2a8a06361d4e9a000006?auth=1697329576-0-p02gv1se-7176422fc7a4a857b9c101a73124e2b6http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f2ab59473994bbd000004?auth=1697333471-0-4rti8284-d5c715c235a1333f7e8c7436f8ddf091http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f2ab59473994bbd000004?auth=1697333471-0-4rti8284-d5c715c235a1333f7e8c7436f8ddf091http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f2ab59473994bbd000004?auth=1697333471-0-4rti8284-d5c715c235a1333f7e8c7436f8ddf091http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f2ab59473994bbd000004?auth=1697333471-0-4rti8284-d5c715c235a1333f7e8c7436f8ddf091http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f2ab59473994bbd000004?auth=1697333471-0-4rti8284-d5c715c235a1333f7e8c7436f8ddf091http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f2ab59473994bbd000004?auth=1697333471-0-4rti8284-d5c715c235a1333f7e8c7436f8ddf091http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f2ab59473994bbd000004?auth=1697333471-0-4rti8284-d5c715c235a1333f7e8c7436f8ddf091http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f2a8a06361d4e9a000006?auth=1697329576-0-p02gv1se-7176422fc7a4a857b9c101a73124e2b6http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f2a4b06361d4f61000001?auth=1697329514-0-91vlus2u-9bc5e0b053138bf3b3ae952169631da7http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f29a194a6f634a6000005?auth=1697329368-0-7u4c40v1-90f601882ffe8e7e2b1041f701c62105http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525f291d94a6f635a5000000?auth=1697329251-0-z4mod2j0-3d0e0c6e78c08076a64f7a9052b17fd1
  • 7/27/2019 Calculus Video Book

    10/35

    1 0 S e t s , R e l a t i o n s a n d F u n c t i o n s

  • 7/27/2019 Calculus Video Book

    11/35

    2 . L i m i t s a n d

    D e r i v a t i v e s

    B e r t r a n d R u s s e l l

    M a t h e m a t i c s t a k e s u s i n t o t h e r e g i o n o f a b -

    s o l u t e n e c e s s i t y , t o w h i c h n o t o n l y t h e a c t u a l

    w o r d , b u t e v e r y p o s s i b l e w o r d , m u s t c o n f o r m .

    1 1

  • 7/27/2019 Calculus Video Book

    12/35

    1 2 L i m i t s a n d D e r i v a t i v e s

    1 . Introduction to Limits W e s t a r t t h e t h i s c h a p t e r b y d i s c u s s i n g s o m e i n s t a n c e s o f

    l i m i t s a n d t h e c a s e s w h e r e t h e y e x i s t .

    2 .Examples

    I n t h i s l e c t u r e i m p a c t o f d i s c o n -

    t i n u i t y i s s h o w n o n t h e l i m i t w i t h r e l e v a n t

    e x a m p l e s .

    3 .Algebra of Limits

    T h i s i s a n i m p o r t a n t l e c -

    t u r e w h e r e s u m , d i e r e n c e , p r o d u c t a n d

    q u o t i e n t o f l i m i t s i s d i s c u s s e d .

    4 .Limits of Fractions

    W e d e s c r i b e t e c h n i q u e s

    o f o b t a i n i n g l i m i t s o f f r a c t i o n s a n d d i s -

    c u s s s o m e e x a m p l e s w h i c h m i g h t g i v e u n -

    d e n e d l i m i t s i n t h e f o r m

    0

    0

    .

    5 . A theorem on limits W e s t a t e t h e t h e o r e m

    l i m

    x 3 a

    =

    x

    n

    a

    n

    x a

    = n x

    n 1

    ; n PR

    a n d d i s c u s s s o m e e x a m p l e s .

    6 .Squeeze Theorem on Limits

    W e d i s c u s s a

    v e r y i m p o r t a n t r e s u l t h e r e c a l l e d t h e ` S q u e e z e

    T h e o r e m ' .

    http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253052294a6f603bf000004?auth=1696555046-0-i3kvfuab-041b8e3d3c7b1cf9c6bcffe11152f862http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253052294a6f603bf000004?auth=1696555046-0-i3kvfuab-041b8e3d3c7b1cf9c6bcffe11152f862http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253052294a6f603bf000004?auth=1696555046-0-i3kvfuab-041b8e3d3c7b1cf9c6bcffe11152f862http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253052294a6f603bf000004?auth=1696555046-0-i3kvfuab-041b8e3d3c7b1cf9c6bcffe11152f862http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253052294a6f603bf000004?auth=1696555046-0-i3kvfuab-041b8e3d3c7b1cf9c6bcffe11152f862http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253057b9473991de5000004?auth=1696555084-0-mdkmtayu-ead6a6e80e6f35ddb0cbf02ae4e47407http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253057b9473991de5000004?auth=1696555084-0-mdkmtayu-ead6a6e80e6f35ddb0cbf02ae4e47407http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253057b9473991de5000004?auth=1696555084-0-mdkmtayu-ead6a6e80e6f35ddb0cbf02ae4e47407http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253057b9473991de5000004?auth=1696555084-0-mdkmtayu-ead6a6e80e6f35ddb0cbf02ae4e47407http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253057b9473991de5000004?auth=1696555084-0-mdkmtayu-ead6a6e80e6f35ddb0cbf02ae4e47407http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525305e694a6f6039f000007?auth=1696555136-0-wfzr7tzq-a3fd27aeb2a55d8b3ebd3cee8b2a02eahttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525305e694a6f6039f000007?auth=1696555136-0-wfzr7tzq-a3fd27aeb2a55d8b3ebd3cee8b2a02eahttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525305e694a6f6039f000007?auth=1696555136-0-wfzr7tzq-a3fd27aeb2a55d8b3ebd3cee8b2a02eahttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525305e694a6f6039f000007?auth=1696555136-0-wfzr7tzq-a3fd27aeb2a55d8b3ebd3cee8b2a02eahttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525305e694a6f6039f000007?auth=1696555136-0-wfzr7tzq-a3fd27aeb2a55d8b3ebd3cee8b2a02eahttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525306409473991d8c000005?auth=1696555227-0-7p59dd1l-92c947981384af06204a7931e7b0ffc0http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525306409473991d8c000005?auth=1696555227-0-7p59dd1l-92c947981384af06204a7931e7b0ffc0http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525306409473991d8c000005?auth=1696555227-0-7p59dd1l-92c947981384af06204a7931e7b0ffc0http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525306409473991d8c000005?auth=1696555227-0-7p59dd1l-92c947981384af06204a7931e7b0ffc0http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525306409473991d8c000005?auth=1696555227-0-7p59dd1l-92c947981384af06204a7931e7b0ffc0http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525306409473991d8c000005?auth=1696555227-0-7p59dd1l-92c947981384af06204a7931e7b0ffc0http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525306409473991d8c000005?auth=1696555227-0-7p59dd1l-92c947981384af06204a7931e7b0ffc0http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525306409473991d8c000005?auth=1696555227-0-7p59dd1l-92c947981384af06204a7931e7b0ffc0http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253067f06361d21c3000003?auth=1696807445-0-fj4e8rg0-ff6df7e202cbc89efd203942f13d0995http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253067f06361d21c3000003?auth=1696807445-0-fj4e8rg0-ff6df7e202cbc89efd203942f13d0995http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253067f06361d21c3000003?auth=1696807445-0-fj4e8rg0-ff6df7e202cbc89efd203942f13d0995http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253067f06361d21c3000003?auth=1696807445-0-fj4e8rg0-ff6df7e202cbc89efd203942f13d0995http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253067f06361d21c3000003?auth=1696807445-0-fj4e8rg0-ff6df7e202cbc89efd203942f13d0995http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253067f06361d21c3000003?auth=1696807445-0-fj4e8rg0-ff6df7e202cbc89efd203942f13d0995http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253067f06361d21c3000003?auth=1696807445-0-fj4e8rg0-ff6df7e202cbc89efd203942f13d0995http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253067f06361d21c3000003?auth=1696807445-0-fj4e8rg0-ff6df7e202cbc89efd203942f13d0995http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253067f06361d21c3000003?auth=1696807445-0-fj4e8rg0-ff6df7e202cbc89efd203942f13d0995http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253067f06361d21c3000003?auth=1696807445-0-fj4e8rg0-ff6df7e202cbc89efd203942f13d0995http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253067f06361d21c3000003?auth=1696807445-0-fj4e8rg0-ff6df7e202cbc89efd203942f13d0995http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253067f06361d21c3000003?auth=1696807445-0-fj4e8rg0-ff6df7e202cbc89efd203942f13d0995http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253067f06361d21c3000003?auth=1696807445-0-fj4e8rg0-ff6df7e202cbc89efd203942f13d0995http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253067f06361d21c3000003?auth=1696807445-0-fj4e8rg0-ff6df7e202cbc89efd203942f13d0995http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253067f06361d21c3000003?auth=1696807445-0-fj4e8rg0-ff6df7e202cbc89efd203942f13d0995http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253067f06361d21c3000003?auth=1696807445-0-fj4e8rg0-ff6df7e202cbc89efd203942f13d0995http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253067f06361d21c3000003?auth=1696807445-0-fj4e8rg0-ff6df7e202cbc89efd203942f13d0995http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253067f06361d21c3000003?auth=1696807445-0-fj4e8rg0-ff6df7e202cbc89efd203942f13d0995http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253067f06361d21c3000003?auth=1696807445-0-fj4e8rg0-ff6df7e202cbc89efd203942f13d0995http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253067f06361d21c3000003?auth=1696807445-0-fj4e8rg0-ff6df7e202cbc89efd203942f13d0995http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253067f06361d21c3000003?auth=1696807445-0-fj4e8rg0-ff6df7e202cbc89efd203942f13d0995http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253067f06361d21c3000003?auth=1696807445-0-fj4e8rg0-ff6df7e202cbc89efd203942f13d0995http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253067f06361d21c3000003?auth=1696807445-0-fj4e8rg0-ff6df7e202cbc89efd203942f13d0995http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253067f06361d21c3000003?auth=1696807445-0-fj4e8rg0-ff6df7e202cbc89efd203942f13d0995http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253067f06361d21c3000003?auth=1696807445-0-fj4e8rg0-ff6df7e202cbc89efd203942f13d0995http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525306cb9473991de5000008?auth=1696555361-0-kwl6r59e-ba900e50c4aad5f5ace806e220db8dcbhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525306cb9473991de5000008?auth=1696555361-0-kwl6r59e-ba900e50c4aad5f5ace806e220db8dcbhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525306cb9473991de5000008?auth=1696555361-0-kwl6r59e-ba900e50c4aad5f5ace806e220db8dcbhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525306cb9473991de5000008?auth=1696555361-0-kwl6r59e-ba900e50c4aad5f5ace806e220db8dcbhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525306cb9473991de5000008?auth=1696555361-0-kwl6r59e-ba900e50c4aad5f5ace806e220db8dcbhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253067f06361d21c3000003?auth=1696807445-0-fj4e8rg0-ff6df7e202cbc89efd203942f13d0995http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525306409473991d8c000005?auth=1696555227-0-7p59dd1l-92c947981384af06204a7931e7b0ffc0http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525305e694a6f6039f000007?auth=1696555136-0-wfzr7tzq-a3fd27aeb2a55d8b3ebd3cee8b2a02eahttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253057b9473991de5000004?auth=1696555084-0-mdkmtayu-ead6a6e80e6f35ddb0cbf02ae4e47407http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253052294a6f603bf000004?auth=1696555046-0-i3kvfuab-041b8e3d3c7b1cf9c6bcffe11152f862
  • 7/27/2019 Calculus Video Book

    13/35

    1 3

    7 . l i m

    x 3 0

    s i n x n x = 1 T h i s r e s u l t i s a n i m p o r -

    t a n t a p p l i c a t i o n o f t h e s q u e e z e t h e o r e m ,

    a n d i s i m p o r t a n t f o r s o l v i n g o t h e r l i m i t s

    i n v o l v i n g t r i g o n o m e t r i c f u n c t i o n s .

    8 .Some applications of

    l i m

    x 3 0

    s i n x n x W e n d

    l i m i t s o f s o m e t r i g o n o m e t r i c f u n c t i o n s u s -

    i n g t h e r e s u l t l i m

    x 3 0

    s i n x n x = 1

    9 .Existence of Limits

    H e r e w e e x p l o r e q u e s -

    t i o n s o n e x i s t e n c e o f l i m i t s v i a c e r t a i n e x -

    a m p l e s .

    1 0 .Infinite Limits

    W e d i s c u s s e x a m p l e s w h e r e

    l i m 3 I :

    1 1 . Definition of a Derivative with examplesW e g i v e a d e n i t i o n o f a d e r i v a t i v e v i a l i m -

    i t s a n d c o m p u t e d e r i v a t i v e s u s i n g l i m i t s .

    1 2 .

    Derivative as a function with ExamplesA

    d e r i v a t i v e c a n a l s o b e d e n e d a s a f u n c -

    t i o n , w e e x p l o r e t h i s d e n i t i o n i n t h i s l e c -

    t u r e .

    1 3 .A theorem on Derivatives

    A n i m p o r t a n t

    t h e o r e m o n d e r i v a t i v e s o f s u m , p r o d u c t ,

    http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525307cf9473991ddd000004?auth=1696556415-0-871qo9x1-fceaf5345a7ed1f31f7ee27a2c7fbda5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525307cf9473991ddd000004?auth=1696556415-0-871qo9x1-fceaf5345a7ed1f31f7ee27a2c7fbda5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525307cf9473991ddd000004?auth=1696556415-0-871qo9x1-fceaf5345a7ed1f31f7ee27a2c7fbda5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525307cf9473991ddd000004?auth=1696556415-0-871qo9x1-fceaf5345a7ed1f31f7ee27a2c7fbda5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525307cf9473991ddd000004?auth=1696556415-0-871qo9x1-fceaf5345a7ed1f31f7ee27a2c7fbda5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525307cf9473991ddd000004?auth=1696556415-0-871qo9x1-fceaf5345a7ed1f31f7ee27a2c7fbda5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525307cf9473991ddd000004?auth=1696556415-0-871qo9x1-fceaf5345a7ed1f31f7ee27a2c7fbda5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525307cf9473991ddd000004?auth=1696556415-0-871qo9x1-fceaf5345a7ed1f31f7ee27a2c7fbda5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525307cf9473991ddd000004?auth=1696556415-0-871qo9x1-fceaf5345a7ed1f31f7ee27a2c7fbda5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525307cf9473991ddd000004?auth=1696556415-0-871qo9x1-fceaf5345a7ed1f31f7ee27a2c7fbda5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525307cf9473991ddd000004?auth=1696556415-0-871qo9x1-fceaf5345a7ed1f31f7ee27a2c7fbda5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525307cf9473991ddd000004?auth=1696556415-0-871qo9x1-fceaf5345a7ed1f31f7ee27a2c7fbda5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525307cf9473991ddd000004?auth=1696556415-0-871qo9x1-fceaf5345a7ed1f31f7ee27a2c7fbda5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525307cf9473991ddd000004?auth=1696556415-0-871qo9x1-fceaf5345a7ed1f31f7ee27a2c7fbda5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525307cf9473991ddd000004?auth=1696556415-0-871qo9x1-fceaf5345a7ed1f31f7ee27a2c7fbda5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525307cf9473991ddd000004?auth=1696556415-0-871qo9x1-fceaf5345a7ed1f31f7ee27a2c7fbda5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525307cf9473991ddd000004?auth=1696556415-0-871qo9x1-fceaf5345a7ed1f31f7ee27a2c7fbda5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525307cf9473991ddd000004?auth=1696556415-0-871qo9x1-fceaf5345a7ed1f31f7ee27a2c7fbda5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525307cf9473991ddd000004?auth=1696556415-0-871qo9x1-fceaf5345a7ed1f31f7ee27a2c7fbda5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525307cf9473991ddd000004?auth=1696556415-0-871qo9x1-fceaf5345a7ed1f31f7ee27a2c7fbda5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525307cf9473991ddd000004?auth=1696556415-0-871qo9x1-fceaf5345a7ed1f31f7ee27a2c7fbda5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525307cf9473991ddd000004?auth=1696556415-0-871qo9x1-fceaf5345a7ed1f31f7ee27a2c7fbda5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253082e06361d2215000002?auth=1696556547-0-hji2ihel-3ab6eae2954cf936efe00ba638a7fe2ahttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253082e06361d2215000002?auth=1696556547-0-hji2ihel-3ab6eae2954cf936efe00ba638a7fe2ahttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253082e06361d2215000002?auth=1696556547-0-hji2ihel-3ab6eae2954cf936efe00ba638a7fe2ahttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253082e06361d2215000002?auth=1696556547-0-hji2ihel-3ab6eae2954cf936efe00ba638a7fe2ahttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253082e06361d2215000002?auth=1696556547-0-hji2ihel-3ab6eae2954cf936efe00ba638a7fe2ahttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525308839473991e8a000001?auth=1696557515-0-74v1zrld-21556ebb1035ee99cbec3c832360d111http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525308839473991e8a000001?auth=1696557515-0-74v1zrld-21556ebb1035ee99cbec3c832360d111http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525308839473991e8a000001?auth=1696557515-0-74v1zrld-21556ebb1035ee99cbec3c832360d111http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525308839473991e8a000001?auth=1696557515-0-74v1zrld-21556ebb1035ee99cbec3c832360d111http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525308839473991e8a000001?auth=1696557515-0-74v1zrld-21556ebb1035ee99cbec3c832360d111http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525308b094a6f6047c000002?auth=1696557563-0-d4xxcmzz-9afbb30a53497a0e9d71a65e95f76593http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525308b094a6f6047c000002?auth=1696557563-0-d4xxcmzz-9afbb30a53497a0e9d71a65e95f76593http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525308b094a6f6047c000002?auth=1696557563-0-d4xxcmzz-9afbb30a53497a0e9d71a65e95f76593http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525308b094a6f6047c000002?auth=1696557563-0-d4xxcmzz-9afbb30a53497a0e9d71a65e95f76593http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525308ff9473991ea6000002?auth=1696557632-0-bgtg0w3h-b43b75b87e179b77fbbfef5e508ffd0ehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525308ff9473991ea6000002?auth=1696557632-0-bgtg0w3h-b43b75b87e179b77fbbfef5e508ffd0ehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525308ff9473991ea6000002?auth=1696557632-0-bgtg0w3h-b43b75b87e179b77fbbfef5e508ffd0ehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525308ff9473991ea6000002?auth=1696557632-0-bgtg0w3h-b43b75b87e179b77fbbfef5e508ffd0ehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525308ff9473991ea6000002?auth=1696557632-0-bgtg0w3h-b43b75b87e179b77fbbfef5e508ffd0ehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525308ff9473991ea6000002?auth=1696557632-0-bgtg0w3h-b43b75b87e179b77fbbfef5e508ffd0ehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253096d94a6f6044a000007?auth=1696557689-0-nimvu6tm-834954588d18bc26c665a86ae2378bcfhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253096d94a6f6044a000007?auth=1696557689-0-nimvu6tm-834954588d18bc26c665a86ae2378bcfhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253096d94a6f6044a000007?auth=1696557689-0-nimvu6tm-834954588d18bc26c665a86ae2378bcfhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253096d94a6f6044a000007?auth=1696557689-0-nimvu6tm-834954588d18bc26c665a86ae2378bcfhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525308ff9473991ea6000002?auth=1696557632-0-bgtg0w3h-b43b75b87e179b77fbbfef5e508ffd0ehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525308b094a6f6047c000002?auth=1696557563-0-d4xxcmzz-9afbb30a53497a0e9d71a65e95f76593http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525308839473991e8a000001?auth=1696557515-0-74v1zrld-21556ebb1035ee99cbec3c832360d111http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253082e06361d2215000002?auth=1696556547-0-hji2ihel-3ab6eae2954cf936efe00ba638a7fe2ahttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525307cf9473991ddd000004?auth=1696556415-0-871qo9x1-fceaf5345a7ed1f31f7ee27a2c7fbda5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253077d06361d2149000002?auth=1696555408-0-zzze5zbr-4a1641823b342af0d7d803a9e3d860de
  • 7/27/2019 Calculus Video Book

    14/35

    1 4 L i m i t s a n d D e r i v a t i v e s

    d i e r e n c e a n d q u o t i e n t o f f u n c t i o n s i s s t a t e d

    i n t h i s l e c t u r e .

    1 4 . Derivative of a trigonometric Function W ec o m p u t e d e r i v a t i v e o f t r i g o n o m e t r i c f u n c -

    t i o n s u s i n g j u s t l i m i t s .

    http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525309b294a6f60497000003?auth=1696557739-0-hmsxr5ik-ab654d7098a86bd0673b338fe537d974http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525309b294a6f60497000003?auth=1696557739-0-hmsxr5ik-ab654d7098a86bd0673b338fe537d974http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525309b294a6f60497000003?auth=1696557739-0-hmsxr5ik-ab654d7098a86bd0673b338fe537d974http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525309b294a6f60497000003?auth=1696557739-0-hmsxr5ik-ab654d7098a86bd0673b338fe537d974http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525309b294a6f60497000003?auth=1696557739-0-hmsxr5ik-ab654d7098a86bd0673b338fe537d974
  • 7/27/2019 Calculus Video Book

    15/35

    3 . C o n t i n u i t y a n d

    D i e r e n t i a b i l i t y

    I s s a c N e w t o n

    T r u t h i s e v e r t o b e f o u n d i n t h e s i m p l i c -

    i t y , a n d n o t i n t h e m u l t i p l i c i t y a n d c o n f u s i o n o f

    t h i n g s .

    1 .Introduction to Continuous and Discontin-uous Functions

    W e g i v e a n i n t u i t i v e u n d e r -

    s t a n d i n g o f c o n t i n u o u s a n d d i s c o n t i n u o u s

    1 5

    http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530b7306361d218b00000a?auth=1696557910-0-ev5yrq1m-0786fbb9963ca3395bfdc85558251108http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530b7306361d218b00000a?auth=1696557910-0-ev5yrq1m-0786fbb9963ca3395bfdc85558251108http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530b7306361d218b00000a?auth=1696557910-0-ev5yrq1m-0786fbb9963ca3395bfdc85558251108http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530b7306361d218b00000a?auth=1696557910-0-ev5yrq1m-0786fbb9963ca3395bfdc85558251108http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530b7306361d218b00000a?auth=1696557910-0-ev5yrq1m-0786fbb9963ca3395bfdc85558251108http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530b7306361d218b00000a?auth=1696557910-0-ev5yrq1m-0786fbb9963ca3395bfdc85558251108http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530b7306361d218b00000a?auth=1696557910-0-ev5yrq1m-0786fbb9963ca3395bfdc85558251108
  • 7/27/2019 Calculus Video Book

    16/35

    1 6 C o n t i n u i t y a n d D i e r e n t i a b i l i t y

    f u n c t i o n s .

    2 . Continuous Functions Definition and Exam-ples H e r e w e d e n e c o n t i n u o u s f u n c t i o n s v i a l i m i t s a n d g i v e s o m e e x a m p l e s .

    3 . Polynomial Functions are Continuous W ee x p l o r e t h e f a c t t h a t p o l y n o m i a l s a r e c o n -

    t i n u o u s f u n c t i o n s .

    4 .Algebra of Continuous Functions

    W e e x -

    p l o r e s u m , d i e r e n c e , p r o d u c t a n d q u o -

    t i e n t o f c o n t i n u o u s f u n c t i o n s .

    5 . Continuous Trigonometric Functions W e s e e s o m e e x a m p l e s o f c o n t i n u o u s t r i g o n o m e t -

    r i c f u n c t i o n s l i k e s i n x ; c o s x .

    6 .Composition of Continuous Functions

    W e

    s t a t e t h e t h e o r e m t h a t c o m p o s i t i o n o f t w o

    c o n t i n u o u s f u n c t i o n s i s c o n t i n u o u s .

    7 .Differentiability and Continuity

    D i e r e n t i a -

    b i l i t y i m p l i e s c o n t i n u i t y .

    8 .Chain Rule

    W e s t a t e t h e c h a i n r u l e a n d

    g i v e s o m e i m m e d i a t e a p p l i c a t i o n s .

    http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530bb194a6f6047c00000c?auth=1696557970-0-9xmgnxca-0bfcedf891d2e42b5da9409198e96857http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530bb194a6f6047c00000c?auth=1696557970-0-9xmgnxca-0bfcedf891d2e42b5da9409198e96857http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530bb194a6f6047c00000c?auth=1696557970-0-9xmgnxca-0bfcedf891d2e42b5da9409198e96857http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530bb194a6f6047c00000c?auth=1696557970-0-9xmgnxca-0bfcedf891d2e42b5da9409198e96857http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530bb194a6f6047c00000c?auth=1696557970-0-9xmgnxca-0bfcedf891d2e42b5da9409198e96857http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530bde9473991eef000005?auth=1696558109-0-a0hbl1rr-929b4e57c5cf99492b92dc9b60afd41dhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530bde9473991eef000005?auth=1696558109-0-a0hbl1rr-929b4e57c5cf99492b92dc9b60afd41dhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530bde9473991eef000005?auth=1696558109-0-a0hbl1rr-929b4e57c5cf99492b92dc9b60afd41dhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530bde9473991eef000005?auth=1696558109-0-a0hbl1rr-929b4e57c5cf99492b92dc9b60afd41dhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530bde9473991eef000005?auth=1696558109-0-a0hbl1rr-929b4e57c5cf99492b92dc9b60afd41dhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530c2506361d22b6000000?auth=1696558136-0-5gnwmok8-23cc401095f4ff77840d199744ab4520http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530c2506361d22b6000000?auth=1696558136-0-5gnwmok8-23cc401095f4ff77840d199744ab4520http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530c2506361d22b6000000?auth=1696558136-0-5gnwmok8-23cc401095f4ff77840d199744ab4520http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530c2506361d22b6000000?auth=1696558136-0-5gnwmok8-23cc401095f4ff77840d199744ab4520http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530c2506361d22b6000000?auth=1696558136-0-5gnwmok8-23cc401095f4ff77840d199744ab4520http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530c4d06361d2280000003?auth=1696558175-0-kb9ech0i-6254e1ee0944abc5a72906f1c63d5b9chttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530c4d06361d2280000003?auth=1696558175-0-kb9ech0i-6254e1ee0944abc5a72906f1c63d5b9chttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530c4d06361d2280000003?auth=1696558175-0-kb9ech0i-6254e1ee0944abc5a72906f1c63d5b9chttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530c4d06361d2280000003?auth=1696558175-0-kb9ech0i-6254e1ee0944abc5a72906f1c63d5b9chttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530c4d06361d2280000003?auth=1696558175-0-kb9ech0i-6254e1ee0944abc5a72906f1c63d5b9chttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530c4d06361d2280000003?auth=1696558175-0-kb9ech0i-6254e1ee0944abc5a72906f1c63d5b9chttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530c4d06361d2280000003?auth=1696558175-0-kb9ech0i-6254e1ee0944abc5a72906f1c63d5b9chttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530c4d06361d2280000003?auth=1696558175-0-kb9ech0i-6254e1ee0944abc5a72906f1c63d5b9chttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530c4d06361d2280000003?auth=1696558175-0-kb9ech0i-6254e1ee0944abc5a72906f1c63d5b9chttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530c8d9473991eef000006?auth=1696558232-0-52r50oq6-db76099fe950f81516d8df3fa6d8ec52http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530c8d9473991eef000006?auth=1696558232-0-52r50oq6-db76099fe950f81516d8df3fa6d8ec52http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530c8d9473991eef000006?auth=1696558232-0-52r50oq6-db76099fe950f81516d8df3fa6d8ec52http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530c8d9473991eef000006?auth=1696558232-0-52r50oq6-db76099fe950f81516d8df3fa6d8ec52http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530c8d9473991eef000006?auth=1696558232-0-52r50oq6-db76099fe950f81516d8df3fa6d8ec52http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530d2094a6f60527000000?auth=1696558423-0-d7ffrvhn-f3127c59b6178b266036e6e676efe830http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530d2094a6f60527000000?auth=1696558423-0-d7ffrvhn-f3127c59b6178b266036e6e676efe830http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530d2094a6f60527000000?auth=1696558423-0-d7ffrvhn-f3127c59b6178b266036e6e676efe830http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530d2094a6f60527000000?auth=1696558423-0-d7ffrvhn-f3127c59b6178b266036e6e676efe830http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530d649473991f2f000001?auth=1696558458-0-vyr24h98-2f6587c5bbc59b6e7003223f2d7fc16bhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530d649473991f2f000001?auth=1696558458-0-vyr24h98-2f6587c5bbc59b6e7003223f2d7fc16bhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530d649473991f2f000001?auth=1696558458-0-vyr24h98-2f6587c5bbc59b6e7003223f2d7fc16bhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530d649473991f2f000001?auth=1696558458-0-vyr24h98-2f6587c5bbc59b6e7003223f2d7fc16bhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530d2094a6f60527000000?auth=1696558423-0-d7ffrvhn-f3127c59b6178b266036e6e676efe830http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530c8d9473991eef000006?auth=1696558232-0-52r50oq6-db76099fe950f81516d8df3fa6d8ec52http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530c4d06361d2280000003?auth=1696558175-0-kb9ech0i-6254e1ee0944abc5a72906f1c63d5b9chttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530c2506361d22b6000000?auth=1696558136-0-5gnwmok8-23cc401095f4ff77840d199744ab4520http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530bde9473991eef000005?auth=1696558109-0-a0hbl1rr-929b4e57c5cf99492b92dc9b60afd41dhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530bb194a6f6047c00000c?auth=1696557970-0-9xmgnxca-0bfcedf891d2e42b5da9409198e96857http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530bb194a6f6047c00000c?auth=1696557970-0-9xmgnxca-0bfcedf891d2e42b5da9409198e96857
  • 7/27/2019 Calculus Video Book

    17/35

    1 7

    9 . Implicit Differentiation W e c o m p u t e e x - a m p l e s i n v o l v i n g i m p l i c i t d i e r e n t i a t i o n .

    1 0 .Implicit Differentiation of Trigonometric func-tions

    S o m e t r i g o n o m e t r i c f u n c t i o n s a r e i m -

    p l i c i t l y d i e r e n t i a t e d .

    1 1 .

    Exponential FunctionD e n i n g e x p o n e n t i a l

    f u n c t i o n .

    1 2 .Logarithmic Function

    D e n i n g L o g f u n c -

    t i o n .

    1 3 .

    Properties of LogarithmS o m e p r o p e r t i e s

    o f L o g a r i t h m i c f u n c t i o n s .

    1 4 . Derivative of Log and Exponential W e d e - n e d e r i v a t i v e o f L o g a r i t h m i c a n d E x p o -

    n e n t i a l f u n c t i o n s .

    1 5 . Logarithmic Differentiation W e c o m p u t e d e r i v a - t i v e o f g ( x )

    f ( x )

    :

    1 6 .Example of Logarithmic Differentiation

    S o m e

    f u r t h e r e x a m p l e s o f L o g a r i t h m i c D i e r e n -

    t i a t i o n .

    http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530da69473991f58000002?auth=1696558486-0-irxlfd8y-833473ae505b4336a8ea3eaab8f9fde4http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530da69473991f58000002?auth=1696558486-0-irxlfd8y-833473ae505b4336a8ea3eaab8f9fde4http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530da69473991f58000002?auth=1696558486-0-irxlfd8y-833473ae505b4336a8ea3eaab8f9fde4http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530da69473991f58000002?auth=1696558486-0-irxlfd8y-833473ae505b4336a8ea3eaab8f9fde4http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530dfa06361d222b000008?auth=1696558525-0-tuohojzz-d57827e514a17e9ce5ecc6d5628caf67http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530dfa06361d222b000008?auth=1696558525-0-tuohojzz-d57827e514a17e9ce5ecc6d5628caf67http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530dfa06361d222b000008?auth=1696558525-0-tuohojzz-d57827e514a17e9ce5ecc6d5628caf67http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530dfa06361d222b000008?auth=1696558525-0-tuohojzz-d57827e514a17e9ce5ecc6d5628caf67http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530dfa06361d222b000008?auth=1696558525-0-tuohojzz-d57827e514a17e9ce5ecc6d5628caf67http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530e389473991f2f000002?auth=1696558567-0-kc9vxfgi-2ee4716443b548ebcfe56faa5ef21364http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530e389473991f2f000002?auth=1696558567-0-kc9vxfgi-2ee4716443b548ebcfe56faa5ef21364http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530e389473991f2f000002?auth=1696558567-0-kc9vxfgi-2ee4716443b548ebcfe56faa5ef21364http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530e389473991f2f000002?auth=1696558567-0-kc9vxfgi-2ee4716443b548ebcfe56faa5ef21364http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530e8c9473991f58000003?auth=1696558603-0-kzc7co8u-8a3cd48dc23e741b17423b0ccfeff2bbhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530e8c9473991f58000003?auth=1696558603-0-kzc7co8u-8a3cd48dc23e741b17423b0ccfeff2bbhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530e8c9473991f58000003?auth=1696558603-0-kzc7co8u-8a3cd48dc23e741b17423b0ccfeff2bbhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530e8c9473991f58000003?auth=1696558603-0-kzc7co8u-8a3cd48dc23e741b17423b0ccfeff2bbhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530edd94a6f60573000000?auth=1696558818-0-y2eto6be-fc02d9722b1275253d759fdeaa95364ehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530edd94a6f60573000000?auth=1696558818-0-y2eto6be-fc02d9722b1275253d759fdeaa95364ehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530edd94a6f60573000000?auth=1696558818-0-y2eto6be-fc02d9722b1275253d759fdeaa95364ehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530edd94a6f60573000000?auth=1696558818-0-y2eto6be-fc02d9722b1275253d759fdeaa95364ehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530f1d06361d222b00000c?auth=1696558633-0-agbuckdz-25185f26bd5484e1f782248fd9c26221http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530f1d06361d222b00000c?auth=1696558633-0-agbuckdz-25185f26bd5484e1f782248fd9c26221http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530f1d06361d222b00000c?auth=1696558633-0-agbuckdz-25185f26bd5484e1f782248fd9c26221http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530f1d06361d222b00000c?auth=1696558633-0-agbuckdz-25185f26bd5484e1f782248fd9c26221http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530f1d06361d222b00000c?auth=1696558633-0-agbuckdz-25185f26bd5484e1f782248fd9c26221http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530f7906361d223f000007?auth=1696558678-0-41u755v2-51a0ce5c56182905bb594cf3e4eeffa1http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530f7906361d223f000007?auth=1696558678-0-41u755v2-51a0ce5c56182905bb594cf3e4eeffa1http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530f7906361d223f000007?auth=1696558678-0-41u755v2-51a0ce5c56182905bb594cf3e4eeffa1http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530f7906361d223f000007?auth=1696558678-0-41u755v2-51a0ce5c56182905bb594cf3e4eeffa1http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530f7906361d223f000007?auth=1696558678-0-41u755v2-51a0ce5c56182905bb594cf3e4eeffa1http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530f7906361d223f000007?auth=1696558678-0-41u755v2-51a0ce5c56182905bb594cf3e4eeffa1http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530f7906361d223f000007?auth=1696558678-0-41u755v2-51a0ce5c56182905bb594cf3e4eeffa1http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530f7906361d223f000007?auth=1696558678-0-41u755v2-51a0ce5c56182905bb594cf3e4eeffa1http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530f7906361d223f000007?auth=1696558678-0-41u755v2-51a0ce5c56182905bb594cf3e4eeffa1http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530f7906361d223f000007?auth=1696558678-0-41u755v2-51a0ce5c56182905bb594cf3e4eeffa1http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530f7906361d223f000007?auth=1696558678-0-41u755v2-51a0ce5c56182905bb594cf3e4eeffa1http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530f7906361d223f000007?auth=1696558678-0-41u755v2-51a0ce5c56182905bb594cf3e4eeffa1http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530fba06361d2268000008?auth=1696558850-0-02cn0k2b-7a4b5cc1d3b0a84ed0a85d11949fc821http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530fba06361d2268000008?auth=1696558850-0-02cn0k2b-7a4b5cc1d3b0a84ed0a85d11949fc821http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530fba06361d2268000008?auth=1696558850-0-02cn0k2b-7a4b5cc1d3b0a84ed0a85d11949fc821http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530fba06361d2268000008?auth=1696558850-0-02cn0k2b-7a4b5cc1d3b0a84ed0a85d11949fc821http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530fba06361d2268000008?auth=1696558850-0-02cn0k2b-7a4b5cc1d3b0a84ed0a85d11949fc821http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530f7906361d223f000007?auth=1696558678-0-41u755v2-51a0ce5c56182905bb594cf3e4eeffa1http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530f1d06361d222b00000c?auth=1696558633-0-agbuckdz-25185f26bd5484e1f782248fd9c26221http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530edd94a6f60573000000?auth=1696558818-0-y2eto6be-fc02d9722b1275253d759fdeaa95364ehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530e8c9473991f58000003?auth=1696558603-0-kzc7co8u-8a3cd48dc23e741b17423b0ccfeff2bbhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530e389473991f2f000002?auth=1696558567-0-kc9vxfgi-2ee4716443b548ebcfe56faa5ef21364http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530dfa06361d222b000008?auth=1696558525-0-tuohojzz-d57827e514a17e9ce5ecc6d5628caf67http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530dfa06361d222b000008?auth=1696558525-0-tuohojzz-d57827e514a17e9ce5ecc6d5628caf67http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530da69473991f58000002?auth=1696558486-0-irxlfd8y-833473ae505b4336a8ea3eaab8f9fde4
  • 7/27/2019 Calculus Video Book

    18/35

    1 8 C o n t i n u i t y a n d D i e r e n t i a b i l i t y

    1 7 . Derivatives of Functions in parametric FormW e c o m p u t e t h e d e r i v a t i v e s o f f u n c t i o n s i n

    t h e p a r a m e t r i c f o r m .

    1 8 . Second Order Derivative D e n e s e c o n d o r - d e r d e r i v a t i v e a n d c o m p u t e s o m e e x a m p l e s

    1 9 .

    Rolles Theorem and Mean Value Theo-remW e s t a t e t h e t w o i m p o r t a n t t h e o r e m s

    R o l l e ' s T h e o r e m a n d t h e M e a n V a l u e T h e -

    o r e m a n d i l l u s t r a t e t h e m v i a e x a m p l e s .

    http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530ffb9473991f83000002?auth=1696558884-0-kgbxs3x0-c6ebb2522fbe1a544458bf663e4eef93http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530ffb9473991f83000002?auth=1696558884-0-kgbxs3x0-c6ebb2522fbe1a544458bf663e4eef93http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530ffb9473991f83000002?auth=1696558884-0-kgbxs3x0-c6ebb2522fbe1a544458bf663e4eef93http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530ffb9473991f83000002?auth=1696558884-0-kgbxs3x0-c6ebb2522fbe1a544458bf663e4eef93http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525310579473991f68000004?auth=1696558927-0-h4eyfs9m-5bd7dae79fefaf4b6ca118ae5399e342http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525310579473991f68000004?auth=1696558927-0-h4eyfs9m-5bd7dae79fefaf4b6ca118ae5399e342http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525310579473991f68000004?auth=1696558927-0-h4eyfs9m-5bd7dae79fefaf4b6ca118ae5399e342http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525310579473991f68000004?auth=1696558927-0-h4eyfs9m-5bd7dae79fefaf4b6ca118ae5399e342http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525310b39473991f2f000005?auth=1696558975-0-hhncalvn-e057bcca9064d8544e34e6e2f16b7336http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525310b39473991f2f000005?auth=1696558975-0-hhncalvn-e057bcca9064d8544e34e6e2f16b7336http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525310b39473991f2f000005?auth=1696558975-0-hhncalvn-e057bcca9064d8544e34e6e2f16b7336http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525310b39473991f2f000005?auth=1696558975-0-hhncalvn-e057bcca9064d8544e34e6e2f16b7336http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525310b39473991f2f000005?auth=1696558975-0-hhncalvn-e057bcca9064d8544e34e6e2f16b7336http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525310b39473991f2f000005?auth=1696558975-0-hhncalvn-e057bcca9064d8544e34e6e2f16b7336http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525310b39473991f2f000005?auth=1696558975-0-hhncalvn-e057bcca9064d8544e34e6e2f16b7336http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525310579473991f68000004?auth=1696558927-0-h4eyfs9m-5bd7dae79fefaf4b6ca118ae5399e342http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52530ffb9473991f83000002?auth=1696558884-0-kgbxs3x0-c6ebb2522fbe1a544458bf663e4eef93
  • 7/27/2019 Calculus Video Book

    19/35

    4 . A p p l i c a t i o n s o f

    D i e r e n t i a t i o n

    I s s a c N e w t o n

    W i t h C a l c u l u s a s t h e k e y m a t h e m a t i c s c a n

    b e s u c c e s s f u l l y a p p l i e d t o t h e e x p l a n a t i o n o f t h e

    c o u r s e o f n a t u r e .

    D e r i v a t i v e s f o r m a p o w e r f u l t o o l i n d i s p e n s a b l e

    f o r s o l v i n g o p t i m i z a t i o n p r o b l e m s i n a l l q u a n t i -

    t a t i v e s u b j e c t s .

    1 9

  • 7/27/2019 Calculus Video Book

    20/35

    2 0 A p p l i c a t i o n s o f D i e r e n t i a t i o n

    1 . Volume of Cube and area of circle W e i n t e r - p r e t d e r i v a t i v e a s r a t e o f c h a n g e a n d u s e

    i t t o c o m p u t e v o l u m e o f a c u b e a n d a r e a

    o f a c i r c l e .

    2 . Area of Rectangle, Marginal Cost and MarginalRevenue Examples U s i n g d e r i v a t i v e s t o s o l v e p r o b l e m s a r i s i n g i n E c o n o m i c s .

    3 .Definition of Increasing and decreasing func-

    tionsW e d e n e i n c r e a s i n g a n d d e c r e a s i n g

    f u n c t i o n s .

    4 .Increasing Function

    A s t r i c t l y i n c r e a s i n g f u n c -

    t i o n f ( x ) c a n b e c h a r a c t e r i s e d b y i t s d e r i v a -

    t i v e f

    H

    ( x ) > 0 .

    5 .Decreasing Function

    A s t r i c t l y d e c r e a s i n g

    f u n c t i o n f ( x ) c a n b e c h a r a c t e r i s e d b y i t s

    d e r i v a t i v e f

    H

    ( x )

  • 7/27/2019 Calculus Video Book

    21/35

    2 1

    7 . Graphing a Trigonometric Function W e c a n g r a p h a t r i g o n o m e t r i c f u n c t i o n b y j u s t l o o k -

    i n g a t i t s d e r i v a t i v e s i n t h e r i g h t i n t e r v a l s .

    8 .Tangent Line to a Curve

    D e n i n g t a n g e n t

    t o a c u r v e w i t h e x a m p l e s .

    9 .Multiple Tangent Lines

    M u l t i p l e t a n g e n t

    l i n e s w i t h t h e s a m e s l o p e .

    1 0 .Normal Line to a Curve

    D e n i n g n o r m a l

    l i n e t o a c u r v e .

    1 1 . Approximation Using Derivatives T h i s l e c - t u r e d e m o n s t r a t e s h o w d e r i v a t i v e s a r e u s e -

    f u l f o r f a s t c o m p u t a t i o n s .

    1 2 .Approximation of Volume and Error Esti-mation

    W e u s e d e r i v a t i v e s t o c o m p u t e a p -

    p r o x i m a t e c h a n g e i n v o l u m e w i t h o u t e x -

    t e n s i v e c o m p u t a t i o n .

    1 3 .Global Maximum and Minimum and Lo-cal Maximum and Minimum

    T h i s l e c t u r e

    o u t l i n e s t h e t e c h n i q u e o f n d i n g g l o b a l m a x -

    i m u m a n d g l o b a l m i n i m u m u s i n g d e r i v a -

    t i v e s .

    http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253125694a6f60596000007?auth=1696625627-0-7usod4qu-f6e1a23483dc17c58ba69df17d0c1265http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253125694a6f60596000007?auth=1696625627-0-7usod4qu-f6e1a23483dc17c58ba69df17d0c1265http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253125694a6f60596000007?auth=1696625627-0-7usod4qu-f6e1a23483dc17c58ba69df17d0c1265http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253125694a6f60596000007?auth=1696625627-0-7usod4qu-f6e1a23483dc17c58ba69df17d0c1265http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253125694a6f60596000007?auth=1696625627-0-7usod4qu-f6e1a23483dc17c58ba69df17d0c1265http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525312be94a6f60573000005?auth=1696625661-0-0btvzgqv-f995a70394e4ef383c31311b4b441594http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525312be94a6f60573000005?auth=1696625661-0-0btvzgqv-f995a70394e4ef383c31311b4b441594http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525312be94a6f60573000005?auth=1696625661-0-0btvzgqv-f995a70394e4ef383c31311b4b441594http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525312be94a6f60573000005?auth=1696625661-0-0btvzgqv-f995a70394e4ef383c31311b4b441594http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525312fa94a6f605ce000006?auth=1696625688-0-r67ao16z-bb364838809bded7b3a62d2722f0cba5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525312fa94a6f605ce000006?auth=1696625688-0-r67ao16z-bb364838809bded7b3a62d2722f0cba5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525312fa94a6f605ce000006?auth=1696625688-0-r67ao16z-bb364838809bded7b3a62d2722f0cba5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525312fa94a6f605ce000006?auth=1696625688-0-r67ao16z-bb364838809bded7b3a62d2722f0cba5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253135806361d239a000001?auth=1696625718-0-z496dfw2-66ca871a2a87244b45e7f39e598e3796http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253135806361d239a000001?auth=1696625718-0-z496dfw2-66ca871a2a87244b45e7f39e598e3796http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253135806361d239a000001?auth=1696625718-0-z496dfw2-66ca871a2a87244b45e7f39e598e3796http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253135806361d239a000001?auth=1696625718-0-z496dfw2-66ca871a2a87244b45e7f39e598e3796http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525313db947399203a000000?auth=1696625748-0-8xgvbrka-88d92a979f3c9779b6055f3c39576dc9http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525313db947399203a000000?auth=1696625748-0-8xgvbrka-88d92a979f3c9779b6055f3c39576dc9http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525313db947399203a000000?auth=1696625748-0-8xgvbrka-88d92a979f3c9779b6055f3c39576dc9http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525313db947399203a000000?auth=1696625748-0-8xgvbrka-88d92a979f3c9779b6055f3c39576dc9http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525313db947399203a000000?auth=1696625748-0-8xgvbrka-88d92a979f3c9779b6055f3c39576dc9http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253141c94a6f605bf000005?auth=1696625791-0-2n6nb6ni-5b4017aee370022f8bb0a488dd27585ehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253141c94a6f605bf000005?auth=1696625791-0-2n6nb6ni-5b4017aee370022f8bb0a488dd27585ehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253141c94a6f605bf000005?auth=1696625791-0-2n6nb6ni-5b4017aee370022f8bb0a488dd27585ehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253141c94a6f605bf000005?auth=1696625791-0-2n6nb6ni-5b4017aee370022f8bb0a488dd27585ehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253141c94a6f605bf000005?auth=1696625791-0-2n6nb6ni-5b4017aee370022f8bb0a488dd27585ehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253141c94a6f605bf000005?auth=1696625791-0-2n6nb6ni-5b4017aee370022f8bb0a488dd27585ehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253146594a6f605bf000007?auth=1696625834-0-uggkocyj-bd11ded6cc1b06d9c44fa5c8f8a548dbhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253146594a6f605bf000007?auth=1696625834-0-uggkocyj-bd11ded6cc1b06d9c44fa5c8f8a548dbhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253146594a6f605bf000007?auth=1696625834-0-uggkocyj-bd11ded6cc1b06d9c44fa5c8f8a548dbhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253146594a6f605bf000007?auth=1696625834-0-uggkocyj-bd11ded6cc1b06d9c44fa5c8f8a548dbhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253146594a6f605bf000007?auth=1696625834-0-uggkocyj-bd11ded6cc1b06d9c44fa5c8f8a548dbhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253146594a6f605bf000007?auth=1696625834-0-uggkocyj-bd11ded6cc1b06d9c44fa5c8f8a548dbhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253146594a6f605bf000007?auth=1696625834-0-uggkocyj-bd11ded6cc1b06d9c44fa5c8f8a548dbhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253146594a6f605bf000007?auth=1696625834-0-uggkocyj-bd11ded6cc1b06d9c44fa5c8f8a548dbhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253141c94a6f605bf000005?auth=1696625791-0-2n6nb6ni-5b4017aee370022f8bb0a488dd27585ehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253141c94a6f605bf000005?auth=1696625791-0-2n6nb6ni-5b4017aee370022f8bb0a488dd27585ehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525313db947399203a000000?auth=1696625748-0-8xgvbrka-88d92a979f3c9779b6055f3c39576dc9http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253135806361d239a000001?auth=1696625718-0-z496dfw2-66ca871a2a87244b45e7f39e598e3796http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525312fa94a6f605ce000006?auth=1696625688-0-r67ao16z-bb364838809bded7b3a62d2722f0cba5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525312be94a6f60573000005?auth=1696625661-0-0btvzgqv-f995a70394e4ef383c31311b4b441594http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253125694a6f60596000007?auth=1696625627-0-7usod4qu-f6e1a23483dc17c58ba69df17d0c1265
  • 7/27/2019 Calculus Video Book

    22/35

    2 2 A p p l i c a t i o n s o f D i e r e n t i a t i o n

    1 4 . Critical Points, Extreme Value T heorem and

    Fermats T heorem W e d e n e c r i t i c a l p o i n t s ( a l s o c a l l e d c r i t i c a l n u m b e r s ) a n d s t a t e E x -

    t r e m e V a l u e T h e o r e m a n d F e r m a t ' s t h e o -

    r e m .

    1 5 .First Derivative Test

    W e o u t l i n e t h e r s t

    d e r i v a t i v e t e s t a n d u s e i t t o c o m p u t e c r i t -

    i c a l p o i n t s .

    1 6 .Failure of First derivative

    T h e r s t d e r i v a -

    t i v e c a n f a i l t o d e s c r i b e m a x i m u m o r m i n -

    i m u m , w e s h o w t h i s u s i n g e x a m p l e s .

    1 7 .Applications of First Derivative

    F i r s t d e r i v a -

    t i v e c a n b e u s e d t o n d c r i t i c a l p o i n t s a n d

    d e s c r i b e l o c a l m a x i m a o r m i n i m a .

    1 8 . Second Derivative TestW e d e s c r i b e t h e s e c - o n d d e r i v a t i v e t e s t w i t h e x a m p l e s .

    1 9 .Inflection Point

    W e d e n e i n e c t i o n p o i n t

    a n d c o m p u t e i t f o r s o m e e x a m p l e s .

    2 0 .L Hopitals Rule

    D e r i v a t i v e s a r e d e n e d v i a

    l i m i t s , b u t l i m i t s c a n a l s o b e c o m p u t e d u s -

    http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525314b806361d233b000007?auth=1696625876-0-8pyep0r9-b37c85dc39620f6569d600aae8ab2ab5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525314b806361d233b000007?auth=1696625876-0-8pyep0r9-b37c85dc39620f6569d600aae8ab2ab5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525314b806361d233b000007?auth=1696625876-0-8pyep0r9-b37c85dc39620f6569d600aae8ab2ab5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525314b806361d233b000007?auth=1696625876-0-8pyep0r9-b37c85dc39620f6569d600aae8ab2ab5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525314b806361d233b000007?auth=1696625876-0-8pyep0r9-b37c85dc39620f6569d600aae8ab2ab5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525314b806361d233b000007?auth=1696625876-0-8pyep0r9-b37c85dc39620f6569d600aae8ab2ab5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525314b806361d233b000007?auth=1696625876-0-8pyep0r9-b37c85dc39620f6569d600aae8ab2ab5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253152294a6f605a700000d?auth=1696625916-0-744fp9q1-c2afa3843c04cf84790815a593b0f1cbhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253152294a6f605a700000d?auth=1696625916-0-744fp9q1-c2afa3843c04cf84790815a593b0f1cbhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253152294a6f605a700000d?auth=1696625916-0-744fp9q1-c2afa3843c04cf84790815a593b0f1cbhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253152294a6f605a700000d?auth=1696625916-0-744fp9q1-c2afa3843c04cf84790815a593b0f1cbhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253152294a6f605a700000d?auth=1696625916-0-744fp9q1-c2afa3843c04cf84790815a593b0f1cbhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525315569473991feb000006?auth=1696625945-0-n5vhoruf-54946cd528dcd2ab5826cf575456e5achttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525315569473991feb000006?auth=1696625945-0-n5vhoruf-54946cd528dcd2ab5826cf575456e5achttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525315569473991feb000006?auth=1696625945-0-n5vhoruf-54946cd528dcd2ab5826cf575456e5achttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525315569473991feb000006?auth=1696625945-0-n5vhoruf-54946cd528dcd2ab5826cf575456e5achttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525315569473991feb000006?auth=1696625945-0-n5vhoruf-54946cd528dcd2ab5826cf575456e5achttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253158d06361d23f5000000?auth=1696625984-0-90vjk5n0-d8727956b2ffa8c18a3cc1e4d942b343http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253158d06361d23f5000000?auth=1696625984-0-90vjk5n0-d8727956b2ffa8c18a3cc1e4d942b343http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253158d06361d23f5000000?auth=1696625984-0-90vjk5n0-d8727956b2ffa8c18a3cc1e4d942b343http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253158d06361d23f5000000?auth=1696625984-0-90vjk5n0-d8727956b2ffa8c18a3cc1e4d942b343http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253158d06361d23f5000000?auth=1696625984-0-90vjk5n0-d8727956b2ffa8c18a3cc1e4d942b343http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525315c89473992068000001?auth=1696626022-0-sty475ti-65d415026d8dd3d3f5097b358de32bachttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525315c89473992068000001?auth=1696626022-0-sty475ti-65d415026d8dd3d3f5097b358de32bachttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525315c89473992068000001?auth=1696626022-0-sty475ti-65d415026d8dd3d3f5097b358de32bachttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525315c89473992068000001?auth=1696626022-0-sty475ti-65d415026d8dd3d3f5097b358de32bachttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253161306361d232c000007?auth=1696626050-0-1csbum3i-ef8599bb2fd2a8dc3410727da35526cchttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253161306361d232c000007?auth=1696626050-0-1csbum3i-ef8599bb2fd2a8dc3410727da35526cchttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253161306361d232c000007?auth=1696626050-0-1csbum3i-ef8599bb2fd2a8dc3410727da35526cchttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253161306361d232c000007?auth=1696626050-0-1csbum3i-ef8599bb2fd2a8dc3410727da35526cchttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525316789473991feb00000a?auth=1696626076-0-bwgfpp97-22d17dc4581b4b408b4baa1a13ba3935http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525316789473991feb00000a?auth=1696626076-0-bwgfpp97-22d17dc4581b4b408b4baa1a13ba3935http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525316789473991feb00000a?auth=1696626076-0-bwgfpp97-22d17dc4581b4b408b4baa1a13ba3935http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525316789473991feb00000a?auth=1696626076-0-bwgfpp97-22d17dc4581b4b408b4baa1a13ba3935http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253161306361d232c000007?auth=1696626050-0-1csbum3i-ef8599bb2fd2a8dc3410727da35526cchttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525315c89473992068000001?auth=1696626022-0-sty475ti-65d415026d8dd3d3f5097b358de32bachttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253158d06361d23f5000000?auth=1696625984-0-90vjk5n0-d8727956b2ffa8c18a3cc1e4d942b343http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525315569473991feb000006?auth=1696625945-0-n5vhoruf-54946cd528dcd2ab5826cf575456e5achttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253152294a6f605a700000d?auth=1696625916-0-744fp9q1-c2afa3843c04cf84790815a593b0f1cbhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525314b806361d233b000007?auth=1696625876-0-8pyep0r9-b37c85dc39620f6569d600aae8ab2ab5http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525314b806361d233b000007?auth=1696625876-0-8pyep0r9-b37c85dc39620f6569d600aae8ab2ab5
  • 7/27/2019 Calculus Video Book

    23/35

    2 3

    i n g d e r i v a t i v e s . L H o p i t a l ' s r u l e a l l o w s u s

    t o u s e d i e r e n t i a t i o n t o c o m p u t e l i m i t s .

    2 1 . Failure of L Hopitals Rule I n c e r t a i n c a s e s L H o p i t a l r u l e c a n f a i l . W e d i s c u s s t h e s e

    c a s e s h e r e .

    http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525316d306361d23f5000004?auth=1696626123-0-jx0zknn1-b1e0bb0948e0533da58cc24fdf72566bhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525316d306361d23f5000004?auth=1696626123-0-jx0zknn1-b1e0bb0948e0533da58cc24fdf72566bhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525316d306361d23f5000004?auth=1696626123-0-jx0zknn1-b1e0bb0948e0533da58cc24fdf72566bhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525316d306361d23f5000004?auth=1696626123-0-jx0zknn1-b1e0bb0948e0533da58cc24fdf72566bhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525316d306361d23f5000004?auth=1696626123-0-jx0zknn1-b1e0bb0948e0533da58cc24fdf72566b
  • 7/27/2019 Calculus Video Book

    24/35

    2 4 A p p l i c a t i o n s o f D i e r e n t i a t i o n

  • 7/27/2019 Calculus Video Book

    25/35

    5 . I n t e g r a t i o n

    G . K . C h e s t e r t o n

    T h e d i e r e n c e b e t w e e n t h e p o e t a n d t h e

    m a t h e m a t i c i a n i s t h a t t h e p o e t t r i e s t o g e t h i s

    h e a d i n t o t h e h e a v e n s w h i l e t h e m a t h e m a t i c i a n

    t r i e s t o g e t t h e h e a v e n s i n t o h i s h e a d .

    1 .Integration as Antiderivative

    I n t e g r a t i o n s

    i s d e n e d a s a n i n v e r s e p r o c e s s o f d i e r -

    e n t i a t i o n .

    2 5

    http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525326ce9473992244000005?auth=1696716680-0-l86htikl-56d7c584c076430bb69fba3ba9e63f20http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525326ce9473992244000005?auth=1696716680-0-l86htikl-56d7c584c076430bb69fba3ba9e63f20http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525326ce9473992244000005?auth=1696716680-0-l86htikl-56d7c584c076430bb69fba3ba9e63f20http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525326ce9473992244000005?auth=1696716680-0-l86htikl-56d7c584c076430bb69fba3ba9e63f20http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525326ce9473992244000005?auth=1696716680-0-l86htikl-56d7c584c076430bb69fba3ba9e63f20http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525326ce9473992244000005?auth=1696716680-0-l86htikl-56d7c584c076430bb69fba3ba9e63f20
  • 7/27/2019 Calculus Video Book

    26/35

    2 6 I n t e g r a t i o n

    2 . Important Formulae S o m e i m p o r t a n t f o r - m u l a e a r e l i s t e d i n t h i s l e c t u r e . T h e s e a r e

    d e r i v e d d i r e c t l y b y t h i n k i n g o f i n t e g r a t i o n

    a s a n i n v e r s e p r o c e s s o f d i e r e n t i a t i o n .

    3 .Integration with two answers

    F o r s o m e t r i g o n o -

    m e t r i c f u n c t i o n s w e m i g h t g e t t w o d i e r -

    e n t a n s w e r s f o r t h e s a m e i n t e g r a l .

    4 .Important Properties of Indefinite Inte-grals

    S o m e i m p o r t a n t p r o p e r t i e s o f i n d e -

    n i t e i n t e g r a l s a r e d i s c u s s e d h e r e .

    5 . Some Examples W e c o m p u t e s o m e e l e m e n - t a r y e x a m p l e s .

    6 . Integration by Substitution H e r e w e l e a r n i n t e g r a t i o n b y s u b s t i t u t i o n .

    7 . Integrating Trigonometric Functions usingSubstitution

    W e c a n i n t e g r a t e t r i g o n o m e t -

    r i c f u n c t i o n s u s i n g s u b s t i t u t i o n s .

    8 .Integrating Exponential Functions

    W e l e a r n

    h o w t o i n t e g r a t e e x p o n e n t i a l f u n c t i o n s .

    http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253275294739921c8000007?auth=1696716714-0-658cblez-7c4d1601944e3a51b90b00994df4dbf2http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253275294739921c8000007?auth=1696716714-0-658cblez-7c4d1601944e3a51b90b00994df4dbf2http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253275294739921c8000007?auth=1696716714-0-658cblez-7c4d1601944e3a51b90b00994df4dbf2http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253275294739921c8000007?auth=1696716714-0-658cblez-7c4d1601944e3a51b90b00994df4dbf2http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253275294739921c8000007?auth=1696716714-0-658cblez-7c4d1601944e3a51b90b00994df4dbf2http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253275294739921c8000007?auth=1696716714-0-658cblez-7c4d1601944e3a51b90b00994df4dbf2http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253277806361d259e000005?auth=1696716740-0-zjqzjo95-5845ca98d47dad84852675eaec6a597dhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253277806361d259e000005?auth=1696716740-0-zjqzjo95-5845ca98d47dad84852675eaec6a597dhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253277806361d259e000005?auth=1696716740-0-zjqzjo95-5845ca98d47dad84852675eaec6a597dhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253277806361d259e000005?auth=1696716740-0-zjqzjo95-5845ca98d47dad84852675eaec6a597dhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253277806361d259e000005?auth=1696716740-0-zjqzjo95-5845ca98d47dad84852675eaec6a597dhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525327bf9473992263000004?auth=1696716770-0-wzdpbfpo-a731340279f8863af6e22b90d8eecc04http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525327bf9473992263000004?auth=1696716770-0-wzdpbfpo-a731340279f8863af6e22b90d8eecc04http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525327bf9473992263000004?auth=1696716770-0-wzdpbfpo-a731340279f8863af6e22b90d8eecc04http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525327bf9473992263000004?auth=1696716770-0-wzdpbfpo-a731340279f8863af6e22b90d8eecc04http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525327bf9473992263000004?auth=1696716770-0-wzdpbfpo-a731340279f8863af6e22b90d8eecc04http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525327f89473992263000005?auth=1696716812-0-bb4n3u7d-355066128ea50e3765bda3a3f43a7785http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525327f89473992263000005?auth=1696716812-0-bb4n3u7d-355066128ea50e3765bda3a3f43a7785http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525327f89473992263000005?auth=1696716812-0-bb4n3u7d-355066128ea50e3765bda3a3f43a7785http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525327f89473992263000005?auth=1696716812-0-bb4n3u7d-355066128ea50e3765bda3a3f43a7785http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52532851947399220e000009?auth=1696716836-0-lh06s0y6-3313246383c66d61827cad5f04540ee6http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52532851947399220e000009?auth=1696716836-0-lh06s0y6-3313246383c66d61827cad5f04540ee6http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52532851947399220e000009?auth=1696716836-0-lh06s0y6-3313246383c66d61827cad5f04540ee6http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52532851947399220e000009?auth=1696716836-0-lh06s0y6-3313246383c66d61827cad5f04540ee6http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525328b406361d258b000004?auth=1696716863-0-63iuwms4-c08614d96e4cf4f81ae7bd772fe34ccehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525328b406361d258b000004?auth=1696716863-0-63iuwms4-c08614d96e4cf4f81ae7bd772fe34ccehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525328b406361d258b000004?auth=1696716863-0-63iuwms4-c08614d96e4cf4f81ae7bd772fe34ccehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525328b406361d258b000004?auth=1696716863-0-63iuwms4-c08614d96e4cf4f81ae7bd772fe34ccehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525328b406361d258b000004?auth=1696716863-0-63iuwms4-c08614d96e4cf4f81ae7bd772fe34ccehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525328e994a6f60838000009?auth=1696716916-0-c7qeggrf-967fce5bac22bc28883d13ec00421bbfhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525328e994a6f60838000009?auth=1696716916-0-c7qeggrf-967fce5bac22bc28883d13ec00421bbfhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525328e994a6f60838000009?auth=1696716916-0-c7qeggrf-967fce5bac22bc28883d13ec00421bbfhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525328e994a6f60838000009?auth=1696716916-0-c7qeggrf-967fce5bac22bc28883d13ec00421bbfhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525328b406361d258b000004?auth=1696716863-0-63iuwms4-c08614d96e4cf4f81ae7bd772fe34ccehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525328b406361d258b000004?auth=1696716863-0-63iuwms4-c08614d96e4cf4f81ae7bd772fe34ccehttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52532851947399220e000009?auth=1696716836-0-lh06s0y6-3313246383c66d61827cad5f04540ee6http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525327f89473992263000005?auth=1696716812-0-bb4n3u7d-355066128ea50e3765bda3a3f43a7785http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525327bf9473992263000004?auth=1696716770-0-wzdpbfpo-a731340279f8863af6e22b90d8eecc04http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525327bf9473992263000004?auth=1696716770-0-wzdpbfpo-a731340279f8863af6e22b90d8eecc04http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253277806361d259e000005?auth=1696716740-0-zjqzjo95-5845ca98d47dad84852675eaec6a597dhttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/5253275294739921c8000007?auth=1696716714-0-658cblez-7c4d1601944e3a51b90b00994df4dbf2
  • 7/27/2019 Calculus Video Book

    27/35

    2 7

    9 . Trigonometric Identities for Integration C o m - m o n l y u s e d t r i g o n o m e t r i c i d e n t i t i e s a r e d i s -

    c u s s e d h e r e .

    1 0 .Some Standard Results I

    W e d i s c u s s s o m e

    s t a n d a r d i n t e g r a l s h e r e .

    1 1 . Standard Results II T h i s i s c o n t i n u a t i o n o f t h e p r e v i o u s l e c t u r e .

    1 2 .Integration by Partial Fractions I

    W e d e -

    s c r i b e h o w t o u s e p a r t i a l f r a c t i o n s f o r i n -

    t e g r a t i o n .

    1 3 . Integration by Partial Fractions II W e d i s - c u s s e x a m p l e s d e s c r i b i n g u s e o f p a r t i a l f r a c -

    t i o n s i n d e t a i l .

    1 4 .Integration by Partial Fractions III

    T h i s

    i s a c o n t i n u a t i o n o f p r e v i o u s l e c t u r e a n d

    w e c o n t i n u e d i s c u s s i n g m o r e e x a m p l e s o f

    i n t e g r a t i o n b y p a r t i a l f r a c t i o n s .

    1 5 .Integration by Parts I

    W e d e s c r i b e i n t e -

    g r a t i o n b y p a r t s .

    http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525329289473992253000007?auth=1696716957-0-528bulc7-8433d677f0f2502faefdf6de86ccdccchttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525329289473992253000007?auth=1696716957-0-528bulc7-8433d677f0f2502faefdf6de86ccdccchttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525329289473992253000007?auth=1696716957-0-528bulc7-8433d677f0f2502faefdf6de86ccdccchttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525329289473992253000007?auth=1696716957-0-528bulc7-8433d677f0f2502faefdf6de86ccdccchttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525329289473992253000007?auth=1696716957-0-528bulc7-8433d677f0f2502faefdf6de86ccdccchttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525329a79473992253000008?auth=1696717004-0-v2mhvq4m-e1f3369ea4ab902c550b5748d67043fchttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525329a79473992253000008?auth=1696717004-0-v2mhvq4m-e1f3369ea4ab902c550b5748d67043fchttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525329a79473992253000008?auth=1696717004-0-v2mhvq4m-e1f3369ea4ab902c550b5748d67043fchttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525329a79473992253000008?auth=1696717004-0-v2mhvq4m-e1f3369ea4ab902c550b5748d67043fchttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525329f094739922f8000002?auth=1696717031-0-ill37a4l-2997cfad6327529c18fb2eac5c766bf9http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525329f094739922f8000002?auth=1696717031-0-ill37a4l-2997cfad6327529c18fb2eac5c766bf9http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525329f094739922f8000002?auth=1696717031-0-ill37a4l-2997cfad6327529c18fb2eac5c766bf9http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/525329f094739922f8000002?auth=1696717031-0-ill37a4l-2997cfad6327529c18fb2eac5c766bf9http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52532a4894739921fc000006?auth=1696718945-0-c4dql0fu-f97e1c41c129becf7df0a6c5db71e8a6http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52532a4894739921fc000006?auth=1696718945-0-c4dql0fu-f97e1c41c129becf7df0a6c5db71e8a6http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52532a4894739921fc000006?auth=1696718945-0-c4dql0fu-f97e1c41c129becf7df0a6c5db71e8a6http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52532a4894739921fc000006?auth=1696718945-0-c4dql0fu-f97e1c41c129becf7df0a6c5db71e8a6http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52532a4894739921fc000006?auth=1696718945-0-c4dql0fu-f97e1c41c129becf7df0a6c5db71e8a6http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52532a8e06361d25ff000006?auth=1696718966-0-td12n1kw-c9975752ba0f792074104d1aa14ea352http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52532a8e06361d25ff000006?auth=1696718966-0-td12n1kw-c9975752ba0f792074104d1aa14ea352http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52532a8e06361d25ff000006?auth=1696718966-0-td12n1kw-c9975752ba0f792074104d1aa14ea352http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52532a8e06361d25ff000006?auth=1696718966-0-td12n1kw-c9975752ba0f792074104d1aa14ea352http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52532a8e06361d25ff000006?auth=1696718966-0-td12n1kw-c9975752ba0f792074104d1aa14ea352http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52532ad094a6f608df000004?auth=1696718980-0-g1llw9ka-706fb3d6dbf0a6e498fc2344d82b5943http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52532ad094a6f608df000004?auth=1696718980-0-g1llw9ka-706fb3d6dbf0a6e498fc2344d82b5943http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52532ad094a6f608df000004?auth=1696718980-0-g1llw9ka-706fb3d6dbf0a6e498fc2344d82b5943http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52532ad094a6f608df000004?auth=1696718980-0-g1llw9ka-706fb3d6dbf0a6e498fc2344d82b5943http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52532ad094a6f608df000004?auth=1696718980-0-g1llw9ka-706fb3d6dbf0a6e498fc2344d82b5943http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52532ad094a6f608df000004?auth=1696718980-0-g1llw9ka-706fb3d6dbf0a6e498fc2344d82b5943http://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52532b2006361d25c9000009?auth=1696717126-0-kxjzwg6x-3353c0df405f0de4a54e613dfd61ad2ahttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52532b2006361d25c9000009?auth=1696717126-0-kxjzwg6x-3353c0df405f0de4a54e613dfd61ad2ahttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52532b2006361d25c9000009?auth=1696717126-0-kxjzwg6x-3353c0df405f0de4a54e613dfd61ad2ahttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52532b2006361d25c9000009?auth=1696717126-0-kxjzwg6x-3353c0df405f0de4a54e613dfd61ad2ahttp://api.dmcloud.net/player/embed/524b31a094a6f62bc6000006/52532ad094a6f608df000004?

Recommended