Post on 02-Jan-2016
transcript
MAT 1234Calculus I
Section 2.4
Derivatives of Tri. Functions
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Exam 1 Tutoring Record
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HW and Quiz
WebAssign HW 2.4 Quiz: 2.3, 2.4
Preview
Skills• Formulas for the derivatives of tri. functions
• Find limits by change of variables
Concepts• Find limits by simple geometric insights
• an application of the squeeze theorem
Formulas
xxdx
d
xxxdx
d
xxxdx
d
xxdx
d
xxdx
d
xxdx
d
22 csccot
tansecsec
cotcsccsc
sectan
sincos
cossin
Formulas
sin cosd
x xdx
Why?
Let ( ) sin
( ) ( )
f x x
f x h f x
h
0 0
( ) ( )lim limh h
f x h f x
h
Formulas
0
cos( ) 1lim 0h
h
h
0
sin( )lim 1h
h
h
We are going to look at the first limit later.
Example 1
t
ttf
sin1
cos)(
sin cos
cos sin
dx x
dxd
x xdx
( )f t
Example 2
tan)( h
2tan secd
x xdx
( )
( )
h
h
Important Limit
1sin
lim0
Use to find the formulas for the derivatives of the tri. functions
Use to find other limits Use often in physics for approximations
e.g. mechanical system, optics
Example Simple Pendulum
When the angle is small, the motion can be modeled by
l
02
2
l
g
dt
d
2
2sin 0
d g
dt l
0
sinlim 1
sin
Important Limit
1sin
lim0
Evidence: Graphs Proofs (a) Geometric proof (Section 2.4)
(b) L’ hospital Rule (Section 6.8)
(c) Taylor Series (Section 11.10)
Why?
Important Limit 0
sinlim 1
Evidence: Graphs
Important Limit
Evidence: Graphs
0
sinlim 1
Important Limit
Proofs (a) Geometric proof (Section 2.4)
(b) L’ hospital Rule (Section 6.8)
(c) Taylor Series (Section 11.10)
0
sinlim 1
Example 3
x
xx
2
0
sinlim
0
sinlim 1
2
0
sinlimx
x
x
Example 4
xx
xx sin
sinlim
0
0
sinlim 1
0
sinlim
sinx
x
x x
Generalized Formula
0
sinlim 1 , where 0x
kxk
kx
0
sinlim 1
Why?
Example 5
0
sin 7lim
3x
x
x
0
sinlim 1x
kx
kx
0
sin 7lim
3x
x
x
Remark
It is incorrect to use the limit laws and write
0 0
sin 7 1 sin 7lim lim
3 3x x
x x
x x
since we do not know the existence of
0
sin 7limx
x
x
Example 6
x
xx sin
7sinlim
0
0
sinlim 1x
kx
kx
0
sin 7lim
sinx
x
x
Purposes (Skip if …)
Look at the interesting power of geometry.
Look at an application of the squeeze theorem.
Geometric Proof (Idea)
1sin
lim0
1sin
cos
Simplified Proof: 1
sincos
1
1?
angleradiusArc
Simplified Proof:
1
1sin
1sin
cos
1sin
sin
angleradiusArc
Simplified Proof:
1
?
1sin
cos
Simplified Proof:
1
tan
1sin
cos
sin cos
cos
sin
tan
Important Limit1
sinlim
0
1sin
lim
theorem,squeeze By the
11lim and 1coslim
1sin
cos
,small and 0 If
0
00
Important Limit1
sinlim
0
1
sinlim
)(
)sin(lim
sinlim
)(
)sin(sinsin ,0 If
0
00
0,0 As
-θ