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1 COLLECTION OF SUPPLEMENTARY PROBLEMS CALCULUS I I. CHAPTER 0 --- PRELIMINARIES (real number system, inequalities and absolute values, functions): A. J. STEWART CALCULUS TEXTBOOK: 1. (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c ) How can you tell whether a given curve is the graph of a function? 2. (a) What is an even function? How can you tell if a function is even by looking at its graph? (b) What is an odd function? How can you tell if a function is odd by looking at its graph? 3. Sketch by hand, on the same axes, the graphs of the following functions: (a) () fx x (b) 2 () gx x (c ) 3 () gx x (d) 4 () hx x 4. Draw, by hand, a rough sketch of the graph of each function: (a) sin( ) y x (b) tan( ) y x (c) x y e (d) ln( ) y x (e) 1/ y x (f) y x (g) y x (h) 1 tan () y x 5. Suppose the graph of f is given. Write an equation for each of the graphs that are obtained from the graph of f as follows: (a) Shift 2 units upward (b) Shift 2 units downward (c) Shift 2 units to the right (d) Shift 2 units to the left (e) Reflect about the x-axis (f) Reflect about the y-axis (g) Stretch vertically by a factor of 2 (h) Shrink vertically by a factor of 2 (i) Stretch horizontally by a factor of 2 (h) Shrink horizontally by a factor of 2 6. Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or/and give an example that disproves the statement. a) If f is a function, then ( ) () () fs t fs ft b) If () () fs ft , then s t
Transcript
  • 1

    COLLECTION OF SUPPLEMENTARY PROBLEMS

    CALCULUS I

    I. CHAPTER 0 --- PRELIMINARIES (real number system, inequalities and absolute values, functions):

    A. J. STEWART CALCULUS TEXTBOOK:

    1. (a) What is a function? What are its domain and range?

    (b) What is the graph of a function?

    (c ) How can you tell whether a given curve is the graph of a function?

    2. (a) What is an even function? How can you tell if a function is even by looking at its graph?

    (b) What is an odd function? How can you tell if a function is odd by looking at its graph?

    3. Sketch by hand, on the same axes, the graphs of the following functions:

    (a) ( )f x x (b) 2( )g x x (c ) 3( )g x x (d) 4( )h x x

    4. Draw, by hand, a rough sketch of the graph of each function:

    (a) sin( )y x (b) tan( )y x (c) xy e (d) ln( )y x

    (e) 1/y x (f) y x (g) y x (h) 1tan ( )y x

    5. Suppose the graph of f is given. Write an equation for each of the graphs that are obtained from the graph of

    f as follows:

    (a) Shift 2 units upward (b) Shift 2 units downward (c) Shift 2 units to the right

    (d) Shift 2 units to the left (e) Reflect about the x-axis (f) Reflect about the y-axis

    (g) Stretch vertically by a factor of 2 (h) Shrink vertically by a factor of 2

    (i) Stretch horizontally by a factor of 2 (h) Shrink horizontally by a factor of 2

    6. Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or/and

    give an example that disproves the statement.

    a) If f is a function, then ( ) ( ) ( )f s t f s f t

    b) If ( ) ( )f s f t , then s t

  • 2

    c) A vertical line intersects the graph of a function at most once.

    7. Let f be the function whose graph is given.

    (a) Estimate the value of f(2). (b) Estimate the values of x such that f(x)=3

    (c ) State the domain of f (d) State the range of f (e) On what interval is f increasing

    (f) Is f one-to-one? Explain. (g) Is f even, odd, or neither even nor odd? Explain.

    8. Find the domain and range of the function:

    (a) 2( ) 4 3f x x (b) ( ) 1/ ( 1)g x x (c) 1 siny x (d) ln lny x

    9. Find an expression for the function whose graphs consists of the line segment from the point (-2,2) to the

    point (-1,0) together with the top half of the circle with center the origin and radius 1.

    PROBLEMS PLUS:

    10. One of the legs of a right triangle has length 4 cm. Express the length of the altitude perpendicular to the

    hypotenuse as a function of the length of the hypotenuse. (Sketch the triangle and label its elements as part of

    your solution).

    11. The altitude perpendicular to the hypotenuse of a right triangle is 12 cm. Express the length of the

    hypotenuse as a function of the perimeter. (Sketch the triangle and label its elements as part of your solution).

    12. Solve the equation: 2 1 5 3x x .

    13. Solve the inequality: 1 3 5x x (give your solution in interval form).

  • 3

    14. Sketch the graph of the function 2( ) 4 3f x x x .

    15. Sketch the graph of the function 2 2( ) 1 4g x x x .

    16. Draw the graph of the equation x x y y .

    17. Draw the graph of the equation4 2 2 2 24 4 0x x x y y .

    18. Sketch the region in plane consisting of all the points ,x y such that 1x y .

    19. Sketch the region in plane consisting of all the points ,x y such that 2x y x y .

    20. Evaluate 2 3 4 31log 3 log 4 log 5 log 32 .

    21. (a) Show that the function 2( ) ln 1f x x x is an odd function.

    (b) Show that f is invertible over its domain and find its inverse.

    22. Solve the inequality 2ln 2 2 0x x .

    23. Use indirect reasoning to prove that 2log 5 is an irrational number.

    24. A driver sets out on a journey. For the first half of the distance she drives at a leisurely pace of 30 mi/h; she

    drives the second half at 60 mi/h. What is her average speed on this trip?

    25. Is it true that f g h f g f h ?

    26. Prove that if n is an arbitrary positive integer, then 7 1n is divisible by 6.

    27. Prove that 21 3 5 2 1n n .

    28. If 20 ( )f x x and 1 0( ) ( )n nf x f f x , for 0,1,2,n , find a formula for ( )nf x and use mathematical

    induction to prove it.

    29. (a) If 01

    ( )2

    f xx

    and 1 0n nf f f for 0,1,2,n , find a formula for ( )nf x and use mathematical

    induction to prove it.

    (b) Graph 0 1 2 3, , ,f f f f on the same screen and describe the effects of repeated composition.

  • 4

    II. CHAPTER 1: Limits:

    A. J. STEWART CALCULUS TEXTBOOK:

    1. Explain (using both the intuitive and the rigorous definition of limit) what each of the following means and

    illustrate with a sketch:

    (a) lim ( )x a

    f x L

    (b) lim ( )x a

    f x L

    (c) lim ( )x a

    f x L

    (d) lim ( )x a

    f x

    (e) lim ( )x

    f x L

    .

    2. Describe several ways in which a limit can fail to exist. Illustrate with sketches.

    3. (a) What does it mean for f to be continuous at a?

    (b) What does it mean for f to be continuous on the interval , ? What can you say about the graph of

    such a function?

    4. Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or/and

    give an example that disproves the statement.

    (a) 4 4 4

    2 8 2 8lim lim lim

    4 4 4 4x x x

    x x

    x x x x

    .

    (b) If 0

    lim ( )x

    f x

    and 0

    lim ( )x

    g x

    , then 0

    lim ( ) ( ) 0x

    f x g x

    .

    (c ) A function can have two different horizontal asymptotes.

    (d) If the line 1x is a vertical asymptote for ( )y f x , then f is not defined at 1.

    5. The graph of f is given.

    (a) Find each limit, or explain why it does not exist:

    (i) 2

    lim ( )x

    f x

    (ii) 3

    lim ( )x

    f x

    (iii) 3

    lim ( )x

    f x

    (iv) 4

    lim ( )x

    f x

    (v) 0

    lim ( )x

    f x

    (vi) 2

    lim ( )x

    f x

    (vii) lim ( )x

    f x

    (viii) lim ( )x

    f x

    (b) State the equations of the horizontal asymptotes;

    (c ) State the equations of the vertical asymptotes;

    (d) At what numbers is f discontinuous? Explain.

  • 5

    6. Find the limit:

    (a) 3

    1lim x xx

    e

    (b) 2

    23

    9lim

    2 3x

    x

    x x

    (c)

    2

    23

    9lim

    2 3x

    x

    x x

    (d)

    2

    21

    9lim

    2 3x

    x

    x x

    (e) 2

    32

    4lim

    8t

    t

    t

    (f)

    49

    lim9r

    r

    r (g)

    4

    4lim

    4v

    v

    v

    (h)

    2

    2

    1 2lim

    1 2x

    x x

    x x

    (i) 3lim x

    xe

    7. Prove that 2 20

    lim cos 1/ 0x

    x x

    .

    8. Prove the following statements using the precise (rigorous) definition of a limit:

    (a) 5

    lim 7 27 8x

    x

    (b) 30

    lim 0x

    x

    (c) 22

    lim 3 2x

    x x

    9. Let

    2

    if 0

    ( ) 3 if 0 3

    3 if 3

    x x

    f x x x

    x x

  • 6

    (a) Evaluate each limit, if it exists:

    (i)0

    lim ( )x

    f x

    (ii) 0

    lim ( )x

    f x

    (iii) 0

    lim ( )x

    f x

    (iv) 3

    lim ( )x

    f x

    (v) 3

    lim ( )x

    f x

    (vi) 3

    lim ( )x

    f x

    (b) Where is f discontinuous?

    (c ) Sketch (by hand) the graph of f .

    10. Show that the function is continuous on its domain. State the domain.

    (a) ( ) cos sin( )h x x x

    (b) 2

    2

    9( )

    2

    xg x

    x

    11. Use the intermediate value theorem to show that there is a root of the equation in the given interval:

    3 22 2 0, 2, 1x x

    12. Find the asymptotes of the graph of 4

    ( )3

    xf x

    x

    and use them to sketch the graph of f .

    PROBLEMS PLUS:

    13. Evaluate 3

    1

    1lim

    1x

    x

    x

    .

    14. Find numbers a and b such that 0

    2lim 1x

    ax b

    x

    .

    15. Evaluate 0

    2 1 2 1limx

    x x

    x

    .

  • 7

    16.

    The figure shows a point P on the parabola 2y x and the point Q where the perpendicular bisector of OP

    intersects the y-axis. As P approaches the origin along the parabola, what happens to Q? Does it have a limiting

    position? If so, find it.

    17. Find all values of a such that f is continuous on : 2

    1, if ( )

    , if

    x x af x

    x x a

    .

    18. If lim ( ) and lim ( )x a x a

    f x g x

    exist, and if lim ( ) ( ) 2x a

    f x g x

    and lim ( ) ( ) 1x a

    f x g x

    , find lim ( ) ( )x a

    f x g x

    .

    19. Suppose that f is a function that satisfies the equation 2 2( ) ( )f x y f x f y x y xy for all real

    numbers and x y . Find (0)f .

    20. A

    P

    B C

    (a) The figure shows an isosceles triangle ABC with B C . The bisector of angle B intersects the side AC

    at the point P. Suppose that the base BC remains fixed but the altitude |AM| of the triangle approaches 0, so A

    approaches the midpoint M of BC. What happens to P during the process? Does it have a limiting position? If

    so, find it.

    (b) Try to sketch the path traced out by P during the process. Then find the equation of this curve and use this

    equation to sketch the curve (you may need to use a calculator or a software for the sketching the curve from its

    equation).

    M

    °

    °

  • 8

    III. CHAPTER 2: The derivative:

    A. J. STEWART CALCULUS TEXTBOOK:

    1. Write / State each of the following differentiation rules both in symbols and in words:

    (a) The Power Rule (b) The Constant Multiple Rule (c) The Sum Rule (d) The Difference Rule

    (e) The Product Rule (f) The Quotient Rule (g) The Chain Rule

    2. Give the derivative of each function:

    (a) ry x (b) sin( )y x (c) cos( )y x (d) tan( )y x (e) cot( )y x

    3. Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or/and

    give an example that disproves the statement.

    (a) ( ) ( ) '( ) '( )d

    f x g x f x g xdx

    (b) If and f g are differentiable, then ( ) ' ( ) '( )d

    f g x f g x g xdx

    (c ) If f is differentiable, then '

    ( )2 ( )

    f xdf x

    dx f x

    (d) If f is differentiable, then '

    2

    f xdf x

    dx x

    (e) 2 2tan ( ) sec ( )d d

    x xdx dx

    (f) 2 2 1d

    x x xdx

    (g) If 52

    ( ) (2)( ) , then lim 80

    2x

    g x gg x x

    x

    . (h)

    22

    2

    d y dy

    dx dx

    (i) An equation of the tangent line to the parabola 2y x at (-2,4) is 4 2 2y x x .

    4. Calculate 'y for:

    (a) 3

    4 23 5y x x (b) cos tan( )y x (c) 3 4

    1y x

    x

    (d) 21

    ty

    t

    (e) 4 2 3xy x y x y (f)

    sec 2

    1 tan 2y

  • 9

    (g) 2 cos( ) sin 2x y y xy (h) 1

    11y x

    (i) 3

    1y

    x x

    (j) siny x (j)

    5

    7

    1 2

    3

    x xy

    x

    (k)

    4

    4 4

    xy

    x

    5. If ( ) 4 1f t t , find ''(2)f .

    6. Find 6 6'' if 1y x y .

    7. Find ( )1

    ( ) if ( )2

    nf x f xx

    .

    8. Evaluate

    3

    30lim

    tan 2t

    t

    t.

    9. Find an equation of the tangent to the curve at the given point:

    (a) 24sin ( ), ,16

    y x

    (b) 2

    2

    1, 0, 1

    1

    xy

    x

    (c ) 1 4sin( ), 0,1y x

    10. At what points on the curve sin( ) cos( ), 0 2y x x x is the tangent line horizontal?

    11. Find the points on the ellipse 2 22 1x y where the tangent line has slope 1.

    12. If ( )f x x a x b x c , show that '( ) 1 1 1

    ( )

    f x

    f x x a x b x c

    .

    13. (a) By differentiating the double angle formula for Cosine: 2 2cos 2 cos sinx x x obtain the double angle

    formula for Sine.

    (b) By differentiating the addition formula for Sine: sin sin cos cos sinx a x a x a obtain the addition

    formula for Cosine.

    14. Suppose that ( ) ( ) ( )h x f x g x and ( ) ( )F x f g x , where

    (2) 3, (2) 5, '(2) 4, '(2) 2, and '(5) 12f g g f f . Find (a) '(2)h and (b) '(2)F .

  • 10

    15. The volume of a cube is increasing at a rate of 310 / mincm . How fast is the surface area increasing when

    the length of an edge is 30cm?

    16. A paper cup has the shape of a cone with height 10cm radius 3cm (at the top). If water is poured into the cup

    at a rate of 32 / seccm , how fast is the water level rising when the water is 5cm deep?

    17. Express the limit as a derivative and evaluate:

    (a) 17

    1

    1lim

    1x

    x

    x

    (b)

    4

    0

    16 2limh

    h

    h

    (c)

    /3

    cos 0.5lim

    / 3

    18. Evaluate 30

    1 tan 1 sinlimx

    x x

    x

    .

    19. Suppose that f is a differentiable function such that ( )f g x x and 2

    '( ) 1 ( )f x f x . Show that

    2

    1'( )

    1g x

    x

    .

    PROBLEMS PLUS:

    20. If f is a differentiable function and ( ) ( )g x xf x , use the definition of a derivative to show that

    '( ) '( ) ( )g x xf x f x .

    21. Suppose that f is a function that satisfies the equation 2 2( ) ( ) ( )f x y f x f y x y xy for all real

    numbers and x y . Suppose also that 0

    ( )lim 0x

    f x

    x .

    (a) Find (0)f . (b) Find '(0)f . (c) Find '( )f x .

    22. Suppose f is a function with the property that 2( )f x x for all x . Show that (0) 0f . Then show that

    '(0) 0f .

    23.

  • 11

    Find points P and Q on the parabola 21y x so that the triangle ABC formed by the x-axis and the tangent

    lines to the parabola at P and Q is an equilateral triangle.

    24. Prove that 4 4 1sin cos 4 cos 4 / 2n

    n

    n

    dx x x n

    dx (use mathematical induction).

    25. Find the nth derivative of the function ( )1

    nxf x

    x

    .

    26.

    The figure above shows a circle with radius 1 inscribed in the parabola 2y x . Find the center of the circle.

    27. If f is differentiable at a , where 0a , evaluate the following limit in terms of

    '( )f a : ( ) ( )

    limx a

    f x f a

    x a

    .

    28. Tangent lines 1T and 2T are drawn at two points 1P and 2P on the parabola 2y x and they intersect at a point

    P. Another tangent line T is drawn at a point between 1P and 2P ; it intersects 1T at 1Q and 2T at 2Q . Show that:

    1 2

    1 2

    1PQ PQ

    PP PP .

    29. Evaluate 2

    0

    sin 3 sin 9limx

    x

    x

    .

  • 12

    30. (a) Use the trigonometric identity: tan tan

    tan( )1 tan tan

    x yx y

    x y

    to show that if two lines

    1L and 2L intersect at

    an angle , then 2 1

    1 2

    tan1

    m m

    m m

    , where 1m and 2m are the slopes of 1L and 2L , respectively.

    (b) The angle between the curves 1C and 2C at a point of intersection P is defined to be the angle between

    the tangent lines to 1C and 2C at P (if these tangent lines exist). Use part (a) to find, correct to the nearest degree,

    the angle between each pair of the following curves at an arbitrary point of intersection (a,b):

    (i) 2y x and 2

    2y x

    (ii) 2 2 3x y and 2 24 3 0x x y .

    31. If and f g are differentiable functions with 0 0 0f g and ' 0 0g , show that 0

    ( ) '(0)lim

    ( ) '(0)x

    f x f

    g x g .

    32. Evaluate

    20

    sin 2 2sin sinlimx

    a x a x a

    x

    .

  • 13

    IV. CHAPTER 3: Applications of the derivative:

    A. J. STEWART CALCULUS TEXTBOOK:

    1. Explain the difference between an absolute maximum and a local maximum. Illustrate with a sketch.

    2. (a) State the first derivative test.

    (b) State the second derivative test.

    (c) What are the relative advantages and disadvantages of these tests?

    3. Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or/and

    give an example that disproves the statement.

    (a) If '( ) 0f c , then f has a local maximum or a local minimum at c .

    (b) If f has an absolute minimum value at c , then '( ) 0f c .

    (c) If f is continuous on ,a b , then f attains an absolute maximum value ( )f c and an absolute minimum

    value ( )f d at some numbers c and d in ,a b .

    (d) If '( ) 0f x for 1 6x , then f is decreasing on 1,6 .

    (e) If ''(2) 0f , then 2, (2)f is an inflection point of the curve ( )y f x .

    (f) If '( ) '( )f x g x for 0 1x , then ( ) ( )f x g x for 0 1x .

    (g) There exists a function f such that ( ) 0, '( ) 0f x f x and ''( ) 0f x for all x .

    (h) If f and g are increasing on an interval I , then f g is increasing on I .

    (i) If f and g are increasing on an interval I , then f g is increasing on I .

    (j) If f is increasing and ( ) 0 on f x I , then 1

    ( )( )

    g xf x

    is decreasing on I .

    (k) If '( )f x exists and is nonzero for all x , then (1) (0)f f .

    4. In the following problems, a function ( )f x and its domain D are given. Determine the critical points,

    evaluate f at these critical points, and using the critical point theorem find the global maximum and minimum

    values of ( )f x on D . For all these critical points found above, use your test of choice ( either (a) the first

    derivative test, or (if possible) (b) the second derivative test ) to decide which of these points give a local

    maximum and which give a local minimum:

  • 14

    (a) 3( ) 10 27 , = 0,4f x x x D

    (b) ( ) , 0,4f x x x D

    (c) 2( ) , 2,01x

    f x Dx x

    (d) 3

    2( ) 2 , 2,1f x x x D

    (e) ( ) sin 2 , 0,f x x x D

    5. Sketch the graph of a function that satisfies the following conditions:

    (a)

    6

    (0) 0, '( 2) '(1) '(9) 0

    lim ( ) 0, lim ( )

    '( ) 0 on , 2 , 1,6 and 9,

    '( ) 0 on 2,1 ,and 6,9

    ''( ) 0 on ,0 ,and 12,

    ''( ) 0 on 0,6 ,and 6,12

    x x

    f f f f

    f x f x

    f x

    f x

    f x

    f x

    (b)

    (0) 0, is continuous and even

    '( ) 2 if 0 1, '( ) 1, if 1 3

    '( ) 1, if 3

    f f

    f x x x f x x

    f x x

    6. The figure below shows a graph of the derivative '( )f x of a function ( )f x .

    (a) On what intervals is f increasing and decreasing?

    (b) For what values of x does f has a local maximum or minimum?

    (c) Sketch the graph of ''f .

    (d) Sketch a possible graph of f .

  • 15

    7. Use the guidelines (method) in Section 3.5 to sketch the curve: (DO NOT use – or at least do not start with –

    a calculator. You can use a calculator for verification purposes at most):

    (a) 32 2y x x

    (b) 4 3 23 3y x x x x

    (c) 2

    1

    1y

    x

    (d) 1 1

    1y

    x x

    (e) 1y x x

    (f) 3y x x

    (g) 2y x x

    (h) 2

    8

    xy

    x

    (i) 2sin 2cosy x x

    (j) 4 tany x x

    8. Show that the equation 101 51 1 0x x x has exactly one real root.

  • 16

    9. By applying the mean value theorem to the function 1/5( )f x x on the interval [32,33] , show that

    52 33 2.0125 .

    10. For what values of the constants a and b is (1,6) a point of inflection of the curve 3 2 1y x ax bx ?

    11. Find two positive integers such that the sum of the first number and four times the second number is 1000

    and the product of the numbers is as large as possible.

    12. Using Calculus, show that the shortest distance from the point 0 0,x y to the straight line 0Ax By C is

    0 0

    2 2

    Ax By C

    A B

    (this is the formula for the distance between a point and a line in coordinate geometry).

    13. Find the point on the hyperbola 8xy that is closest to the point (3,0) .

    14. Find the smallest possible area of an isosceles triangle that is circumscribed about a circle of given fixed

    radius r .

    15. Find the volume of the largest cone that can be inscribed in a sphere of given fixed radius r .

    16. In ABC , D lies on AB, CD AB , 4AD BD cm , and 5CD cm . Where should a point P be

    chosen on CD, such that the sum PA PB PC is a minimum?

    17. The velocity of a wave of length L in deep water is L C

    v KC L

    , where and K L are assumed known

    positive constants. What is the length of the wave that gives the minimum velocity?

    18. Find f is

    (a) 55

    4'( )f x x

    x

    (b) '( ) 2 3sin , (0) 5f x x x f

    (c ) 2

    '( ) , (1) 3u u

    f u fu

    (d) 3 2''( ) 2 3 4 5, (0) 2, (1) 0f x x x x f f

    19. A rectangular beam will be cut from a cylindrical log of radius 10 inches. Show that the beam of maximal

    cross-sectional area is a square.

  • 17

    20. Show that, for 0x :

    (a) sin( )x x

    (b) tan( )x x

    21. Sketch the graph of a function f such that '( ) 0f x for all x , ''( ) 0f x for 1x , ''( ) 0f x for 1x , and

    lim ( ) 0x

    f x x

    .

    PROBLEMS PLUS:

    22. Show that sin cos 2x x for all x .

    23. Show that 2 2 2 24 4 16x y x y for all numbers x and y such that 2x and 2y .

    HINT: Simplify the problem by setting to prove an appropriate inequality for an expression of x only, and a

    similar inequality for an expression of y only, and combine these two results to prove the inequality above.

    24. Let a and b be positive numbers. Show that not both of the numbers 1a b and 1b a can be greater

    than 1

    4.

    25. Find the point on the parabola 21y x at which the tangent line cuts from the first quadrant the triangle

    with the smallest area.

    26. Find the highest and the lowest points on the curve 2 2 12x xy y .

    27. Find a function f such that 1

    '( 1) , '(0) 0, and ''( ) 02

    f f f x for all x , or prove that such a function

    cannot exist.

    28. Determine the values of the number a for which the function 2( ) 6 cos 2 2 cos1f x a a x a x has no critical point.

    29. Let ABC be a triangle with 120 and 1BAC AB AC

    (a) Express the length of the angle bisector AD in terms of x AB .

    (b) Find the largest possible value of AD .

  • 18

    30. For what values of c is there a straight line that intersects the curve 4 3 212 5 2y x cx x x in four

    distinct points?

    31. Given a sphere with radius r, find in terms of r the height h of a pyramid of minimum volume whose base is

    a square and whose base and triangular faces are all tangent to the sphere. (Use the fact that the volume of a

    pyramid is 1

    3V Ah , where A is the area of the base).

    32. Assume that a spherical snowball melts so that its volume decreases at a rate proportional to its surface area.

    If it takes three hours for the snowball to decrease to half its original volume, how much longer will it take for

    the snowball to melt completely?

  • 19

    V. CHAPTER 4: The Definite Integral:

    A. J. STEWART CALCULUS TEXTBOOK:

    1. (a) Write an expression for a Riemann sum of a function f . Explain the meaning of the notation you use.

    (b) If ( ) 0f x , what is the geometric interpretation of its Riemann sum? Illustrate with a diagram.

    (c ) If ( )f x takes on both positive and negative values, what is the geometric interpretation of its Riemann

    sum? Illustrate with a diagram.

    2. (a) Write the definition of the definite integral of a continuous function from to a b (its formula in terms of

    the Riemann sum).

    (b) What is the geometric interpretation of ( )b

    a

    f x dx if ( ) 0f x ?

    (c ) What is the geometric interpretation of ( )b

    a

    f x dx if ( )f x takes on both positive and negative values?

    Illustrate with a diagram.

    3. State/Write the first Fundamental Theorem of Calculus.

    4. State/Write the second Fundamental Theorem of Calculus.

    5. (a) Explain the meaning of the indefinite integral ( )f x dx .

    (b) What is the connection between the definite integral ( )b

    a

    f x dx and the indefinite integral ( )f x dx ?

    6. Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or/and

    give an example that disproves the statement.

    (a) If and f g are continuous on ,a b , then ( ) ( ) ( ) ( )b b b

    a a a

    f x g x dx f x dx g x dx .

    (b) If and f g are continuous on ,a b , then ( ) ( ) ( ) ( )b b b

    a a a

    f x g x dx f x dx g x dx

    .

    (c ) If f is continuous on ,a b , then 5 ( ) 5 ( )b b

    a a

    f x dx f x dx .

  • 20

    (d) If f is continuous on ,a b and ( ) 0f x , then ( ) ( )b b

    a a

    f x dx f x dx .

    (e) If 'f is continuous on 1,3 , then 3

    1

    '( ) (3) (1)f x dx f f .

    (f) If and f g are continuous on ,a b and ( ) ( ) for f x g x a x b then ( ) ( )b b

    a a

    f x dx g x dx .

    (g) If and f g are differentiable on ,a b and ( ) ( ) for f x g x a x b , then '( ) '( ) for f x g x a x b .

    (h)

    1

    5 9

    24

    1

    sin( )6 0

    1

    xx x dx

    x

    .

    (i) 5 5

    2 2

    5 0

    2ax bx c dx ax c dx

    .

    (j)

    1

    4

    2

    1 3

    8dx

    x

    .

    (k) 2

    3

    0

    x x dx represents the area under the curve 3y x x from 0 to 2.

    (l) All continuous functions have derivatives.

    (m) All continuous functions have antiderivatives.

    7. (a) Evaluate the Riemann sum for 2( )f x x x , 1 3x

    with four subintervals, taking the sample points to be the right endpoints. Explain, with the aid of a diagram,

    what the Riemann sum represents. Then evaluate and explain the Riemann sum with 10 subintervals.

    (b) Use the definition of a definite integral (with right endpoints) to calculate the value of the integral

    3

    2

    1

    x x dx . Compare with your results in part (a) and explain the difference.

    (c ) Use the Fundamental Theorem of Calculus to check your answer to part (b).

    (d) Draw a diagram to explain the geometric meaning of the integral in part (b).

  • 21

    8. Repeat problem 7 for ( ) 2 3f x x , 0 1x .

    9. Repeat problem 7 for ( ) 10f x x , 1 1x .

    10. Repeat problem 7 for 2( )f x x x , 0 2x .

    11. Repeat problem 7 for 3 2( ) 6 11 6f x x x x , 1 1x .

    12. Repeat problem 7 for 3 2( ) 6 11 6f x x x x , 2 2x .

    13. Evaluate 1

    2

    0

    1x x dx by interpreting in terms of areas.

    14. Express 1

    lim sinn

    in

    i

    x x

    , where 1i ix x x ,as a definite integral on the interval 0, , and then evaluate

    the integral to calculate this limit.

    15. If 6

    0

    ( ) 10f x dx , and 4

    0

    ( ) 7f x dx , find 6

    4

    ( )f x dx .

    16. Evaluate:

    (a) 1

    2

    0

    sin 1d

    x dxdx

    (b) 1

    2

    0

    sin 1d

    x dxdx

    (c) 2

    0

    sin 1

    xd

    t dtdx

    17. Evaluate the integral, if it exists:

    (a) 1

    9

    0

    1 x dx (b) 1

    9

    0

    1 x dx (c) 9 2

    1

    2x xdx

    x

    (d) 1

    24

    0

    1u du (e) 2

    2 3

    0

    1y y dy (f)

    3

    2

    1 4

    dt

    t

    (g)

    5

    2

    1 4

    dt

    t (h)

    1

    0

    sin 3 t dt (i) 1

    2

    1

    sin( )

    1

    xdx

    x

    (j)

    4 2

    2

    1 x xdx

    x

    (k)

    10

    2

    3

    4x x dx (l) 10

    2

    1

    4x x dx

    (m) 2

    2

    4

    xdx

    x x

    (n) sin cost t dt (o)

    3

    2

    0

    4x dx

  • 22

    (p) 4

    0

    1x dx

    18. Find the derivative of the function:

    (a) 2

    1

    1

    x

    t dt (b) 2

    sin( )x

    x

    tdt

    t

    19. Use properties of integrals to verify the inequality:

    (a) 1

    2

    0

    1cos( )

    3x x dx (b)

    /2

    /4

    sin( ) 2

    2

    xdx

    x

    20. Let 2

    1, if 3 0( )

    1 if 0 1

    x xf x

    x x

    .

    Evaluate 1

    3

    ( )f x dx

    by interpreting the integral as a difference of areas.

    21. Find the value of the number c such that the area under the curve siny cx between 0x and 1x is

    equal to 1.

    22. Find

    2

    3

    02

    1lim 1

    h

    ht dt

    h

    .

    23. Evaluate

    9 9 9 91 1 2 3

    limn

    n

    n n n n n

    .

    PROBLEMS PLUS:

    24. Evaluate 3

    sin( )

    3

    xd x t

    dtdx x t

    .

    25. Evaluate 3

    3

    sin( )lim

    3

    x

    x

    x tdt

    x t .

    26. If 2

    0

    sin ( )

    x

    x x f t dt , where f is a continuous function, find (4)f .

  • 23

    27. Suppose that the curve ( )y f x passes through the origin and the point 1,1 . Find the value of the integral 1

    0

    '( )f x dx .

    28. Show that

    2

    4

    1

    1 1 7

    17 1 24dx

    x

    .

    29. (can use calculator/Mathematica)

    (a) Graph several members of the family of functions 2

    3

    2( )

    cx xf x

    c

    for 0c and look at the regions enclosed

    by these curves and the x-axis. Make a conjecture about how the areas of these regions are related.

    (b) Prove your conjecture in part (a).

    (c ) Take another look at the graphs in part (a) and use them to sketch the curve traced by the vertices (highest

    points) of the family of functions. Can you guess what kind of curve this is?

    (d) Find the equation of the curve you sketched in part (c ).

    30. If

    ( )

    30

    1( )

    1

    g x

    f x dtt

    , where cos

    2

    0

    ( ) 1 sin

    x

    g x t dt , find ' 2

    f

    .

    31. If 2 20

    ( ) sin

    x

    f x x t dt , find '( )f x .

    32. Evaluate 1/

    00

    1lim 1 tan 2

    tx

    xt dt

    x .

    33. Find the interval [ , ]a b for which the value of the integral 22b

    a

    x x dx is a maximum.

    34. Use an integral to estimate 10000

    1i

    i

    .

    35. If f is a differentiable function such that 2

    0

    ( ) ( )

    x

    f t dt f x for all x , find ( )f x .

    36. Evaluate 1 1 1 1

    lim1 2 3n n n n n n n n n n

    .

  • 24

    37. Suppose that f is continuous, 1

    0

    1(0) 0, (1) 1, '( ) 0, and ( )

    3f f f x f x dx . Find the value of the integral

    1

    1

    0

    ( )f y dy .

  • 25

    VI. CHAPTER 5: Applications of integration:

    A. J. STEWART CALCULUS TEXTBOOK:

    1. a) Draw two typical curves ( ) and ( )y f x y g x where ( ) ( ) for f x g x a x b . Show how to

    approximate the area between these curves by a Riemann sum and sketch the corresponding approximating

    rectangles. Then write an expression for the exact area. (For clarity, you can treat separately the cases

    ( ) ( ) 0f x g x , ( ) 0 ( )f x g x and 0 ( ) ( )f x g x ).

    b) Explain how the situation changes if the curves have equations ( ) and ( )x f y x g y where

    ( ) ( ) for f y g y c y b .

    2. Suppose that Sue runs faster than Kathy throughout a 1500 meter race. What is the physical meaning of the

    area between their velocity curves for the first minute of the race?

    3. a) Suppose S is a solid with known cross-sectional areas. Explain how to approximate the volume of S by a

    Riemann sum. Then write an expression for the exact volume.

    b) If S is a solid of revolution, how do you find the cross sectional areas?

    4. a) What is the average value of a function f on an interval ,a b ?

    b) What does the Mean Value Theorem for integrals say? What is its geometric interpretation?

    5. Find the area of the region bounded by the given curves (you MUST plot the curves first, in order to apply

    the correct formula):

    a) 2 6, 0y x x y

    b) 2 220 , 12y x y x

    c) 20, 3x y x y y

    d) 2sin 22

    xy y x x

    e) 2, , 2y x y x x

    6. Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified

    axis:

    a) 22 , y x y x about the x axis

    b) 21 , 3x y y x about the y axis

  • 26

    c) 2 21, 9y x y x about 1y

    d) 2 2 2 , x y a x a h (where 0, 0a h ) about the y axis .

    7. Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by

    the given curves about the specified axis:

    3 5cos , 0, ,

    2 2y x y x x

    about the y axis

    8. Let be the region in the first quadrant bounded by the curves 3 2 and 2y x y x x . Calculate the

    following quantities:

    a) The area of .

    b) The volume obtained by rotating about the x axis.

    c) The volume obtained by rotating about the y axis.

    9. Each integral represents the volume of a solid. Describe the solid:

    a)

    /2

    2

    0

    2 cos xdx

    b) 2

    2

    0

    2 4y y dy

    c) 1

    222

    0

    2 2x x dx

    10. The base of a solid is a circular disc with radius 3. Find the volume of the solid if parallel cross-sections

    perpendicular to the base are isosceles right triangles with hypotenuse lying along the base.

    11. The base of a solid is the region bounded by the parabolas 2y x and 22y x . Find the volume of the

    solid if the cross-sections perpendicular to the x - axis are square with one side lying along the base.

    12. The base of a solid is a square with vertices located at 1,0 , 0,1 , 1,0 and 0, 1 . Each cross-section

    perpendicular to the x - axis is a semi-circle. Find the volume of the solid.

  • 27

    13. The height of a monument is 20 m. A horizontal cross-section at a distance x meters from the top is an

    equilateral triangle with side / 4x meters. Find the volume of the monument.

    14. Let 1 be the region bounded by 2 , 0, and y x y x b , where 0b . Let 2 be the region bounded by

    2 2, 0, and y x x y b .

    a) Is there a value of b such that 1 and 2 have the same area?

    b) Is there a value of b such that 1 sweeps out the same volume when rotated about the x axis and the y axis ?

    c) Is there a value of b such that 1 and 2 sweep out the same volume when rotated about the x axis ?

    d) Is there a value of b such that 1 and 2 sweep out the same volume when rotated about the y axis ?

    PROBLEMS PLUS:

    15. a) Find a positive continuous function f such that the area under the graph of f with x from 0 to t is

    3( )A t t for all 0t .

    b) A solid is generated by rotating about the x axis the region under the curve ( )y f x , where f is a

    positive function and 0x . The volume generated by the part of the curve from 0x to x b is 2b for all

    0b . Find the function f .

    16. There is a line through the origin that divides the region bounded by the parabola 2y x x and the x -axis

    into two regions with equal area. What is the slope of that line?

    17. The figure above shows a horizontal line y c intersecting the curve 38 27y x x . Find the number c

    such that the areas of the shaded regions are equal.

  • 28

    18. a) Show that the volume of a segment of height h of a sphere of radius r is 2

    33

    hV r h

    .

    b) Show that if a sphere of radius 1 is sliced by a plane at a distance x from the center in such a way that the

    volume of one segment is twice the volume of the other, then x is a solution of the equation:

    33 9 2 0x x

    where 0 1x . Use Newton’s method to find x accurate to four decimal places.

    c) Using the formula for the volume of a segment of a sphere, it can be shown that the depth x to which a

    floating sphere of radius r sinks in the water is a root of the equation:

    3 2 33 4 0x rx r s

    where s is the specific gravity of the sphere. Suppose a wooden sphere of radius 0.5 has a specific gravity of

    0.75. Calculate, to four decimal places accuracy, the depth to which the sphere will sink.

    d) A hemispherical bowl has radius 5 inches and water is running into the bowl at a rate of 30.2 / secin .

    i) How fast is the water level in the bowl rising at the instant the water is 3 inches deep?

    ii) At a certain instant, the water is 4 inches deep. How long will it take to fill the bowl?

    19. The figure shows a curve C with the property that, for every point P on the middle curve 22y x , the areas

    A and B are equal. Find an equation for the curve C.

    20. A sphere of radius 1 overlaps a smaller sphere of radius r in such a way that their intersection is a circle of

    radius r . (In other words, they intersect in a great circle of the small sphere). Find r such that the volume inside

    the small sphere and outside the large sphere is as large as possible.


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