AP ExamAP ExamReview Review

CompetitionCompetition•First to finish calls “15 seconds”. First to finish calls “15 seconds”.

•Answers must be written down & then Answers must be written down & then revealed.revealed.

•Add 1 pt to your score for each correct Add 1 pt to your score for each correct response.response.

•Come back later for review & practice! Come back later for review & practice!

1.Find the 1.Find the limit.limit.

Recall key stepRecall key step: :

divide all terms by the highest power xdivide all terms by the highest power x33

Answer: Answer:

*green terms *green terms 00

3

30

4 7

1

1

5lim 1

5xx x

x

3 2

34 7

5 1limx

x x

x

2.Find the derivative.2.Find the derivative.

Answer: Answer:

1

212 21 1 12 2

22x xe or e x xx x

2 ln 0xde x x for x

dx

3. Evaluate:3. Evaluate:2csc x dx

Answer: Answer: cot x C

4. 4. Fill in the Fill in the blanks:blanks:Since polynomial functions are Since polynomial functions are continuous over the reals and for continuous over the reals and for f(x) = xf(x) = x33 -1, we know f(0) = -1 and -1, we know f(0) = -1 and

f(2) = 7, there exists a value f(2) = 7, there exists a value cc in in the interval ________such that f(the interval ________such that f(cc))

== 5 by the ___________________ 5 by the ___________________ Theorem. Theorem.

Answer: (0, 2) Intermediate Answer: (0, 2) Intermediate Value Value

5. Given f(x) and g(x) are diff. 5. Given f(x) and g(x) are diff. over over RR and g(x) = f and g(x) = f -1-1(x). If (x). If f(5)=7,f(5)=7,f ’(5)=2, f(9)=5 and f ’(9)=6, f ’(5)=2, f(9)=5 and f ’(9)=6, find g’(5).find g’(5).

Answer: 1/6 Answer: 1/6

Slopes on inverses are reciprocals at Slopes on inverses are reciprocals at corresponding pts. Since (9,5) is on corresponding pts. Since (9,5) is on f, then (5, 9) is on g . . . so we simply f, then (5, 9) is on g . . . so we simply take the reciprocal of f’(9) to get take the reciprocal of f’(9) to get g’(5).g’(5).

6. Evaluate:6. Evaluate:

Answer: Fundamental Thrm of Answer: Fundamental Thrm of Calculus Calculus

7. Evaluate f 7. Evaluate f ’’(x):(x):

8. Name the theorem used in 8. Name the theorem used in problem 7 above.problem 7 above.

1

0

3xe dx Answer: Answer:

3 31 1 11

3 3 3e OR e

5

2( )

x tf x e dtAnswer: Answer:

55 xe

9. Evaluate:9. Evaluate:5

12x dx

Answer: 9Answer: 9

(Two triangles with (Two triangles with

b = 3 and h = 3.)b = 3 and h = 3.)(2,0)

(5,3)(-1,3)

10. Differentiate with 10. Differentiate with respect to t (time):respect to t (time):

PV = cPV = cwhere c is a where c is a

constantconstant

Answer: Answer: 0

dV dPP Vdt dt

11. For s(t) = t11. For s(t) = t22 + 1, what is the + 1, what is the average velocityaverage velocity over the time over the time interval (0,4) seconds if distance is interval (0,4) seconds if distance is given in ft?given in ft?

12. For s(t) = t12. For s(t) = t22 + 1 above, what is the + 1 above, what is the instantaneous velocityinstantaneous velocity at t = 4? at t = 4?

Answer: Answer:

( ) '( ) 2

(4) 8sec

v t s t t

ftv

Answer: Answer:

17 1

4 0 sec4 ft

13. Evaluate. (You 13. Evaluate. (You must have both must have both

correct!)correct!)

Answer: tanAnswer: tan-1-1x + C and sinx + C and sin-1-1x x + C+ C

2 2

1 1

1 1dx and dx

x x

14. Find the derivative.14. Find the derivative.

Recall key stepRecall key step: : apply the quotient apply the quotient rulerule

32 1d x

dx x

Answer: Answer:

2

3 2

6

3 3 2 2 34

6 6 43

2 2 1 3

2 6 3 3 4 3 4

'

3

'

4

x x x

x

x x x x x xOR x x

x

g f f

x

g

x

g

15. Given the graph of f 15. Given the graph of f ‘‘ (x) shown, (x) shown, give the x-coordinate(s) where f(x) give the x-coordinate(s) where f(x) has has

local minima.local minima.

Answer: Answer:

0 and 30 and 3

(where slopes (where slopes change fromchange fromneg to pos)neg to pos)

2 4321-1-2-3f ' (x)0 321(a,b) (c,d)

16. Given the graph of f 16. Given the graph of f ‘‘ (x) shown, (x) shown, give the x-coordinates where f(x) give the x-coordinates where f(x) has has

points of points of inflection.inflection. Answer: Answer:

a and ca and c

(where f ‘(where f ‘ changes changes from incr from incr decr, decr, the concavity will the concavity will change)change)

2 4321-1-2-3f ' (x)0 321(a,b) (c,d)

17. If f17. If f (x) = g ( h(x) = g ( h (x) (x) ), then f ’(x) = ), then f ’(x) = __?____?__Answer: Answer:

f ’(x) = g’(h(x)) f ’(x) = g’(h(x)) ● ● h’(x)h’(x)

*Derivative of a composite function requires the chain rule

18.By the 218.By the 2ndnd Derivative Derivative Test, if f ”(x) is Test, if f ”(x) is continuous,continuous,f ’(2) =0, and f ”(2) > 0, f ’(2) =0, and f ”(2) > 0, then (2, f(2) ) is a ____ then (2, f(2) ) is a ____ ____.____.Answer: local minimum.Answer: local minimum.

*horiz. tangent in a concave up *horiz. tangent in a concave up interval interval local min local min

19. Find f19. Find f ’(x) if:’(x) if:7

23( )

xf x t dt

Answer: Answer:

(by FTC 2)(by FTC 2)

7 2

2 7

3

3 3

3

( )

'( ) 2 2 16

x

xf x t dt t dt

f x x x

20. Evaluate:20. Evaluate:

2 3 3

sin cos

1 1sin

3 3

u x du x dx

u du u C x C

Answer: Answer:

2sin cosx x dx

21. If f21. If f (x) = (x) =

sinsin22(3x), find f(3x), find f

’(x).’(x).Answer: Answer:

f ’(x) = 2sin(3x) f ’(x) = 2sin(3x) ● ● cos(3x) cos(3x) ● ● 33

*Power rule and two chain rules on the inside functions.

22.Name the type of 22.Name the type of discontinuity at x = 3 fordiscontinuity at x = 3 for

3( )

3

xf x

x

Answer: jump Answer: jump discontinuitydiscontinuity

(3, 1)

(3, -1)

23. Find f23. Find f ’(x) if:’(x) if:

2

77 3( )x

f x t dtAnswer: Answer:

(by FTC 2)(by FTC 2)

2

2

2

77

77

3

3 3

7

( )

'( ) 2 2

x

xf x t dt t dt

f x x x x

24. Evaluate:24. Evaluate:

2csc 2x dxAnswer: Answer:

1cot(2 )

2x C

25. Find the average value of f(x) 25. Find the average value of f(x) over the interval (2,7) given:over the interval (2,7) given:

7

2( ) 14f x dx Answer: Answer:

Recall this is the height of a Recall this is the height of a rectangle that has the same area rectangle that has the same area as the area under the curve.as the area under the curve.

142.8

5or

26. Find the critical #s of26. Find the critical #s of f(x) if:f(x) if:

3' ( )

5

x xf x

x

Answer: -5, -1, 0, 1Answer: -5, -1, 0, 1*Crit # are where f ’(x) = 0 -1, 0, 1

or f ’(x) is undefined -5( 1)( 1)

'( )5

x x xf x

x

27. If oil leaks from a tank at a 27. If oil leaks from a tank at a rate of r(t) gallons per rate of r(t) gallons per minute minute

what doeswhat does

represent?represent?Answer: the net number of Answer: the net number of gallons that leaked from the gallons that leaked from the

tank in the first five minutes.tank in the first five minutes.

5

0( )r t dt

28. Evaluate:28. Evaluate:

2sin cosx x dxAnswer: Answer:

2 3 3

sin cos

1 1sin

3 3

u x du x dx

u du u C x C

29. Evaluate:29. Evaluate:

21

2 1dx

xAnswer: Answer:

22

2

1

1 1

1

2 21

1 1 1

2 1 2

dx x dxx

x

30. If f(x) is differentiable over 30. If f(x) is differentiable over the reals & f ”the reals & f ” (x)(x) == (x –(x – 1)(x –1)(x –

2), over which interval(s) is 2), over which interval(s) is f(x) concave down? f(x) concave down?

Answer: (1, 2)Answer: (1, 2)

f ” < 0 f ” < 0 concave down concave down (-∞ ,1) (1, 2)(-∞ ,1) (1, 2) (2, (2,

∞) ∞) ff ”(x)>0”(x)>0 f ”(x) <0 f f ”(x) <0 f

”(x) >0”(x) >0

31. If f is continuous at (c, 31. If f is continuous at (c, f(c)), which of the following f(c)), which of the following could be FALSE?could be FALSE?

A. A. B.B.

C.C. D.D.

Answer: C Answer: C (e.g., a corner is (e.g., a corner is continuous, but not continuous, but not differentiable)differentiable)A, B & D are the very def of A, B & D are the very def of continuouscontinuous

lim ( )x c

f x exists

lim ( ) ( )x c

f x f c

( )f c is defined'( )f c exists

32. A particle moves along the32. A particle moves along thex-axis so that its position at x-axis so that its position at any time t any time t 0 is given by 0 is given by

The particle is at rest when t The particle is at rest when t

= ?= ?

3 21( ) 4 15

3s t t t t

Answer: when t=3 and t=5 Answer: when t=3 and t=5 secondsseconds

When v(t) = 0When v(t) = 02( ) '( ) 8 15 ( 5)( 3)v t s t t t t t

33. P(t) = 520e33. P(t) = 520e570t570t is the model is the model for the number of fruit flies for the number of fruit flies at time t hours in a biology at time t hours in a biology experiment. What do you experiment. What do you know about the population know about the population at t = 0 hours?at t = 0 hours?

Answer: 520 fruit flies Answer: 520 fruit flies

For P(t) = CeFor P(t) = Cektkt, C is the initial , C is the initial poppop

34. Evaluate both:34. Evaluate both:

2

1

1d

1

1dx

xAnswer:Answer:

arcsin ln 1C and x C

35. Given the graph of f (x) shown, 35. Given the graph of f (x) shown, find the interval(s) where f ”(x) < find the interval(s) where f ”(x) < 0.0.

Answer: Answer:

(-∞, c)(-∞, c)(where f is (where f is

concave down)concave down)

2 4

3

2

1

-1

-2

-3

f (x)

0321

(a,b)

(e,f)

(c,d)

36. Given the graph of f (x) shown 36. Given the graph of f (x) shown give the interval(s) where f ’(x) < give the interval(s) where f ’(x) < 0.0.

Answer: Answer:

(a, e)(a, e)(where f is (where f is decreasing)decreasing)

2 4

3

2

1

-1

-2

-3

f (x)

0321

(a,b)

(e,f)

(c,d)

37. Given the graph of f(x), 37. Given the graph of f(x),

evaluate evaluate

Answer:Answer:

– – ½½ Sum of 2 Sum of 2 ΔΔss

½ + -1½ + -1

0

3( )f x dx

f(x)

½½ ––11

38. Evaluate:38. Evaluate:

2

2

3 x dx

Answer: Answer: 00

(it is an odd (it is an odd

function)function)

39. Give the third part of the 39. Give the third part of the definition of continuity: definition of continuity:

““f is continuous at c if:f is continuous at c if:i.i.

ii.ii.

iii. iii. ??????lim ( )x c

f x exists

Answer:Answer: lim ( ) ( )x c

f x f c

( )f c is defined

40. Find the derivative (and factor 40. Find the derivative (and factor the GCF).the GCF).

Answer:Answer:

2

1 2 2

2

22

2 1

x x

x

x e e x

xe x

1 2xdx e

dx

41. Evaluate:41. Evaluate:1

1lim

1x

x

x

Answer:Answer:

1

1

1 1lim

1 11 1

lim21 1

x

x

x x

x xx

x x

42. Find the equation 42. Find the equation of the tangent to of the tangent to y = xy = x33 – 1 at x =1. – 1 at x =1.

Answer:Answer:

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

0 3( 1) 3 3

y x y

y x OR y x

43. Evaluate:43. Evaluate:

22 tan 2 sec 2x x dxAnswer:Answer:

2

2

2

tan(2 ) 2sec 2

1 1tan 2

2 2

u x du x dx

u du u C x C

44. If f is differentiable over (1, 3), 44. If f is differentiable over (1, 3), f(1)=4, and f(3)= 8, what can f(1)=4, and f(3)= 8, what can you conclude by the Mean you conclude by the Mean Value Theorem?Value Theorem?

Answer: Since f is differentiable Answer: Since f is differentiable over (1,3) and the slope of the over (1,3) and the slope of the secant between endpoints issecant between endpoints is

, there exists a value , there exists a value

c in (1,3) such that fc in (1,3) such that f ’(c) = 2.’(c) = 2.

8 42

3 1m

45. Evaluate:45. Evaluate: 3

32

lim5 17x

x

x x Answer:Answer:

When degrees are equal the horiz When degrees are equal the horiz asymptote is at ratio of leading asymptote is at ratio of leading coefficients.coefficients.

2

5

46. Evaluate:46. Evaluate:3

212

1limx

x

x Answer: Answer: - ∞- ∞

Next to vertical asymp, think of Next to vertical asymp, think of the signs of num. over denom.the signs of num. over denom.

47. Find the slope of 47. Find the slope of the normal to the normal to y = xy = x33 – 1 at x =1. – 1 at x =1.

Answer:Answer:

2' 3 '(1) 3

1

3

y x y

slopeof normal m

48. Evaluate:48. Evaluate:

2

2

2

u u x

u x du x dx

C Ce du ee

Answer:Answer:

22 xxe dx

49. If f is differentiable over 49. If f is differentiable over (0,4) and f(1)=7 and (0,4) and f(1)=7 and f(3)=5, then we know there f(3)=5, then we know there exists a c in ___?___ such exists a c in ___?___ such that f(c)=6 by the that f(c)=6 by the _________?_________._________?_________.

Answers: (1,3)Answers: (1,3)

Intermediate Value TheoremIntermediate Value Theorem

50. If f is differentiable over (0,4) 50. If f is differentiable over (0,4) and we know f(2) = 7 and and we know f(2) = 7 and ff ’(2)=3, what is the best ’(2)=3, what is the best approximation we can give for approximation we can give for f(2.1)?f(2.1)?

Answers: 7.3 by Linear Answers: 7.3 by Linear Approx.Approx.

Tangent line is: y – 7 = 3(x – 2)Tangent line is: y – 7 = 3(x – 2)*Find (2.1, *Find (2.1, ??) on tangent ) on tangent

as a close approximation since the as a close approximation since the tangent lies close to the f(x) curve.tangent lies close to the f(x) curve.

51. Evaluate:51. Evaluate: 3 3

0

2 2limh

h

h

Answer: 12Answer: 12 Shortcut: this is def of derivative for Shortcut: this is def of derivative for f ’(2) where f(x) = xf ’(2) where f(x) = x33, so use f ’(x)= , so use f ’(x)= 3x3x22

Or Alg: Or Alg:

2 2 3

22

3 3

0

0 0

3(2 ) 3(2)

12

2

612 6 1

2

2

lim

lim lim

h

h h

h h h

h

h hh h

h

h

52. Evaluate:52. Evaluate:

1

22

1 1

1dx

x

Answer:Answer:

2

11 122

1 1

1

1arcsin(1) arcsin

2 2 6 3

arcsindxx

x

53.Find the limit.53.Find the limit.2

325 14

8limx

x x

x

Answer: Answer: 22

( 2)( 7) 9 3

12 4( 2)( 2 4)limx

x x

x x x

54. Evaluate each:54. Evaluate each:

7 2

3 2lim

x

xx

7 2

3 2lim

x

xx

Answer: -1 and 7/3Answer: -1 and 7/3

For pos exponents the ratio of is -1For pos exponents the ratio of is -1 (while the 7 and 3 become neglible).(while the 7 and 3 become neglible).For neg exponents, the power terms For neg exponents, the power terms become neglible and the ratio is 7/3become neglible and the ratio is 7/3

2

2

x

x

55. Evaluate:55. Evaluate:

Answer:Answer:

2

33

111

1

arctan 3 arctan 13 4 12

arctandxx

x

21

3 1

1dx

x

56. Evaluate:56. Evaluate: 4 4

0

1 1limh

h

h

Answer: 4Answer: 4 Shortcut: this is the def of derivative for Shortcut: this is the def of derivative for

f(x) = xf(x) = x44 when finding f ’(1), so use f ’(x)= 4x when finding f ’(1), so use f ’(x)= 4x33

Or Alg: Or Alg:

3 2 2 3 4

1 2 32

4 4

3

0

0 0

4(1 ) 6(1 ) 4(1)

4 6 44 4

1

4

1

6

lim

lim lim

h

h h

h h h h

h

h h hh h h

h

h

57. Evaluate:57. Evaluate: 3 2

3 202 7

5 14limx

x x

x x

2

20

2 7 7 1

14 25 14limx

x x

x x

Answer: Answer:

58. 58.

Which of the following are true Which of the following are true about f? about f? (may be one or more answers)(may be one or more answers)

I.I. f has a limit at x =3f has a limit at x =3II.II. f is continuous at x=3f is continuous at x=3III.III. f is differentiable at x=3f is differentiable at x=3

Answer: I onlyAnswer: I only

2 93

( ) 31 3

xif x

f x xif x

59. If f(x) is differentiable over 59. If f(x) is differentiable over RR and f ’(x) and f ’(x) == xx22(x –(x – 1)(x –1)(x –

2), at what values of x, does 2), at what values of x, does f have local minima?f have local minima?

Answer: at x = 2Answer: at x = 2Where f ’ changes from neg to Where f ’ changes from neg to pospos((∞∞, 0) (0, 1) (1, 2) , 0) (0, 1) (1, 2)

(2, (2, ∞∞ ) )f ’(x)>0 f ’(x)>0 f ’(x)<0 f f ’(x)>0 f ’(x)>0 f ’(x)<0 f ’(x)>0’(x)>0

+ - - + - - + + - + - - + - - + + - + + ++ + +

60. If f(x) is differentiable over 60. If f(x) is differentiable over RR and and f ’(x)f ’(x) == xx22(x –(x – 1)(x –1)(x – 2), what term 2), what term describes the point (0, f(0)) on the describes the point (0, f(0)) on the graph of f?graph of f?

Answer: Answer: Inflection PtInflection PtHas a horiz tangent Has a horiz tangent there, but graph is there, but graph is increasing on both increasing on both sides. (2sides. (2ndnd deriv will deriv will change signs there.)change signs there.)

0.5

-0.5

2

f ’ (x)

0.5

-0.5

2

f(x)

61. Use the graph of f ’(x) to give the 61. Use the graph of f ’(x) to give the x-coordinates where the tangent to x-coordinates where the tangent to f(x) will be horizontal.f(x) will be horizontal.

Answer: x=0, 2 and 4 Answer: x=0, 2 and 4 Where f ’(x) Where f ’(x) = 0= 0

0.5

-0.5

2

f ’ (x)

1 2 3 4 5

0.5

-0.5

2

f(x)

1 2 3 4 5

62. Use the graph of f ’(x) to give the 62. Use the graph of f ’(x) to give the interval(s) where f(x) will be interval(s) where f(x) will be concave down.concave down.

0.5

-0.5

2

f ’ (x)

1 2 3 4 5

(1.1,0.2)

(3.2,-0.6)

Answer: (-∞, 0) U (1.1, 3.2)Answer: (-∞, 0) U (1.1, 3.2)Where f ’ is decreasing, f ” will be Where f ’ is decreasing, f ” will be negneg

63. If f(x) = ln e63. If f(x) = ln e, , then then f ’(x) = ?f ’(x) = ?Answer: 0Answer: 0

ln eln e = = and deriv of a and deriv of a constant is zeroconstant is zero

64. Evaluate:64. Evaluate: 3

3 sind

dxx

Answer: - cos xAnswer: - cos x2

2

3

3

sin cos sin sin

sin cos

d d

dx dx

d

dx

x x x x

x x

65. Evaluate:65. Evaluate:

35

3limx

x

x

Answer: -Answer: -∞∞Only two possible answers Only two possible answers ∞ ∞ or -or -

∞∞ Here: Here:

66. If f(x) = (x + 5)(x - 66. If f(x) = (x + 5)(x -

1)1)33, then f ’(x) = ?, then f ’(x) = ?

Answer: Answer: Product rule w/ power ruleProduct rule w/ power rule

f ’(x) = (x + 5) f ’(x) = (x + 5) ● ● 3(x - 1)3(x - 1)22 + (x - + (x -

1)1)33(1)(1)

= (x – 1)= (x – 1)22 [3(x+5) + (x-1)] [3(x+5) + (x-1)]

= 2(x – 1)= 2(x – 1)22 (2x +7) (2x +7)

67. Evaluate:67. Evaluate:

Hint: Convert from complex to simple Hint: Convert from complex to simple fractionfraction

0

11

1limx

xx

Answer: Answer:

0 0

0

11

111

11

11

lim lim

lim

x x

x

xxx x x

x

xx

68. Given f(x) and g(x) are 68. Given f(x) and g(x) are

diff. over diff. over RR and g(x) = f and g(x) = f --11(x). If f(1)=6, f ’(1)=4, (x). If f(1)=6, f ’(1)=4, f(7)=2 and f(7)=2 and f ’(7)=3, find g’(2).f ’(7)=3, find g’(2). Answer: 1/3 Answer: 1/3

Slopes on inverses are reciprocals at Slopes on inverses are reciprocals at corresponding pts. Since (7,2) is on corresponding pts. Since (7,2) is on f, then (2, 7) is on g . . . so we simply f, then (2, 7) is on g . . . so we simply take the reciprocal of f ’(7) to get g take the reciprocal of f ’(7) to get g ’(2).’(2).

69. Find a general solution if:69. Find a general solution if:dy x

dx y Answer:Answer:

2

2 21

2 22

1 1

2 2

y dy x dx

C

C

C

y

y

y x

x

x

70. Find a particular solution 70. Find a particular solution if f(-3) = -2if f(-3) = -2

dy x

dx y Answer:Answer:

2

2

2

2 3

4 9 5

5, 5

( )

( )

C

C

c c

for x

f x

f x

x

x

71. For the diff EQ below, if 71. For the diff EQ below, if given f(0) = 3, then find given f(0) = 3, then find f(1).f(1).

1 1

1

2 2

2(0)22

2(1)2 2

ln

12 ln 2

3 3

( ) 3 (1) 3 3

x C C x

x

x

y

dy dx y x Cy

e e y e e

y C e Ce C

f x e f e e

Answer:Answer:

2dy

dxy

72. Find the volume if the region 72. Find the volume if the region bounded by y = 1-xbounded by y = 1-x22 and y=0 and y=0 is revolved about the x-axis.is revolved about the x-axis.

422

1 0

1 1

12

0

2

5

2 1 2

2 12

3 5

2 1 15 10 3 161

3 5 15 15 15 15

1

2 2

V dx OR V x x dx

x x

x

x

Answer: Answer:

73. Let R be a region in quadrant I 73. Let R be a region in quadrant I bounded by f(x) and g(x) as bounded by f(x) and g(x) as shown. shown. Set up an integral to find the Set up an integral to find the volume if R is revolved about the volume if R is revolved about the x-axis.x-axis. Answer: Answer:

washer

2 2

2 2

( ) ( )

A = ( ) ( )

( ) ( )

o i

a

R f x r g x

f x g x

V f x g x dxc

(c,d)

f(x)

g(x)

(a,b)

74. Let R be a region in quadrant I 74. Let R be a region in quadrant I bounded by f(x) and g(x) as bounded by f(x) and g(x) as shown. shown. Set up an integral to find the Set up an integral to find the volume if cross-sections volume if cross-sections perpendicular to the x-axis are perpendicular to the x-axis are squares.squares.

(c,d)

f(x)

g(x)

(a,b)Answer: Answer:

square

2

2

( ) ( )

A = ( ) ( )

( ) ( )a

sideof square f x g x

f x g x

V f x g x dxc

75-76. Let R be the region bounded by 75-76. Let R be the region bounded by y = 1-xy = 1-x22 and y=0. On base R, and y=0. On base R, cross-sections perpendicular to the cross-sections perpendicular to the y-axis are semi-circles. Find the y-axis are semi-circles. Find the volume.volume.

1 pt for correct set-up of integral & limits1 pt for correct set-up of integral & limits1 pt for correct answer for volume1 pt for correct answer for volume

Answer: Answer:

cross-section

0 0

1 1

12

0

3

radius of semi-circlecross-section: r = 1

1 1A = 1 1

2 21 1

1 12 2

1 1 1 11

2 2 2 2 4

2

y

y y

V y dy y dy

y y units

77. The radius of a circular water spill 77. The radius of a circular water spill is increasing at a rate of 3cm/sec. is increasing at a rate of 3cm/sec. Find the rate at which the Area of Find the rate at which the Area of the spill is increasing when the the spill is increasing when the radius is 10cm.radius is 10cm.

Answer: Answer:

2

2

dA

se

=

d

c

2

A=

sec2 10 63 0

dr

dA r

dt

cm

dt

cmcm

tr

78. Solve the integral78. Solve the integral

by substitution with u = cos 2x.by substitution with u = cos 2x.

32cos (2 )sin(2 )x x dxAnswer: Answer:

3 3

4 4

cos(2 )

cos (

2sin(2 )

2s2 )

1 12

4

i

4

n(2 )

cos

du x dx

x dx du

Let

C x C

u x

x u

u

79. Given y = 5x + k is a 79. Given y = 5x + k is a tangent to f(x) = xtangent to f(x) = x33 + 2x in + 2x in quadrant I. Find k.quadrant I. Find k.

Answer: We see the slope of y = 5x + k Answer: We see the slope of y = 5x + k is 5. is 5. So at what pt on f is the slope equal to So at what pt on f is the slope equal to

5? 5? f ’(x)f ’(x) = 3x = 3x22 + 2 + 2

5 5 = 3x= 3x22 + 2 when x = + 2 when x = 1 1Quad I is given so f(1) = 3 is pt of Quad I is given so f(1) = 3 is pt of tangencytangency (1, 3) lies both on f(x) and on the (1, 3) lies both on f(x) and on the tangenttangentSo plug into y = 5x + k So plug into y = 5x + k 3 = 5(1) + k 3 = 5(1) + k

K = -2K = -2

80. If (a,b) is a cusp on f(x), 80. If (a,b) is a cusp on f(x), what do you know about what do you know about the values of the left and the values of the left and right hand derivatives at x right hand derivatives at x = a?= a?

Answer: the two slopes must go to Answer: the two slopes must go to

∞∞ and and - ∞- ∞ (in either (in either order) order)

81. Differentiate the formula for 81. Differentiate the formula for surface area of a sphere surface area of a sphere implicitly with respect to t implicitly with respect to t (time):(time):

A = 4A = 4rr22Answer: Answer:

dA=4 2 8

dr drr r

dt dt dt

82. If f82. If f ’(x)’(x) >> 0 & f0 & f ”(x)”(x) << 0 over [a, 0 over [a, b], which graph could b], which graph could represent the shape of f(x) on represent the shape of f(x) on this interval?this interval?Answer: Answer:

A. B. C.

E.D.

83. Name the type of 83. Name the type of discontinuity fordiscontinuity for

where x = 1.where x = 1.

Answer: Answer: Removable DiscontinuityRemovable DiscontinuityOthers:Others: “ “infinite discontinuity” at vert. asympinfinite discontinuity” at vert. asymp “ “jump discontinuity” where y bumps jump discontinuity” where y bumps upup

1( )

( 1)( 2)

xf x

x x

84. Give an example of any 84. Give an example of any function that is continuous, function that is continuous, but not differentiable, at a but not differentiable, at a specific x-coordinate. specific x-coordinate. Explain your choice.Explain your choice.

SampleSample

Answer:Answer: f(x) = | f(x) = | xx | is continuous at x | is continuous at x = 0, but the left & right = 0, but the left & right slopes do not agree, so it has slopes do not agree, so it has no derivative at this “corner” no derivative at this “corner”

85. Find two derivatives:85. Find two derivatives:

1dsin x

dx 1d

sin xdx

Answers: Answers:

1

2

1

d 1sin

1d d

sin csc csc cot

xdx x

x x x xdx dx

86. Find86. Find 1 12 2

0

(9 ) (9)limh h

h

Answer: this is the definition of the derivative Answer: this is the definition of the derivative of the square root function!of the square root function!

12

1 12 2

1 1 1( ) '( ) '(9)

62 92

f x f x fxx

87. Given f(x) = g ( h(x) );87. Given f(x) = g ( h(x) );

g(x) = xg(x) = x33 and h(x) = 5x. and h(x) = 5x.

Find f ’ (2).Find f ’ (2).Answer: Answer:

g’(x) = 3xg’(x) = 3x22 and h’(x) = and h’(x) = 55So by the CHAIN RULE:So by the CHAIN RULE:

f’(2) = g’ ( h(2) ) f’(2) = g’ ( h(2) ) ● ● h’(2)h’(2) = g’ (10) = g’ (10) ●●5 = 300 5 = 300 ●●5 = 5 =

1500 1500

88. For88. For

Find c & d given that f(x) is Find c & d given that f(x) is differentiable over (0, 8).differentiable over (0, 8).

2

0 4( )

4 8

cx if xf x

x d if x

Answer:Answer: Cont Cont equal values when x =4 equal values when x =4

c(4) = 4c(4) = 42 2 + d+ dDiff Diff equal derivs when x = 4 equal derivs when x = 4

c = 2x c = 2x c = c = 2(4)2(4)

Thus c = 8 and by back-sub, d = 16Thus c = 8 and by back-sub, d = 16

89.Find the 89.Find the derivative.derivative. 2 2ln 0xdx e for x

dx

Answer: Answer:

22 21 2

2 2 2x xx e exx

90. Evaluate:90. Evaluate:27csc x dx

Answer: Answer: 7cot x C

91. Evaluate:91. Evaluate:

Answer: Answer: 00

(it is an odd (it is an odd

function)function)

7

7

5 x dx

92. For92. For

Find c & d given that f(x) is Find c & d given that f(x) is differentiable over (0, 6).differentiable over (0, 6).

Answer:Answer: Cont Cont equal values when x =3 equal values when x =3

c(3) = 3c(3) = 32 2 + d+ dDiff Diff equal derivs when x = 3 equal derivs when x = 3

c = 2x c = 2x c = c = 2(3) = 62(3) = 6

Thus c = 6 and by back-sub, d = 9Thus c = 6 and by back-sub, d = 9

2

0 3( )

3 6

cx if xf x

x d if x

93. If the rate at which ballots 93. If the rate at which ballots are collected (per hour) is are collected (per hour) is

given by r(t) = tgiven by r(t) = t22 + 2t, how + 2t, how many ballots are collected in many ballots are collected in

the first three hours? the first three hours? Answer: Answer:

3

02 3 2 31

9 9 1803

2 dt ballotst t t t

94. The rate at which ballots 94. The rate at which ballots are collected (per hour) is are collected (per hour) is

given by r(t) = tgiven by r(t) = t22 + 2t. Using + 2t. Using correct units, how is this rate correct units, how is this rate

changing at t = 1 hour?changing at t = 1 hour?Answer: Answer:

2

'( ) 2

'(1) 4

2

4ballots

hour ballotsORhour hour

r t

r

t

95. The rate at which ballots are 95. The rate at which ballots are collected (per hour) is given by r(t) = tcollected (per hour) is given by r(t) = t22

+ 2t and b(x) represents the total # + 2t and b(x) represents the total # ballots collected at any time t. Describe ballots collected at any time t. Describe the concavity of b(t) for all times, t > 0.the concavity of b(t) for all times, t > 0.

96. Evaluate:96. Evaluate:2 7xxe dx

Answer: Answer:

2

2 7

7 2

1 1 1

2 2 2uu x

u x du x dx

C Cdu e ee

97. Differentiate implicitly 97. Differentiate implicitly with respect to x:with respect to x:

xx22 + xy + y + xy + y22 = 9 = 9

Answer: Answer: 2 2 0

dy dyx x y y

dx dx

THE ENDTHE ENDCongratulations!Congratulations!

•Repeat as needed until you remember your Repeat as needed until you remember your basics amd build your confidence.basics amd build your confidence.

•Look at the Look at the links on our class website to find on our class website to find more review lessons, videos and practice as more review lessons, videos and practice as needed.needed.

•Repetition and practice are keys to be Repetition and practice are keys to be successful on the AP Exam!successful on the AP Exam!