First Semester Review – Calculus BC Multiple Choice: No Calculator 1. What is the x – coordinate of the point of inflection on the graph of 3 21 5 24

3y x x ?

A 5 B 0

C 103

D –5 E –10

2. The graph of a piecewise-linear function f , for 1 4x , is shown below.

What is the value of 4

1

f x dx ?

A 1 B 2.5 C 4 D 5.5 E 8

3. 2

21

1 dxx

A 1

2 B 7

24 C 1

2 D 1 E 2 ln 2

1 2 3 4

1

2

–1

–2

–1

x

y

4. If f is continuous for a < x < b and differentiable for a < x < b, which of the following could be false?

A

'f b f a

f cb a

for some c such that a < x < b

B ' 0f c for some c such that a < x < b C f has a minimum value on a < x < b D f has a maximum value on a < x < b

E b

a

f x dx exists.

5. 0

sinx

t dt

A sin x B – cos x C cos x D cos x – 1 E 1 – cos x

6. If 2 10x xy , then when x = 2, dydx

A 72

B –2

C 27

D 32

E 72

7. 2

1

1e x dxx

=

A 1ee

B 2e e

C 2 1

2 2e e

D 2 2e

E 2 3

2 2e

8. Let f and g be differentiable functions with the following properties: i) g (x) > 0 for all x ii) f (0) = 1 If h (x) = f (x) g (x) and ' 'h x f x g x , then f x A 'f x B g (x) C xe D 0 E 1

9. What is the instantaneous rate of change at x = 2 of the function f given by 2 2

1xf xx

?

A –2

B 16

C 12

D 2 E 6

10. If f is a linear function and 0 < a < b, then ''b

a

f x dx

A 0 B 1

C 2

ab

D b – a

E 2 2

2b a

11. If ( ) 2

ln for 0 2ln for 2 4x x

f xx x x

< ≤= < ≤

, then ( )2

limx

f x→

=

A ln 2 B ln 8 C ln 16 D 4 E nonexistent

12. The graph of the function f shown in the figure below has a vertical tangent at the point (2, 0) and horizontal tangents at the points (1, –1) and (3, 1). For what values of x, 2 4x− < < , is f not differentiable?

A 0 only B 0 and 2 only C 1 and 3 only D 0, 1, and 3 only E 0, 1, 2, and 3

1

1

x

y

13. A particle moves along the x-axis so that its position at time t is given by 2 6 5x t t t . For what value of t is the velocity of the particle zero? A 1 B 2 C 3 D 4 E 5

14. If 3

0

1x

F x t dt , then ' 2F

A –3 B –2 C 2 D 3 E 18 15. If sin xf x e , then 'f x A cos xe B cos x xe e C cos x xe e D cosx xe e E cosx xe e

16. The graph of a twice-differentiable function f is shown in the figure below. Which of the following is true? A 1 ' 1 '' 1f f f B 1 '' 1 ' 1f f f C ' 1 1 '' 1f f f D '' 1 1 ' 1f f f E '' 1 ' 1 1f f f 17. An equation of the line tangent to the graph of cosy x x at the point (0, 1) is A y = 2x + 1 B y = x + 1 C y = x D y = x – 1 E y = 0 18. If 2'' 1 2f x x x x , then the graph of f has inflection points when x = A –1 only B 2 only C –1 and 0 only D –1 and 2 only E –1, 0, and 2 only

x

y

1

19. What are all values of k for which 2

3

0k

x dx

?

A –3 B 0 C 3 D –3 and 3 E –3, 0, and 3 20. The function f is given by 4 2 2f x x x . On which of the following intervals is f increasing?

A 1,

2

B 1 1,

2 2

C 0, D ,0

E 1,

2

21. The graph of f is shown below. Which of the following could be the graph of the derivative of f ?

a b

a b

B

a b

D

a b

E

a b

A

a b

C

22. The maximum acceleration attained on the interval 0 < t < 3 by the particle whose velocity is given by 3 23 12 4v t t t t is

A 9 B 12 C 14 D 21 E 40

23. Evaluate: 2

2lim

2

x

x

ex x

−

→− ∞

−

+

A DNE

B 0

C ∞

D –1

E 47 and a half

24. If 1tan 2f x x , then ' 1f

A 15

B 23

C undefined

D 13

E 25

First Semester Review – Calculus BC Multiple Choice: Calculator Allowed 25. The graph of a function f is shown below. Which of the following statements about f is false? A f is continuous at x = a. B f has a relative maximum at x = a. C x = a is in the domain of f. D lim

x af x

is equal to lim

x af x

E lim

x af x

exists

26. Let f be the function given by 23 xf x e and let g be the function given by 36g x x . At what value of x do the graph of f and g have parallel tangent lines? A –0.701 B –0.567 C –0.391 D –0.302 E –0.258 27. The radius of a circle is decreasing at a constant rate of 0.1 centimeter per second. In terms of the circumference C, what is the rate of change of the area of the circle in square centimeters per second? A 0.2 C B 0.1 C

C 0.1

2C

D 20.1 C E 20.1 C

x

y

a

28. The graphs of the derivatives of the functions f , g, and h are shown below. Which of the functions f, g, or h have a relative maximum on the open interval a < x < b ? A f only B g only C h only D f and g only E f , g, and h

29. The first derivative of the function f is given by 2cos 1'

5xf x

x . How many critical values does f have on the

open interval (0, 10) ? A One B Three C Four D Five E Seven 30. Let f be the function given by f x x . Which of the following statements about f are true? I. f is continuous at x = 0. II. f is differentiable at x = 0. III. f has an absolute minimum at x = 0. A I only B II only C III only D I and III only E II and III only

x

y

a b

x

y

a b

x

y

a b

31. If 0a ≠ , then 2 2

4 4limx a

x ax a→

−=

−

A 2

1a

B 2

12a

C 2

16a

D 0

E nonexistent

32. The function f is continuous on the closed interval [2, 8] and has values that are given in the table below.

x 2 5 7 8 f (x) 10 30 40 20

Using the subintervals [2, 5], [5, 7], and [7, 8], what is the trapezoidal approximation of 8

2

f x dx ?

A 110 B 130 C 160 D 190 E 210 33. Which of the following is an equation of the line tangent to the graph of 4 22f x x x at the point where ' 1f x ? A y = 8x – 5 B y = x + 7 C y = x + 0.763 D y = x – 0.122 E y = x – 2.146

34. Let F (x) be an antiderivative of 3ln x

x. If F (1) = 0, the F (9) =

A 0.048 B 0.144 C 5.827 D 23.308 E 1,640.250 35. If g is a differentiable function such that g (x) < 0 for all real numbers x and if 2' 4f x x g x , which of the following is true? A f has a relative maximum at x = –2 and a relative minimum at x = 2. B f has a relative minimum at x = –2 and a relative maximum at x = 2. C f has relative minima at x = –2 and at x = 2. D f has relative maxima at x = –2 and at x = 2. E It cannot be determined if f has any relative extrema. 36. If the base b of a triangle is increasing at a rate of 3 inches per minute while its height h is decreasing at a rate of 3 inches per minute, which of the following must be true about the area A of the triangle? A A is always increasing B A is always decreasing C A is decreasing only when b < h D A is decreasing only when b > h E A remains constant 37. Approximate ( )cos 0.01 using linearization. A 0 B 1 C 0.01 D – 0.01 E – 1

38. Let f be a function that is differentiable on the open interval (1, 10). If 2 5f , 5 5f , and 9 5f , which of the following must be true? I. f has at least 2 zeros. II. The graph of f has at least one horizontal tangent. III. For some c, 2 < c < 5, f (c) = 3. A None B I only C I and II only D I and III only E I, II, and III 39. The function f is continuous on the closed interval [0, 2] and has values that are given in the table below.

x 0 1 2 f (x) 1 k 2

The equation ( ) 1

2f x = must have at least two solutions in the interval [0, 2] if k =

A 0

B 12

C 1

D 2

E 3

Solutions: Part A: 1. D 2. B 3. C 4. B 5. E 6. A 7. E 8. E 9. D 10. A 11. E 12. B 13. C 14. D 15. E 16. D

17. B 18. C 19. A 20. C 21. A 22. D 23. A 24. E

Part B: 25. A 26. C 27. B 28. A 29. B 30. D 31. B 32. C 33. D 34. C 35. B 36. D 37. B 38. E 39. A

First Semester Review – Calculus BC Free Response: Calc Allowed 1. An isosceles triangle, whose base is the interval from (0, 0) to (c, 0), has its vertex on the graph of 212f x x . For what value of c does the triangle have maximum area? Justify your answer:

(0, 0) (c, 0) x

y

2. Given the following table of values at x = 1 and x = –2, find the indicated derivatives in parts a – l.

x f x 'f x g x 'g x 1 1 3 –2 –1 –2 –2 –5 1 7

a) 2 2

1

3x

d f x g xdx

b) 1x

d f x g xdx

c)

2x

f xddx g x

d)

2x

g xddx f x

e) 1x

d f g xdx

f) 2x

d f g xdx

g) 2x

d g f xdx

h) 2x

d g g xdx

i) 1

4 6x

d f g xdx

j) 3

1x

d g xdx

k) 1x

d f xdx

l) 2

12

x

d f xdx

3. Let f be a function defined by 2

2

2 for 1for 1

x x xf x

x kx p x

.

a) For what values of k and p will f be continuous and differentiable at x = 1? b) For the values of k and p found in part a, on what interval or intervals is f increasing? c) Using the values of k and p found in part a, find all points of inflection of the graph of f. Support your conclusion.

4. Consider the curve defined by 2 2 27x xy y . a) Write an expression for the slope of the curve at any point (x, y). b) Determine whether the lines tangent to the curve at the x – intercepts of the curve are parallel. Show the analysis that leads to your conclusion. c) Find the points on the curve where the lines tangent to the curve are vertical.

5. The figure below shows the graph of 'f , the derivative of a function f . The domain of the function is the set of all x such that 3 3x . a) For what values of x, –3 < x < 3, does f have a relative minimum? A relative maximum? Justify your answer. b) For what values of x is the graph of f concave up? Justify your answer. c) Use the information found in parts a and b and the fact that f (–3) = 0 to sketch a possible graph of f on the axes provided below.

Note: This is the graph of the derivative of f , not the graph of f.

x

y

x

y

6. Find the value of c that satisfies the Mean Value Theorem for 3 22 5 2f x x x x over [–4, 2]. 7. An object moves along the x-axis with velocity 3 22 5 2v t t t t where 0,2t . a) When is the object stopped? Justify your answer. b) When is the object moving right? Justify your answer. c) What is the acceleration at time t = 1.3 ? Show all your work. d) When is the object speeding up? Justify your answer. e) The object was at a position of +4 when t = 0. Where will it be at t = 2? Show all your work.

8. Given: the graph of f x consisting of two line segments and a semicircle

as shown, and that 1

x

g x f x dx . Domain: [0, 4]

a) Find 0g , 1g , and 4g . b) When does g x have a minimum? Justify your response. c) When does g x have a point of inflection? Justify your response. d) Sketch g x . Label the vertical axis.

1 2 3 4

1

–1

f (x)

x

1 2 3 4

g (x)

x

9. [2002 AP Calculus AB Free Response Question #5] A container has the shape of an open right circular cone, as shown below. The height of the container is 10 cm and the diameter of the opening is 10 cm. Water in the container is evaporating so that

its depth h is changing at the constant rate of 310 cm/hr.

(Note: The volume of a cone of height h and radius r is given by 213

V r h .)

a) Find the volume V of water in the container when h = 5 cm. Indicate units of measure. b) Find the rate of change of the volume of water in the container, with respect to time, when h = 5 cm.

Indicate units of measure. c) Show that the rate of change of the volume of water in the container due to evaporation is directly proportional to the exposed surface area of the water. What is the constant of proportionality?

10 cm

10 cm h

r

10. [2008 AP Calculus AB Free Response Question #2, parts a and b]

t (hours) 0 1 3 4 7 8 9 L (x) (℃) 120 156 176 126 150 80 0

Concert tickets went on sale at noon (t = 0) and were sold out within 9 hours. The number of people waiting in line to purchase tickets at time t is modeled by a twice-differentiable function of L for 0 9t≤ ≤ . Values of L(t) at various times t are shown in the table above. (a) Use the data in the table to estimate the rate at which the number of people waiting in line was changing at 5:30pm (t = 5.5). Show the computations that lead to your answer. Indicate units of measure. (b) Use a trapezoidal sum with three subintervals to estimate the average number of people waiting in line during the first 4 hours that tickets were on sale.

11. Find the linearization of 2

( ) 3x xf x e dx= + ∫ at x = 2