2013-2014 ap calculus ab unit 4 assessmentmrsyoungsmathclasses.weebly.com/uploads/3/7/...unit... ·...

Name: ________________________ Class: ___________________ Date: __________ ID: A

1

2013-2014 AP Calculus AB Unit 4 Assessment

Multiple Choice

Identify the choice that best completes the statement or answers the question.

A calculator may NOT be used on this part of the exam. (36 minutes)

1. The slope field for a certain differential equation shown below. Which of the following could be a solution to the

differential equation with the initial condition y(0) = 1?

a) y = 1 − x2

d) y = cos x

b) y = 1 − x2

e) y =1

1 + x2

c) y = ex

2. (3x2

− 2x + 3) dx∫ =

a) x3

− x2

+ C d)1

2(3x

2− 2x + 3)

2+ C

b) x3

− x2

+ 3x + C e) none of these

c) 3x3

− x2

+ 3x + C

Name: ________________________ ID: A

2

3. Based on the function f shown in the graph below.

On which of the following intervals is f continuous?

a) −1 ≤ x ≤ 0 d) 0 < x < 1

b) 2 ≤ x ≤ 3 e) none of these

c) 1 ≤ x ≤ 2

4. If the substitution u = x + 1 is used, then dx

x x + 10

3

∫ is equivalent to

a) 2du

(u − 1)(u + 1)0

3

∫ d)2du

u2

− 11

2

∫

b) 2du

u(u − 1)0

3

∫ e) 2du

u(u2

− 1)1

2

∫

c)du

u2

− 11

2

∫

Name: ________________________ ID: A

3

5. 3x − 1

3xdx =

1

2

∫

a) −1

3ln2 d) 1

b) 1 −1

3ln2 e)

3

4

c) 1 − ln2

6. The best linear approximation for f(x) = tanx near x =π

4 is

a) 1 + 2(x −π

4) d) 1 + 2(x −

π

4)

b) 1 + (x −π

4) e) 1 +

1

2(x −

π

4)

c) 2 + 2(x −π

4)

7. limx → 0

cos x − 1

x is

a) ∞ d) −1

b) 0 e) 1

c) none of these

8. If sinxy = x, then dy

dx=

a)sec xy − y

xd) sec xy − 1

b) sec xy e) −1 + sec xy

x

c)sec xy

x

Name: ________________________ ID: A

4

9. The function f is given by f(x) =ax

2+ 12

x2

+ b. The figure below shows a portion of the graph of f. Which of the

following could be the values of the constants a and b?

a) a = −3, b = 2 d) a = 3, b = 4

b) a = 3, b = −4 e) a = 2, b = −2

c) a = 2, b = −3

10. The function f is continuous on the closed interval [0,6] and has the values given in the table below. The

trapezoidal approximation for f(x)dx0

6

∫ found with 3 subintervals of equal length is 52. What is the value of k?

x 0 2 4 6

f(x) 4 k 8 12

a) 10 d) 2

b) 6 e) 7

c) 14

11. Let f be the function given by f(x) = (2x − 1)5(x + 1). Which of the following is an equation for the line tangent to

the graph of f at the point where x = 1?

a) y = 10x + 2 d) y = 21x − 19

b) y = 10x − 8 e) y = 21x + 2

c) y = 11x − 9

12. (2t − 1)3dt =

0

1

∫a) 0 d) 6

b)1

2e) 4

c)1

4

Name: ________________________ ID: A

5

13. If f is the function given by f(x) = t2

− t dt4

2x

∫ , then f ′(2) =

a) 2 d) 0

b)7

2 12e) 2 12

c) 12

14. The equation of the curve whose slope at point (x,y) is x2− 2 and which contains the point (1,−3) is

a) y = 2x − 1 d) y =1

3x

3− 2x −

4

3

b) 3y = x3

− 10 e) y =1

3x

2−

10

3

c) y =1

3x

3− 2x

15. The area of the shaded region in the figure is equal exactly to ln3. If we approximate ln3 using LRAM

with n = 2 and RRAM with n = 2, which inequality follows?

a)1

3<

1

xdx

1

3

∫ < 2 d)1

3<

1

xdx

2

3

∫ <1

2

b)1

2<

1

xdx

1

2

∫ < 1 e)5

6<

1

xdx

1

3

∫ <3

2

c)1

2<

1

xdx

0

2

∫ < 2

Name: ________________________ ID: A

6

16. 4 − 2t dt =∫a)

1

2(4 − 2t)

2+ C d) −

1

3(4 − 2t)

3 / 2+ C

b)4

3(4 − 2t)

3 / 2+ C e)

2

3(4 − 2t)

3 / 2+ C

c) −1

6(4 − 2t)

3+ C

17. limx → ∞

2−x

2x

is

a) none of these d) −1

b) 0 e) ∞

c) 1

18. The area of the largest rectangle that can be drawn with one side along the x-axis and two vertices on the

curve of y = e−x

2

is

a)1

2ed)

2

e

b)2

e2

e) 2e

c)2

e

A graphing calculator is REQUIRED for some questions on this part of the exam. (36 minutes)

19. If f(u) = sinu and u = g(x) = x2

− 9, then (f û g)′(3) equals

a) 6 d) none of these

b) 9 e) 0

c) 1

20. The line y = 3x + k is tangent to the curve y = x3 when k is equal to

a) 0 d) 2 or −2

b) 1 or −1 e) 3 or −3

c) 4 or −4

Name: ________________________ ID: A

7

21. 1 − cos 2α0

π / 4

∫ dα =

a) 0.25 d) 0.414

b) 1.414 e) 2.000

c) 1.000

22. If f(x) is continuous on the interval a ≤ x ≤ b and a ≤ c ≤ b, then f(x) dxc

b

∫ is equal to

a) f(x) dxa

c

∫ − f(x) dxb

c

∫ d) f(x) dxa

c

∫ + f(x) dxc

b

∫

b) f(x) dxa

b

∫ − f(x) dxa

c

∫ e) f(x) dxc

a

∫ + f(x) dxb

a

∫

c) f(x) dxa

c

∫ − f(x) dxa

b

∫

23. The function f is differentiable and has values as shown in the table below. Both f and f ′ are strictly increasing on

the interval 0 ≤ x ≤ 5. Which of the following could be the value of f ′(3)?

x 2.5 2.8 3.0 3.1

f(x) 31.25 39.20 45 48.05

a) 20 d) 30

b) 27.5 e) 30.5

c) 29

24. If f(x) =

x2

− x

2x, x ≠ 0

k , x = 0

Ï

Ì

Ó

ÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔÔ

and if f is continuous at x = 0, then k =

a)1

2d) −1

b) −1

2e) 0

c) 1

Name: ________________________ ID: A

8

25. The integral 16 − x2

dx−4

4

∫ gives the area of

a) a quadrant of a circle of radius 4

b) a circle of radius 4

c) a semicircle of radius 4

d) none of these

e) an ellipse whose semi-major axis is 4

26. When 1 + x2

dx0

1

∫ is estimated using n = 5 subintervals, which is (are) true?

I. LRAM = 1 + 1 + 0.22

+ 1 + 0.42

+ 1 + 0.62

+ 1 + 0.82

Ê

Ë

ÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜

II. MRAM = 1 + 0.12

+ 1 + 0.32

+ 1 + 0.52

+ 1 + 0.72

+ 1 + 0.92

Ê

Ë

ÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜(0.2)

III. Trapezoidal =0.2

21 + 2 1 + 0.2

2+ 2 1 + 0.4

2+ 2 1 + 0.6

2+ 2 1 + 0.8

2+ 2

Ê

Ë

ÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜

a) II and III only d) III only

b) I and II only e) II only

c) I and III only

27. The figure below shows the graph of f ′, the derivative of the function f, on the open interval −7 < x < 7. If f ′ has

four zeros on −7 < x < 7, how many relative maxima does f have on −7 < x < 7?

a) Three d) Two

b) Five e) Four

c) One

Name: ________________________ ID: A

9

28. If f(x) is continuous on the interval a ≤ x ≤ b, if this interval is partitioned into n equal subintervals of

length ∆x, and if x, is a number in the kth subinterval, then limn → ∞

f(x k )∆x1

n

∑ is equal to

a) f(b) − f(a)

b) F(x) + C , where dF(x)

dx= f(x) and C is an arbitrary constant

c) f(x) dxa

b

∫d) none of these

e) F(b − a), where dF(x)

dx= f(x)

29. If f(x) dx0

3

∫ = 6 and f(x) dx3

5

∫ = 4, then 3 + 2f(x)ÊËÁÁ ˆ

¯˜̃ dx

0

5

∫ =

a) 23 d) 50

b) 20 e) 10

c) 35

30. Differentiable function f and g have the values shown in the table below.

If D =1

g, then D ′(1) =

a) −1

2d) −

1

3

b) −1

9e)

1

3

c)1

9

Name: ________________________ ID: A

10

Free Response

A graphing calculator is REQUIRED for some questions on this part of the exam. (15 minutes)

31. The function g is defined for x > 0 with g(1) = 2, g' (x) = sin x +1

x

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃

, and g ″(x) = 1 −1

x2

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃

cos x +1

x

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃.

(a) Find all values of x in the interval 0.12 ≤ x ≤ 1 at which the graph of g has a horizontal tangent line.

(b) On what subintervals of (0.12,1), if any, is the graph of g concave down? Justify your answer.

(c) Write an equation for the line tangent to the graph of g at x = 0.3.

(d) Does the line tangent to the graph of g at x = 0.3 lie above or below the graph of g for 0.3 < x < 1?

Why?

A calculator may NOT be used on this part of the exam. (30 minutes)

32. The function g is defined and differentiable on the closed interval −7,5ÈÎÍÍÍ

˘˚˙̇˙ and satisfies g(0) = 5. The graph of

y = g ′(x), the derivative of g, consists of a semicircle and three line segments, as shown in the figure below.

(a) Find g(3) and g(−2).

(b) Find the x-coordinate of each point of inflection of the graph y = g(x) on the interval −7 < x < 5. Explain

your reasoning.

(c) The function h is defined by h(x) = g(x) −1

2x

2. Find the x-coordinate of each critical point of h, where

−7 < x < 5, and classify each critical point as the location of a relative minimum, relative maximum, or

neither a minimum nor a maximum. Explain your reasoning.

Name: ________________________ ID: A

11

33. Solutions to the differential equation dy

dx= xy

3 also satisfy

d2y

dx2

= y3(1 + 3x

2y

2). Let y = f(x) be a particular

solution to the differential equation dy

dx= xy

3 with f(1) = 2.

(a) Write an equation for the line tangent to the graph of y = f(x) at x = 1.

(b) Use the tangent line equation from part (a) to approximate f(1.1). Given that f(x) > 0 for 1 < x < 1.1, is

the approximation for f(1.1) greater than or less than f(1.1)? Explain your reasoning.

(c) Find the particular solution y = f(x) with initial condition f(1) = 2.

ID: A

1

2013-2014 AP Calculus AB Unit 4 Assessment

Answer Section

MULTIPLE CHOICE

1. ANS: E DIF: DOK.2 STA: C 27.0

2. ANS: B DIF: DOK.1 STA: C 15.0

3. ANS: D DIF: DOK.2 STA: C 2.0



6. ANS: A DIF: DOK.2 STA: C 4.1












18. ANS: C DIF: DOK.2 STA: C 11.0








26. ANS: A DIF: DOK.1 STA: C13.0





ID: A

2

ESSAY

31. ANS:

2010B #2

DIF: DOK.4 STA: C 4.1 / C 4.4

ID: A

3

32. ANS:

2010 #5

DIF: DOK.4 STA: C 4.1 / C4.3 / C 6.0

ID: A

4

33. ANS:

2010 #6

DIF: DOK.4 STA: C 4.1 / C4.3 / C 6.0

2013-2014 ap calculus ab unit 4 assessmentmrsyoungsmathclasses.weebly.com/uploads/3/7/...unit... ·...

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