Free Printable AP Calculus AB Final Exam Review

1.

The function f is twice differentiable with f(2) = 1, f'(2)=4, and f''(2)=3. What is the value of the approximation of f(1.9) using the line tangent to the graph of f at x = 2?

a)

0.4

b)

0.6

c)

0.7

d)

1.4

2.

The radius of a circle is increasing at a constant rate of 0.2 meters per second. What is the rate of increase in the area of the circle at the instant when the circumference of the circle is 20π meters?

a)

0.04π m²/sec

b)

0.4π m²/sec

c)

4π m²/sec

d)

20π m²/sec

3.

What are all values of x for which the function f defined by f(x)=(x² - 3)e^-x is increasing? (No calculator.)

a)

x < -1 and x > 3

b)

-3 < x < 1

c)

-1 < x < 3

d)

All values of x

4.

Let f be the function given by f(x) = x³. What are all values of c that satisfy the conclusion of the Mean Value Theorem on the closed interval [-1, 2]? (No calculator)

a)

0 only

b)

1 only

c)

√3 only

d)

-1 and 1

5.

A rectangular solid that has a square base has a surface area of 150 square inches. Find the maximum volume of the solid. (use Calculus)

a)

5

b)

125

c)

150

d)

250

6.

Where is the point of inflection for the function $f\left(x\right)\ =\ x^{3\ }+6x^2$ ?

a)

$x=0$

b)

$x=-4$

c)

$x=-2$

d)

$x=4$

7.

On what interval(s) is the function
$f\left(x\right)\ =\ x^{3\ }+6x^2$
concave up?

a)

$\left(-\infty,\infty\right)$

b)

$\left(-\infty,-2\right)$

c)

$\left(-2,\infty\right)$

d)

$\left(0,\infty\right)$

8.

A square piece of green origami paper that is 6 inches on a side is being made into a gift box (with no lid) by cutting congruent squares out of each corner, folding up the sides, and taping the edges.

What size squares should you cut out for maximum volume? (do the whole problem)

a)

I should cut out squares that are 1/2 in by 1/2 in

b)

I should cut out squares that are 1 in by 1 in

c)

I should cut out squares that are 3 in by 3 in

d)

I should not cut out any squares

9.

A farmer wants to construct a rectangular pigpen using 400 ft of fencing. The pen will be built next to an existing stone wall, so only three sides of fencing need to be constructed to enclose the pen. What dimensions should the farmer use to construct the pen with the largest possible area?

a)

100ft x 200ft

b)

102ft x 196 ft

c)

50 ft x 300 ft

d)

50 ft x 175 ft

10.

Rachel is standing atop a 13 ft ladder. The ladder is leaning against a vertical wall. The ladder starts sliding away from the wall at a rate of 3 ft/sec. How fast is the ladder sliding down the wall when the tip of the ladder is 5 ft high?

a)

3 ft/sec

b)

-7.2 ft/sec

c)

7.2 ft/sec

d)

12

11.

$\int\left(6x^5+12x^3+2x\right)\ dx\ \ \ \ \ if\ f\left(1\right)=2$

a)

$30x^4+\frac{9}{4}x^2\ +c$

b)

$x^6+3x^4+x^2$

c)

$x^6+3x^4+x^2-3$

d)

$x^6+3x^4+x^2+5$

12.

∫ 2x cos(x²) dx

a)

sin(x²) + C

b)

2 sin ( 2x ) + C

c)

(1/2) sin(x²) + C

d)

4 cos(2x ) + C

13.

a)

ln(2x²+6) + C

b)

ln(2x²+6)(4x) + C

c)

1/(2x²+6) + C

d)

1/(2x²+6)²+ C

14.

Evaluate the integral $\int4\sec^24x\ \sqrt{\tan4x}\ dx$

a)

$2\left(\tan4x\right)^{\frac{3}{2}}+C$

b)

$\frac{15}{4}\cdot\left(\tan4x\right)^{\frac{4}{3}}+C$

c)

$\frac{4}{3}\cdot\left(\tan4x\right)^{\frac{3}{2}}+C$

d)

$\frac{2}{3}\cdot\left(\tan4x\right)^{\frac{3}{2}}+C$

15.

$\int\ \frac{dx}{x\ln x}$

a)

$\ln\left(\ln x\right)+C$

b)

$\left(\ln x\right)^2+C$

c)

$\frac{1}{x^2}+C$

d)

$\frac{\ln x}{x^2}+C$

16.

Evaluate the indefinite integral.
$\int3x\ e^{2x}\ dx\$

a)

$-\frac{xe^{2x}}{2}+\frac{e^{2x}}{4}+C$

b)

$\frac{3xe^{2x}}{2}-\frac{3e^{2x}}{4}+C$

c)

$xe^{-2x}+\frac{\left(1-x^2\right)^{ }}{2}+C$

d)

$-\frac{xe^{2x}}{2}+\frac{\ln e^{2x}}{4}+C$

17.

Evaluate the indefinite integral.
$\int t^2\ln t\ dt\$

a)

$\frac{2t^2\ln2t-t^2}{4}+C$

b)

$\frac{t^3\ \ln t}{3}-\frac{t^3}{9}+C$

c)

$\frac{e^t}{2t+2}+C$

d)

$\frac{-2t-1}{4e^{2t}}+C$

18.

2. Solve the differential equation

𝑑𝑦/𝑑𝑡 =3𝑡²/𝑦 with initial condition

𝑦(2) = 0.

a)

𝑦 = ln(15t)

b)

𝑦 = 16t³

c)

𝑦 = (2𝑡³− 16)^1/2

d)

𝑦 = (2𝑡³−16)

19.

Let $y=f\left(x\right)$ be the solution to the differential equation $\frac{dy}{dx}=y-10x^2$ with the initial condition $f\left(0\right)=3$ . What is the approximation for $f\left(0.4\right)$
if Euler’s method is used, starting at $x=0$ with steps of size 0.2 ?

a)

4.120

b)

4.200

c)

4.240

d)

4.768

20.

a)

B

b)

C

c)

D

d)

E

21.

What is the carrying capacity in the model $f\left(x\right)=\frac{125}{1+79.3\left(1.07\right)^{-x}}$

a)

125

b)

79.3

c)

1.07

d)

62.5

22.

The decay equation for a radioactive element is y = ae^-0.24t, where t represents years. How long will it take a sample of this element to decay to 70% of the original amount?

a)

0.85 years

b)

1.23 years

c)

1.37 years

d)

1.49 years

e)

1.65 years

23.

The spread of a virus through a student population can be modeled by the following equation, where S is the total number of students infected after t days. When is the virus spreading the fastest?

a)

about 12 days, 3000 students

b)

about 10.5 days, 2500 students

c)

0 days, 1 student

d)

about 20 days, 5000 students

24.

$\int\frac{dx}{x^2+2x-15}$

a)

$\frac{1}{8}\ln\left|x-3\right|-\frac{1}{8}\ln\left|x+5\right|+c$

b)

$\frac{1}{4}\ln\left|x-3\right|-\frac{1}{8}\ln\left|x+5\right|+c$

c)

$\frac{1}{8}\ln\left|x+5\right|-\frac{1}{8}\ln\left|x-3\right|+c$

d)

$\frac{1}{4}\ln\left|x+5\right|-\frac{1}{4}\ln\left|x-3\right|+c$

25.

a)

A

b)

B

c)

C

d)

D

26.

Find the area of the region enclosed by the lines and curves.

y = 3x + 4 and y = x² + 4

a)

9

b)

93/2

c)

9/2

d)

18

27.

a)

A

b)

B

c)

C

d)

D

28.

NO CALCULATOR: Find the volume of the solid generated by revolving the area bounded by y = x² and the x-axis from [0, 2] around the x-axis.

a)

8π/3

b)

32π/5

c)

108π/5

d)

16π/3

29.

NO CALCULATOR: Find the volume of the solid generated by revolving the area bounded by y = x² and the x-axis from [0, 2] around the y-axis.

a)

8π

b)

6π

c)

2π

d)

0

30.

Determine the volume of the region bounded by y = x² - 2x and y = x that is rotated about y = 4.

a)

5.4

b)

30.6

c)

96.133

d)

108.332

31.

The base of a solid is the region in the first quadrant enclosed by the parabola 𝑦 = 4𝑥² , the line 𝑥 = 1, and the 𝑥-axis. Each plane section of the solid perpendicular to the 𝑥-axis is a square. The volume of the solid is…

a)

4/3

b)

16/5

c)

4

d)

16

32.

Find the length of the curve described by $y=\frac{2}{3}x^{\frac{3}{2}}$ for $0\le x\le8.$

a)

$\frac{26}{3}$

b)

$\frac{52}{3}$

c)

$\frac{512\sqrt{2}}{15}$

d)

96

33.

A particle moves along an axis so that at any time t>0, its velocity is given by v(t)=4-6t². If the particle is at position p=7 at t=1, what is the position of the particle at t=2?

a)

-10

b)

-5

c)

-3

d)

3

34.

Based on the table, use a left Riemann sum and 4 sub-intervals to estimate the Area under the curve. (Choose the correct set-up.)

a)

5(3) + 1(4) + 2(5) + 1(7)

b)

5(4) + 1(5) + 2(7) + 1(6)

c)

5(3) + 6(4) + 8(5) + 9(7)

d)

0(3) + 5(4) + 6(5) + 8(7)

35.

Use a trapezoidal sum with four intervals from the table to estimate the integral of 0 to 9 of V(t).

a)

64

b)

67

c)

93

d)

151/2

36.

a)

-1

b)

-2

c)

1/2

d)

-1/2

37.

a)

-4

b)

28

c)

40

d)

20

38.

Let $A\left(x\right)=\int_0^x\sqrt{9-x^2}dx$ find $A'\left(0\right)$

a)

3

b)

0

c)

9

d)

-3

39.

Find the average value of the function $f\left(x\right)=x+3$ on the interval $\left[3,9\right]$

a)

$54$

b)

$67.5$

c)

$7.5$

d)

$9$

40.

$\int_0^{\frac{\pi}{2}}\cos\left(x\right)\ dx$

a)

-1

b)

1

c)

$\pi$

d)

$\frac{\pi}{2}$

41.

Using the areas of each region given
$\int_a^df\left(x\right)=$

a)

6

b)

20

c)

2

d)

24

42.

The velocity of a particle moving along an axis is given by v(t)=2-t² for t>0, what is the average velocity of the particle from t=1 to t=3?

a)

-4

b)

-3

c)

8/3

d)

- 7/3

43.

a)

-cos(x⁶)

b)

sin(x⁶)

c)

2x sin(x³)

d)

2x sin(x⁶)

44.

$\lim_{x\rightarrow-1+}$

a)

-1

b)

0

c)

1

d)

2

45.

$\lim_{x\rightarrow2}$

a)

-1

b)

0

c)

1

d)

DNE

46.

$\lim_{x\rightarrow0+}$

a)

-1

b)

0

c)

1

d)

2

47.

$\lim_{x\rightarrow1}\ x^3-2x^2+1$

a)

-2

b)

0

c)

1

d)

DNE

48.

$\lim_{x\rightarrow-2}\ \frac{x^2+5x+6}{x^2-4}$

a)

-1/4

b)

0

c)

20

d)

DNE

49.

$f\left(x\right)=\frac{x^2-1}{2x+4}$ Find the vertical asymptotes of the function.

a)

1, -1

b)

-2

c)

2

d)

None

50.

$f\left(x\right)=\frac{x-3}{x^2-9}$ For the given function, find the limit as x approaches -3 from the right.

a)

1, -1

b)

$-\infty$

c)

$\infty$

d)

0

51.

$\lim_{x\rightarrow\infty}\frac{2x^2+5x-1}{x^2+2x}$

a)

y = 2

b)

y = 0

c)

y = -1

d)

none

52.

Find the average rate of change of f(x) = 1 + sin(x) over the interval [0, π/2].

a)

2/π

b)

π/2

c)

1

d)

-1

53.

Write the equation of the normal line to the curve y = (x + 2)² at the point x = 0

a)

y - 4 = -2x

b)

y - 4 = 4x

c)

y - 4 = 1/2 x

d)

y - 4 = -1/4 x

54.

Write the equation of the tangent line to y = 2x² at x = -1.

a)

y + 2 = 1/4(x + 1)

b)

y - 2 = 4(x + 1)

c)

y + 2 = -2(x - 1)

d)

y - 2 = -4(x + 1)

55.

State the point of discontinuity and it's type.

a)

x = 3 infinite

b)

x = 1 removable point

c)

x = 1 jump discontinuity

d)

continuous

56.

Find the slope of the curve y = 2x² + x at x = 1.

a)

3

b)

5

c)

7

d)

-3

57.

$\lim_{x\rightarrow\infty}\ \frac{3x^2-2x^3}{2x-4x^3}$

a)

DNE

b)

1/2

c)

3/2

d)

0

58.

a)

-7

b)

0

c)

1

d)

DNE

59.

Which of the following best describes the continuity at x = 5?

a)

Continuous

b)

Removable Point Discontinuity

c)

Non-removable Infinite Discontinuity

d)

Non-removable Jump Discontinuity

60.

Find the derivative of y = x⁴⋅cos x

a)

y' = x⁴cosx - 4x³sinx

b)

y' = 4x³cosx - x⁴sinx

c)

y' = 4x³cosx + x⁴sinx

d)

y' = -4x³cosx

61.

$f\left(x\right)=\sin x$ Find the slope of the tangent line to f(x) at $x=\frac{\pi}{6}$

a)

$\frac{1}{2}$

b)

$\frac{\sqrt{3}}{2}$

c)

$\frac{-\sqrt{3}}{2}$

d)

$\frac{-1}{2}$

62.

Find the derivative f(x) = tanxcosx

a)

f'(x) = sec²xcosx - tanxsinx

b)

f'(x) = sec²xcosx + tanxsinx

c)

f'(x) = sec²xsinx

d)

f'(x) = sec²xcosx - tanxcosx

63.

Find the derivative of f(x) = -3x⁴cscx

a)

f'(x) = 12x³cscxcotx

b)

f'(x) = -3x⁴cscxcotx

c)

f'(x) =-12x³cscx + 3x⁴cotxcscx

d)

f'(x) = -3xcosx

64.

y = (sinx)/x

find y’

a)

y' = (cosx)/x

b)

y' = (xcosx - sinx)/x²

c)

y' = (cosx - sinx)/x

d)

y' = (sinx - xcosx)/x²

65.

Which of the following is the derivative of the function h(x)= (8x⁶ + 2x + 5)^{4 ?}

a)

h^'(x)= 4(8x⁶ + 2x + 5)³(48x⁵ + 2)

b)

h^'(x)= 4(48x⁵ + 2)³

c)

h^'(x)= 4(48x⁵ + 2 + 5)³

d)

h^'(x)= 3(8x⁶ + 2x + 5)⁴(48x⁵ + 2)²

66.

Find $\frac{dp}{dq}\ if\ p=\frac{1}{\sqrt{q+9}}$

HINT: rewrite the square root as an exponent in the numerator

a)

$-\frac{1}{2\left(q+9\right)^{\frac{3}{2}}}$

b)

$\frac{1}{2\left(q+9\right)^{\frac{3}{2}}}$

c)

$-\frac{1}{\left(q+9\right)^{\frac{3}{2}}}$

d)

$-\frac{1}{\sqrt{q+9}}$

67.

Find the derivative: $\sin^2\left(3x+2\right)$

a)

$6\sin\left(3x+2\right)\cos\left(3x+2\right)$

b)

$-6\sin\left(3x+2\right)\cos\left(3x+2\right)$

c)

$6\cos\left(3x+2\right)$

d)

$-6\cos\left(3x+2\right)$

68.

Find dy/dx

a)

A

b)

B

c)

C

d)

D

69.

Find the derivative.
f(x) = -8x^-3 + 5x - e^x

a)

f'(x) = -24x^-4 + 5 - e^x

b)

f'(x) = 24x^-4 + 5 - e^x

c)

f'(x) = 24x^-2 + 5 - e^x

d)

f'(x) = 24x^-4 + 5 - e^x-1

70.

Find the second derivative of
f(x) = x²+ e^x - cosx

a)

f"(x) = 2 + e^x + cosx

b)

f"(x) = 2x + e^x + cosx

c)

f"(x) = 2x + xe^x - cosx

d)

f"(x) = 2x + e^x + sinx

71.

Find the derivative f(x) = x²/e^x

a)

f'(x) = (x²e^x - 2xe^x) / (e^x)²

b)

f'(x) = (2xe^x + x²e^x) / (e^x)²

c)

f'(x) = (2xe^x - x²e^x) / (e^x)²

d)

f'(x) = x²e^x + 2xe^x

72.

a)

1/(cos(5x²))

b)

-10x tan(5x²)

c)

-1/(10xsin(5x²))

d)

cos(5x²)

73.

Find the derivative of $g(x)=\frac{3x-2}{x^2+2}$

a)

$9x^2+2$

b)

$\frac{3\left(x^2+2\right)}{\left(x^2+2\right)^2}$

c)

$\frac{-3x^2+4x+6}{\left(x^2+2\right)^2}$

d)

$\frac{-3x^2+10}{\left(x^2+2\right)^2}$

74.

Find the derivative.

HINT: rewrite the cube root as an exponent of 1/3

a)

b)

c)

d)

75.

Find the derivative:
y=5x²e^3x

a)

y'=10xe^3x(2x+3)

b)

y'=5xe^3x(3x+2)

c)

y'=10ex^3x(3x+2)

d)

y'=5xe^3x(2x+3)

76.

$f\left(x\right)=x-x^3$ . What is the instantaneous rate of change of $f\left(x\right)$ at $x=-1$

a)

-2

b)

4

c)

-4

d)

2

77.

What is the average rate of change of
f(x)= x²+4x+4 from x=1 to x=-3

a)

2

b)

-4

c)

25/4

d)

4

78.

$\lim_{h\rightarrow0}\left(\frac{5\left(x+h\right)^2-5x^2}{h}\right)$

a)

$5x^2$

b)

$10x$

c)

10

d)

DNE

79.

a)

b)

c)

d)

80.

Find $\frac{d^2y^{ }}{dx^2}$ by Implicit Differentiation
$x^3+y^3=36$

a)

$\frac{-x^2}{y^2}$

b)

$\frac{-2xy^2+2xy}{y^2}$

c)

$\frac{-2xy^3-2x^4}{y^5}$

d)

$\frac{-2xy^2-2x^4}{y^4}$

81.

Find dy/dx
xy+y²=2

a)

-y/(x+2y)

b)

y/(x+2y)

c)

-3y/x

d)

-3x/y

82.

Find the derivative:

y=log₄(6x+5))

a)

(ln4)(1/(6x+5))

b)

(6ln4)/(6x+5)

c)

6/((ln4)(6x+5))

d)

6/(6x+5)

83.

Find the derivative:

y=(2^3x+4)

a)

3(ln2)(2^3x+4)

b)

3(2^3x+4)

c)

3(2^3x+4/(ln2))

d)

2^3x+4

84.

Find dy/dx at a given point.

a)

5/4

b)

4/5

c)

1

d)

-5

85.

The first derivative of $y=4e^{\sin5x}$

a)

$\frac{\text{d}y}{\text{d}x}=20\left(e^{\sin5x}\right)\left(\cos5x\right)$

b)

$\frac{\text{d}y}{\text{d}x}=20e^{\cos5x}$

c)

$\frac{\text{d}y}{\text{d}x}=4e^{\sin5x}$

d)

$\frac{\text{d}y}{\text{d}x}=20\left(e^{\sin5x}\right)\left(\sin5x\right)$

86.

Find $\frac{\text{}d^2y}{dx^2}$

a)

A

b)

B

c)

C

d)

D

87.

Find $\frac{\text{d}y}{\text{d}x}$

a)

A

b)

B

c)

C

d)

D

88.

Find $\frac{\text{d}y}{\text{d}x}$

a)

A

b)

B

c)

C

d)

D

89.

a)

A

b)

B

c)

C

d)

D

90.

Let $g\left(x\right)=x^5+3x$ and let h be the inverse function of g. Notice that $g\left(1\right)=4$ . Find $h'\left(4\right)$

a)

-8

b)

-1/8

c)

1/8

d)

8

AP Calculus AB Final Exam Review

Total questions: 90

Worksheet time: 5hrs 5mins

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