Free Printable Review for Spring Calculus Final 2024

1.

Find the limit of the function as x approaches 2⁺.

a)

1

b)

-1

c)

5

d)

DNE

2.

What is the limit of the function as x approaches -4 from the left?

a)

2

b)

-4

c)

DNE

d)

-2

3.

What is the limit of the function as x approaches 1 from the left?

a)

DNE

b)

1

c)

4

d)

-2

4.

Find the limit as x approaches 0-

a)

0

b)

∞

c)

-∞

d)

1

5.

a)

0

b)

-2/9

c)

Infinity

d)

-5/6

6.

Find the derivative of the given function using the definition of derivative.
f(x) = 1 - x²

a)

1 - 2x

b)

-2x

c)

-2

d)

-1

7.

Find the derivative using the definition. y = 3x²- 4x + 2

a)

6x-6

b)

6x-2

c)

4-6x

d)

6x-4

8.

a)

3/2 or 1.5

b)

3

c)

0

d)

D.N.E

9.

a)

1

b)

1.5

c)

0

d)

Does not exist

10.

Evaluate the limit:

a)

0

b)

1

c)

-9/4

d)

∞

11.

What is the limit?

a)

DNE

b)

Infinity

c)

6

d)

12

12.

Evaluate

a)

-7

b)

0

c)

1

d)

DNE

13.

a)

Does Not Exist

b)

9

c)

1

d)

0

14.

Determine the derivative:

a)

A

b)

B

c)

C

d)

D

15.

Find dy/dx for
y = 3^2x+5

a)

dy/dx = 3^(2x+4)

b)

dy/dx = (3^(2x+4)) ln 3

c)

dy/dx = 3^(2x+5) ln3

d)

dy/dx = (3^(2x+5)) (ln 3) (2)

16.

Find dy/dx for y=-(3x²+5x)⁵

a)

y=-5(x+5)⁴

b)

y=-5(6x+5)(3x²+5x)⁴

c)

y=-6x+5(3x²+5x)⁴

d)

y=-6x(3x²+5x)⁴

17.

d/dx (sin²x) = ?

a)

2sin x

b)

2(sin x)(cos x)

c)

2cos x

d)

2x(cos x)

18.

Find the derivative:
y=4e^x²

a)

y'=8xe

b)

y'=8xe^x²

c)

y'=8xe^2x

d)

y'=8xe^4x²

19.

Find the derivative:
y=11^x²

a)

y′=11^x²(2x)(ln11)

b)

y′=2x(ln11)

c)

y′=11^x²(2x)(lnx)

d)

y′=11^x²(x)(ln11)

20.

.

a)

(3x^2-2)cos(x^3-2x)

b)

-(3x^2-2)cos(x^3-2x)

c)

cos(2x^2-2)

d)

sin(2x^2-2)

21.

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

22.

The position function x(t)=t³+6t²+16t-18 is given on the interval of 0<x<9. What is the velocity at t=6?

a)

197

b)

190

c)

18

d)

196

23.

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

24.

a)

1/(cos(5x²))

b)

-(10xsin(5x²))/(cos(5x²))

c)

-1/(10xsin(5x²))

d)

cos(5x²)

25.

a)

1/(2x⁵)

b)

1/(10x⁴)

c)

2x⁵

d)

5/x

26.

a)

cosx

b)

-cosx

c)

-sinx cosx

d)

sinx cosx

27.

a)

sinx

b)

-sinx

c)

-cscx cotx

d)

sec²x

28.

a)

cscx cotx

b)

secx tanx

c)

sec²x

d)

-cscx cotx

29.

a)

secx tanx

b)

sec²x

c)

cscx cotx

d)

-cscx cotx

30.

a)

secx tanx

b)

-secx tanx

c)

tan²x

d)

cscx cotx

31.

a)

-cscx cotx

b)

cscx cotx

c)

-csc²x

d)

csc²x

32.

f(x)= e^x

f ' (x) = ?

a)

e^x

b)

xe^x

c)

x

d)

1

33.

f(x)= Ln(x)

f ' (x) = ?

a)

Log(x)

b)

-1/(xLn(x))

c)

1/(xLn(x))

d)

1/x

34.

f(x)= x³+8x

f ' (x) = ?

a)

3x²

b)

3x²+8

c)

2x³+8

d)

2x³

35.

f(x)= Ln(9x²)

f ' (x) = ?

a)

2/x

b)

18x/9

c)

x/2

d)

2

36.

f(x)= e^{x^2}

f ' (x) = ?

a)

e^2x

b)

2e^{x^2}

c)

2xe^{x^2}

d)

2xe^2x

37.

find y' for y= (2x+1)¹⁰

a)

10(2x+1)⁹

b)

20(2x-1)⁹

c)

20(2x+1)¹⁰

d)

20(2x+1)⁹

38.

a)

b)

c)

d)

39.

Derivative means the same thing as

a)

slope of the tangent line

b)

slope of the normal line

c)

exponent

d)

slope of the secant line

40.

Where is the graph continuous yet NOT differentiable?

a)

x = a, b, c, d

b)

x = b, c, d

c)

x = a, b,

d)

x = b, d

41.

Find h'(3)

a)

-2

b)

0

c)

1

d)

3

42.

a)

0

b)

1

c)

-32/3

d)

32/3

43.

a)

0

b)

5/3

c)

e^5x

d)

DNE

44.

a)

5/3

b)

0

c)

DNE

d)

1

45.

Evaluate the limit.

$\lim_{x\rightarrow2}\ \frac{x^3-8}{x^2-4}$

a)

3

b)

4

c)

6

d)

12

46.

Which of the following velocity time graphs matches the position time graph above?

a)

b)

c)

d)

47.

Which of the following velocity time graphs matches the position time graph above?

a)

b)

c)

d)

48.

Which of the following velocity time graphs matches the position time graph above?

a)

b)

c)

d)

49.

The slope of the velocity-time graph tells us the

a)

acceleration

b)

displacement

c)

velocity

d)

direction of travel

50.

The slope of the position-time graph tells us the

a)

acceleration

b)

displacement

c)

velocity

d)

direction of travel

51.

If the position of a particle is represented by x(t) = -t² + 1, what is its instantaneous velocity at t = 1?

a)

v = 0

b)

v = 1

c)

v = -1

d)

v = -2

52.

The position of an object is given as a function of time by x = 3t² + 5t³- 2t
What is the acceleration of the object at time t = 2 s?

a)

64 m/s/s

b)

60 m/s/s

c)

66 m/s/s

d)

70 m/s/s

53.

s(t) = t² - 20
Find the average velocity from t = 3 to t = 5.

a)

2

b)

4

c)

6

d)

8

54.

v (t) > 0 means

a)

the particle is moving to the right

b)

the particle is speeding up

c)

the particle has positive position

d)

the particle is at rest

55.

The acceleration function is the first derivative of...

a)

position

b)

velocity

c)

calculus

d)

particle motion

56.

What does this calculate?

a)

Average Rate of Change of f over [a,b]

b)

Average Value of f

c)

Intermediate Value Theorem

d)

L'Hopital's Rule for limits

57.

a)

position

b)

acceleration

c)

total distance

d)

speed

58.

The Mean Value Theorem applies to f(x) = 3x - x² on the interval [2, 5]. Find the value of x where the slope of the tangent line is equal to the slope of the secant line

a)

2

b)

-4

c)

3.5

d)

-2

59.

Which of these sums up the Mean Value Theorem (MVT)?

a)

$f'\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

b)

$f\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

c)

$f\left(c\right)=\frac{f'\left(b\right)-f'\left(a\right)}{b-a}$

d)

$f'\left(c\right)=\frac{f'\left(b\right)-f'\left(a\right)}{b-a}$

60.

Which of these is NOT a hypothesis of the Mean Value Theorem (MVT)?

a)

A closed interval

b)

A differentiable function

c)

A continuous function

d)

A twice-differentiable function

61.

If f and g are inverses what is g'(x)?

a)

$g'\left(x\right)=\frac{1}{f'\left(g\left(x\right)\right)}$

b)

$g'\left(x\right)=\frac{1}{f'\left(x\right)}$

c)

$g'\left(x\right)=\frac{-1}{f'\left(x\right)}$

d)

$g'\left(x\right)=\frac{-1}{f'\left(g\left(x\right)\right)}$

62.

Tangent line formula.

a)

$y-f\left(x_1\right)=f'\left(x_1\right)\left(x-x_1\right)$

b)

$y=f'\left(x_1\right)\left(x-x_1\right)$

c)

$y-f\left(y_1\right)=f'\left(x_1\right)\left(x-x_1\right)$

d)

$y=f\left(x_1\right)\left(x-x_1\right)$

63.

The fundamental theorem of calculus.

a)

$\int_a^bf\left(x\right)=F\left(b\right)-F\left(a\right)\$ where F is the antiderivative

b)

$\int_a^bf\left(x\right)=f'\left(b\right)-f'\left(a\right)\$

c)

$\int_a^bF\left(x\right)=f\left(b\right)-f\left(a\right)\$ where F is the antiderivative

d)

$\int_a^bF\left(x\right)=f'\left(b\right)-f'\left(a\right)\$ where F is the antiderivative

64.

The fundamental theorem of calculus.

a)

$\frac{d}{dx}\int_0^xf\left(t\right)dt=f\left(x\right)$

b)

$\frac{d}{dx}\int_0^xf\left(t\right)dt=F\left(x\right)$ where F is the antiderivative

c)

$\frac{d}{dx}\int_0^xF\left(t\right)dt=f\left(x\right)$ where F is the antiderivative

d)

$\frac{d}{dx}\int_0^xf\left(t\right)dt=f'\left(x\right)$

65.

Volume using discs revolving around horizontal line.

a)

$\pi\int_{x=a}^{x=b}\left(top\ -bottom\right)^2dx\ \$

b)

$\pi\int_{x=a}^{x=b}\left(top\ -bottom\right)dx\ \$

c)

$\int_{x=a}^{x=b}\left(top\ -bottom\right)^2dx\ \$

d)

$\int_{x=a}^{x=b}\left(top\ -bottom\right)dx\ \$

66.

Volume using discs revolving around vertical line.

a)

$\pi\int_{y=a}^{y=b}\left(right\ -left\right)^2dy\ \$

b)

$\pi\int_{y=a}^{y=b}\left(right\ -left\right)dy\ \$

c)

$\int_{y=a}^{y=b}\left(right\ -left\right)^2dy\ \$

d)

$\int_{y=a}^{y=b}\left(right\ -left\right)dy\ \$

67.

Volume using washers revolving around horizontal line.

a)

$\pi\int_{x=a}^{x=b}R^2-r^2\ dx\ \$

b)

$\pi\int_{x=a}^{x=b}\left(R-r\right)^{2\ }dx\ \$

c)

$\int_{x=a}^{x=b}\left(R-r\right)^2dx\ \$

d)

$\int_{x=a}^{x=b}R^{2\ }-r^{2\ }dx\ \$

68.

Volume using washers revolving around vertical line.

a)

$\pi\int_{y=a}^{y=b}R^2-r^{2\ }dy\ \$

b)

$\pi\int_{y=a}^{y=b}\left(R-r\right)^2dy\ \$

c)

$\int_{y=a}^{y=b}\left(R-r\right)^2dy\ \$

d)

$\int_{y=a}^{y=b}R^2-r^{2\ }dy\ \$

69.

What is the integrand when finding the volume of a solid with cross sections of semicircles?

a)

$\frac{\pi}{8}L^2$

b)

$\frac{\pi}{4}L^2$

c)

$\frac{\pi}{2}L^2$

d)

$\pi L^2$

70.

How do you calculate the "L" in a cross sectional volume problem when perpendicular to the x-axis?

a)

$L=right-left$

b)

$L=top-bottom$

c)

$L=right^2-left^2$

d)

$L=top^2-bottom^2$

71.

Identify the critical points of the following function:
g(x)=2x³-3x²

a)

x=-1,1

b)

x=0,0

c)

x=0,-1

d)

x=0,1

72.

What is the relative maximum value?

a)

y = 4

b)

y = 2

c)

y = 1

d)

y = -1

73.

Given a function g(x), if g'(x)=0 at a certain value of x, then g(x) has _____________ at x.

a)

an inflection point

b)

a critical point

c)

a minimum

d)

a maximum

74.

For a function g(x), g'(-2)=0 indicates that x=-2 is ________________.

a)

an inflection point

b)

a critical point

c)

a relative maximum

d)

a relative minimum

75.

What is the derivative of ax?

a)

a

b)

x

c)

1

d)

0

76.

$f\left(x\right)=\frac{2}{3}x^3-6x^2+16x+14$
Find the critical numbers of f(x). Select ALL that apply

a)

x=2

b)

x=0

c)

x= -4

d)

x= -2

e)

x = 4

77.

What is the location of the absolute maximum for the graphed function?

a)

x = 1

b)

x = 3

c)

x = 4

d)

x = 5

e)

Does Not Exist

78.

How many extrema are there in this graph?

a)

2 max, 2 min

b)

3 max, 2 min

c)

2 man, 3 min

d)

3 increasing intervals, 2 decreasing intervals

e)

2 increasing intervals, 3 decreasing intervals

79.

What is the relative max?

a)

x = 0

b)

x = -2.55

c)

(-∞, ∞)

d)

None

80.

The concavity of a function is described by its _______________.

a)

first derivative

b)

second derivative

c)

third derivative

d)

expression

81.

Find the point(s) of inflection (if it exists) of the graphed function.

a)

No point of inflection

b)

Points b and g

c)

Points b, f and g

d)

Points a, b, c, d, e, f, g, and h

e)

Points a, c, e, f and h

82.

Find the point(s) of inflection (if it exists) of the graphed function.

a)

No point of inflection

b)

Points K, R, and S.

c)

Points K and S

d)

Points P, Q, K, R, S, and T

e)

Points Q and T

83.

Discuss the concavity and the point(s) of inflection (if any) of the function:

f\left(x\right)=x^3-3x^2+10

a)

Concave up in $\left(-\infty,1\right)$ , Concave down in . And the point of inflection at x = 1

b)

Concave down in $\left(-\infty,1\right)$ , Concave up in . And the point of inflection at x = 1

c)

Concave down in $\left(-\infty,\infty\right)$ . with no point of inflection.

d)

Concave up in $\left(1,\infty\right)$ . with no point of inflection.

84.

If f '(3) = 0 and f"(3) < 0, then which of the following must be true?

a)

There is a local max at x=3

b)

There is a local min at x = 3

c)

There is an inflection point at x = 3

d)

There is an x-intercept at x = 3

85.

Find the point(s) of inflection (if it exists) of the function:

f\left(x\right)=-x^2+7x-10

Select ALL the correct answer(s).

a)

No point of inflection

b)

At x = 2

c)

At x = 5

d)

At x = $\infty$

e)

At x = 3.5

86.

Discuss the concavity and the point(s) of inflection (if any) for the function

f\left(x\right)=x^6-4x+2

a)

Concave down in $\left(0,\infty\right)$ Concave up in ,. And Point of Inflection at x = 0

b)

Concave up in $\left(-\infty,0\right)$ Concave down in ,. And Point of Inflection at x = 0

c)

Concave down in $\left(-\infty,0\right)$ with no inflection point.

d)

Concave up in $\left(0,\infty\right)$ and . with no inflection point.

e)

Concave up in $\left(-\infty,\ \infty\right)$ with no inflection point.

87.

Discuss the concavity and point(s) of inflection (if any) for the function:

f\left(x\right)=x^4-24x^2+11x+40

a)

Concave up in $\left(-\infty,-2\right)$ and , Concave down in (-2, 2). And points of inflection at x = -2 and 2

b)

Concave down in $\left(-2,\ 2\right)$ and , Concave up in (-2, 2). And points of inflection at x = -2 and 2

c)

Concave down in $\left(4,\infty\right)$ ,With no points of inflection.

d)

Concave up in $\left(-\infty,\infty\right)$ ,With no points of inflection.

88.

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)

89.

Find the right-hand Riemann Sum, with three sub-intervals indicated by the table.

a)

28

b)

16

c)

34

d)

21

90.

Use the function values in the following table and the Trapezoidal Rule with n=6 to approximate the integral from x=0 to x=12.
x: 0 2 4 6 8 10 12
f(x): 7 11 20 25 18 14 9

a)

96

b)

102

c)

190

d)

192

91.

Use the function values in the following table and left reimann sum to approximate the integral from x=0 to x=6 with n=6.
x: 0 1 2 3 4 5 6
f(x): 12 10 9 11 13 16 18

a)

71

b)

74

c)

77

d)

78

92.

Given v(t) = x^-9, find the general equation for the antiderivative.

a)

(1/8)x^-8 + C

b)

(-1/8)x^-8+ C

c)

(1/10)x¹⁰

d)

(-1/8)x^-8

93.

∫ 4 dx

a)

0

b)

4t + C

c)

4x + C

d)

2x² + C

94.

Find the antiderivative of
x²

a)

(1/3)x³+C

b)

x³

c)

(1/3)x³

d)

2x

95.

∫cosx dx =

a)

-secx + C

b)

secx + C

c)

sinx + C

d)

-sinx + C

96.

∫sinx dx =

a)

-cscx + C

b)

cscx + C

c)

cosx + C

d)

-cosx + C

97.

∫ t⁶ dt

a)

6t⁵

b)

6t⁵ + C

c)

1/7 t⁷ + C

d)

t⁷ + C

98.

∫(4 - 18x)dx

a)

F(x) = -18

b)

F(x) = 4x - 9x²

c)

F(x) = 4x - 9x²+ C

d)

F(x) = (4 - 18x)² /2 + C

99.

$\int6x^4+3x^2+5x\ +5\ dx$

a)

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

b)

$5x^5+3x^3+5x^2\ +5x$

c)

$\frac{6}{5}x^5+x^3+5x^2\ +5x\ +c$

d)

$\frac{6}{5}x^5+x^3+\frac{5}{2}x^2\ +5x\ +c$

100.

∫ - sinx dx

a)

-cos x + C

b)

cos x + C

c)

tan x + C

d)

1/√1- x²

101.

∫ 3e^x dx

a)

e^x /3

b)

e^x + C

c)

3e^x+1/(x+ 1) + C

d)

3e^x+ C

102.

∫-csc²x dx

a)

cscx + C

b)

cotx + C

c)

secx + C

d)

tanx + C

103.

∫25sec²x dx

a)

25 secx tanx + C

b)

25 tanx + C

c)

-25 tanx + C

d)

-25 sec x + C

104.

dy/dx = 2x/e^2y find the general solution

a)

y = ln (2x/2 + C)

2

b)

y = ln(2x/2)+c

c)

y = e^2x+c

d)

y = mx+b

105.

dy/dx= x √y

Find the general solution

a)

√y = ln|4x| + c

b)

√y = (x)^-1 + c

c)

2y^1/2= ln|x| + c

d)

1/2y^1/2= ln|x| + c

106.

Determine the value of "c" that satisfies the differential equation $\frac{dy}{dx}=\frac{x+1}{y+2}$ if the curve goes through the point (0, -1).

a)

5/2

b)

-3/2

c)

-1/2

d)

1

107.

Which of the following is the solution to the differential equation $\frac{dy}{dx}=\frac{x^2}{y}$ with the initial condition y(3) = -2?

a)

$y=-2e^{\left(-9+\frac{x^3}{3}\right)}$

b)

$y=\sqrt{\frac{2x^3}{3}}$

c)

$y=\sqrt{\frac{2x^3}{3}-14}$

d)

$y=-\sqrt{\frac{2x^3}{3}-14}$

108.

Solve the following differential equations:
$\frac{\text{d}y}{\text{d}x}=\frac{1}{\cos y}$

a)

$\sin y=1+C$

b)

$\sin y=x+C$

c)

$-\sin y=\frac{x^2}{2}+C$

d)

$\sin y=0+C$

109.

Solve the following differential equations:
$\frac{\text{d}y}{\text{d}x}=e^x$

a)

$1=e^x+C$

b)

$y=\frac{e^x}{2}+C$

c)

$y=e^x+C$

d)

$0=e^x+C$

110.

a)

A

b)

B

c)

C

d)

D

111.

a)

A

b)

B

c)

C

d)

D

112.

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

113.

Write the integral that can be used to find the region bounded by x = -3y² + 4 and x = y³.

a)

b)

c)

d)

e)

114.

a)

A

b)

B

c)

C

d)

D

Review for Spring Calculus Final 2024

Total questions: 114

Worksheet time: 10hrs 30mins

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