Free Printable Calculus Final Review #2

1.

What is the derivative of y= 1 - x² + x - 3x⁴

a)

y^/=x+-2x+1-12x³

b)

y^/=x-x³/3+x²/2-3x⁵/5

c)

y^/=-2x+1-12x³

d)

y^/=2x+1-12x³

2.

when is the function f(x)=x²+6x+9 increasing?

a)

(3,∞)

b)

(-∞,3)

c)

(-3,-∞)

d)

(-3,∞)

3.

Let f be the function defined by f(x)=4x³-10x+5. Find the equation of the tangent line to the graph of f at the point where x = 1

a)

y+1=2(x-1)

b)

y-1=2(x-1)

c)

y+1=2(x+1)

d)

y+1=-2(x-1)

4.

Find the area of the region bounded by the curves y=x² and y=√x

a)

.444

b)

.666

c)

.333

d)

.456

5.

Find the derivative. f(x) = 1/(3x²) + 4x³

a)

f'(x) = 6x³ + 12x²

b)

f'(x) = 6x + 12x²

c)

f'(x) = -2/(3x³) + 12x²

d)

f'(x) = -6/x² + 12x²

6.

Derivative means the same thing as

a)

slope of the tangent line

b)

slope of the normal line

c)

exponent

d)

potato

7.

The slope of a function is described by its ____________.

a)

first derivative

b)

second derivative

c)

third derivative

d)

expression

8.

If a function's FIRST derivative is negative at a certain point, what does that tell you?

a)

The function is increasing at that point

b)

The function is decreasing at that point

c)

The concavity of the function is up at that point

d)

The concavity of the function is down at that point

9.

If (a,b) is a local maximum, then what will be true about f'(a)?

a)

It's positive

b)

It's negative

c)

It's zero

d)

Cannot be determined

10.

Find the critical points of f(x) = 2x⁴- 4x² + 1

a)

x= 0

b)

x = -1, 1

c)

x = -1, 0, 1

d)

no critical points

11.

If the 2nd derivative of a function is negative on that interval, then that interval is concave down.

a)

True (Facts)

b)

False (Not Facts)

12.

If the first derivative of a function changes from positive to negative at a certain point, then that point is a known as a relative minimum.

a)

True (Facts)

b)

False (Not Facts)

13.

Find the derivative of f(x)=(x³-2x)²

a)

6x⁵ - 12x³+8x

b)

6x⁵ - 16x³+8x

c)

x⁶-4x⁴+4x²

d)

6x⁵ - 16x^3-8x

14.

Find the derivative of f(x) = (x⁶ + 4)⁵

a)

f '(x) = 5x⁵(x⁴ + 4)⁴

b)

f '(x) = 6x⁵(x⁶ + 4)⁴

c)

f '(x) = 30x⁵(x⁶ + 4)⁴

d)

f '(x) = 30x⁶(x⁶ + 4)⁴

15.

Find dy/dx if y = cos⁵x

a)

5cos⁴x

b)

-5sin⁴x

c)

5cos⁴xsinx

d)

-5cos⁴xsinx

16.

a)

-135

b)

-5/27

c)

-5/729

d)

-5

17.

Find an equation of the tangent line to the graph of f(x) at the point (1, 100)
f(x) = (5x⁵ + 5)²

a)

y = 500x + 400

b)

y = 100x + 400

c)

y = -500 x - 400

d)

y = 500x - 400

18.

Find the derivative.

a)

x⁴cosx - 4x³sinx

b)

x⁴cosx + 4x³sinx

c)

4x³cosx

d)

-4x³cosx

19.

Find the second derivative of the function:
f (x) = 2x - 5x⁶

a)

f ''(x)= 2 - 30x

b)

f ''(x) = 2-30x⁵

c)

f ''(x) = -30x⁵

d)

f ''(x) = -150x⁴

20.

Which of the following is the indefinite integral of $\frac{x^3}{2}+7$ ?

a)

$\frac{3x^2}{2}$

b)

$\frac{3x^2}{2}+7x+c$

c)

$\frac{x^4}{8}+7x+c$

d)

$\frac{x^4}{8}+c$

21.

$\int5x^4dx$

a)

$x^5$

b)

$\frac{5}{4}x^5+C$

c)

$x^5+C$

d)

$20x^3$ + C

22.

$\int\left(x^2-2x\right)dx$

a)

$\frac{1}{3}x^3-x^2+C$

b)

$x^{2\ }-2x+C$

c)

$2x-2$

d)

$\frac{1}{3}x^3-x^2$

23.

$\int\left(5x^2-7x+6\right)dx$

a)

$\frac{5}{3}x-\frac{7}{2}x+6x+C$

b)

$x+x+x+x+x+C$

c)

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

d)

$\frac{5}{3}x^3-\frac{7}{2}x^2+6x+C$

24.

∫(3x⁴+2)⁴(12x³)dx

a)

½(3x⁴+2)⁴+C

b)

(3x⁴+2)⁴+C

c)

⅕(3x⁴+2)⁵+C

d)

⅓(3x⁴+2)⁶+C

25.

Find the integral:

a)

1/3(x⁴ + 3)³ + c

b)

1/12(x⁴ + 3)³ + c

c)

3/4(x⁴ + 3)³ + c

d)

1/4(x⁴ + 3)³ + c

26.

What does C represent in an antiderivative?

a)

A variable

b)

A constant

c)

None of these

d)

Unknown

27.

∫_₀^²(3x² - 2x)dx ?

a)

8

b)

4

c)

-8

d)

-4

28.

a)

⅝

b)

-⅝

c)

⅜

d)

45/16

29.

Find the area under the curve y =3x²-2x from x= 1 to x =5. Do not use your calculator.

a)

100

b)

99

c)

150

d)

152

30.

Find the area of a shaded region

a)

80/3

b)

40/3

c)

26/3

d)

80/6

31.

Find the area under the curve

a)

^17/5

b)

6/5

c)

13/3

d)

1/3

32.

Write the integral that would be used to find the area bounded by y = x² + 1, y = x - 2, x = -2 and x = 2.

a)

b)

c)

d)

e)

33.

Find the area of the region bounded by the graphs of y = x² and y = 4x.

a)

32/3

b)

64/3

c)

32

d)

64

e)

32/5

34.

if this triangle is rotated about the x axis, what figure is formed?

a)

pyramid

b)

cylinder

c)

cone

d)

frustum

35.

what is the volume of the cone formed by rotating this triangle about the x axis?

a)

113.04 u³

b)

339.12 u³

c)

169.56 u³

d)

56.52 u³

36.

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

37.

Find the volume of the solid generated by revolving about the y-axis the area enclosed by $y=x^3$ , and the lines $x=0\ and\ y=8.$

a)

$\frac{96\pi}{3}$

b)

$\frac{25\pi}{4}$

c)

$\frac{54\pi}{3}$

d)

$\frac{37\pi}{4}$

38.

Which integral would be used to find the volume of the solid that results when the region enclosed by the curves is revolved about the indicated axis?

a)

A

b)

B

c)

C

d)

D

39.

Find the volume of the region that revolves around the y axis.

a)

b)

c)

d)

40.

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

41.

The concavity of a function is described by its _______________.

a)

first derivative

b)

second derivative

c)

third derivative

d)

expression

42.

For a function g(x), g''(3)=-8 indicates that g(x) is ____________ at x=3.

a)

increasing

b)

decreasing

c)

concave up

d)

concave down

43.

For a function f(x), f'(-3) = 5 indicates f(x) is ___________ at x=-3.

a)

increasing

b)

decreasing

c)

concave up

d)

concave down

44.

f' is given, which could be f?

a)

A

b)

B

c)

C

45.

List all of the critical values (x-locations of the critical points) of the function whose derivative is graphed.

a)

{-3, -1, 1}

b)

{-3, -1}

c)

{-4, -2, 0}

d)

Cannot be determined

46.

List all of the x-values of the points of inflection of the function whose second derivative is graphed.

a)

{-3, 1}

b)

{-3, -1, 1}

c)

{-4, 2}

d)

Cannot be determined

47.

What is a point of inflection?

a)

When a function goes from increasing to decreasing.

b)

When the concavity changes.

c)

When the derivative changes.

d)

When the function crosses the x-axis.

48.

Use the sign chart for f'(x). There is(are) ...

a)

a local maximum at x = -2.

b)

a local minimum at

x = -2 and a local maximum at x = 4.

c)

a local maximum at

x = -2 and a local minimum at x = 4.

d)

no extrema.

49.

If (a,b) is a local maximum, then what will be true about f''(a)?

a)

It's positive

b)

It's negative

c)

It's zero

d)

Cannot be determined

50.

Where is the point of infection for the function y = x³ + 6x²?

a)

0

b)

-4

c)

-2

d)

4

51.

On what interval(s) is the function y=x³ + 6x²

concave down?

a)

(-∞,-4)

b)

(-∞,-2)

c)

(-2,∞)

d)

(0,∞)

52.

a)

b)

c)

d)

53.

.

a)

e^(2x)

b)

(x^2-2)e^(2x)

c)

2xe^(x^2-2)

d)

(2x-2)e^(x^2-2)

54.

What is the equation of the line tangent to the curve y = x² at x = 1?

a)

y = 2x + 1

b)

y = 2x - 1

c)

y = -2x + 1

d)

y = x - 1

55.

Chris is sitting on the edge of a dock tossing rocks into the water. As each rock hits the water, small circles appear traveling outward from the point of impact. The radius of the circle is changing at a rate of 5 in/sec. How fast is the area changing when the circumference is 4 in?

a)

40 in/sec

b)

20 in/sec

c)

20pi in/sec

d)

40pi in/sec

56.

Find dy/dx by Implicit Differentiation
2xy³- x²y = 2

a)

2xy - 2y³

b)

2y(x-y²)/x(6y²-x)

c)

-2y/x

d)

2

57.

Find dy/dx by Implicit Differentiation
2xy³- x²y = 2

a)

2xy - 2y³

b)

2y(x-y²)/x(6y²-x)

c)

-2y/x

d)

2

58.

Devin set up a toy rocket. For safety, he stands 6 meters from the rocket. He sets off the rocket and it heads straight up at a constant rate of 4 m/s. How fast is the distance between the rocket and Devin changing after 2s?

a)

-2.5 m/s

b)

2.5 m/s

c)

3.2 m/s

d)

-3.2 m/s

59.

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

60.

Find dy/dx at a given point.

a)

5/4

b)

4/5

c)

1

d)

-5

61.

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

62.

The volume of a cylindrical tin can with a top and a bottom is to be 16π cubic inches. If a minimum amount of tin is to be used to construct the can, what must be the height, in inches, of the can?

a)

2 ³√2

b)

2√2

c)

4

d)

2 ³√4

Calculus Final Review #2

Total questions: 62

Worksheet time: 4hrs 3mins

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