Free Printable H. PreCalc Ch 6

1.

Derivative means the same thing as

a)

slope of the tangent line

b)

slope of the normal line

c)

exponent

d)

potato

2.

The derivative of a function is its

a)

Slope

b)

Maximum/Minimum

c)

Instantaneous rate of change

d)

Common Denominator

3.

Given a function, f(x), if f'(x)>0 over a certain interval, then f(x) is __________ over that interval.

a)

increasing

b)

decreasing

c)

concave up

d)

concave down

4.

a)

15x²

b)

(5/4)x⁴

c)

indeterminate

d)

∞

5.

f'(x)=

a)

$\lim_{h\rightarrow0}\left(\frac{f\left(x+h\right)-f\left(x\right)}{h}\right)$

b)

$\lim_{x\rightarrow c}\left(\frac{f\left(x\right)-f\left(c\right)}{x-c}\right)$

c)

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

d)

an equation for the slope of the tangent line

6.

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

a)

$5x^2$

b)

$10x$

c)

10

d)

DNE

7.

$\frac{\text{d}}{\text{d}x}\left[x^3\right]$

a)

$6x$

b)

3

c)

$3x^2$

d)

6

8.

$\lim_{x\rightarrow2}\left(\frac{x^2-4}{x-2}\right)$

a)

2

b)

4

c)

6

d)

8

9.

$\lim_{x\rightarrow\infty}x^2+3x-5$

a)

$-\infty$

b)

2

c)

0

d)

$\infty$

10.

$\lim_{x\rightarrow\infty}\left(\frac{x^2+x-3}{x^4-x^3+1}\right)$

a)

$-\infty$

b)

0

c)

1

d)

$\infty$

11.

It's the average rate of change of y=3x^2+4x-2 on the interval [-2, 1].

a)

0

b)

1

c)

2

d)

3

12.

This is NOT a correct notation for a derivative

a)

y ' (say it as "y prime")

b)

f '(x) (say it as "f prime of x")

c)

dy/dx

d)

dx/dy

13.

Use the definition of the derivative (aka the limit process) to find the derivative of y=x^2-2x+3 (be ready to show work on the test).

a)

y ' = 2x - 2

b)

y ' = 2x^2

c)

y ' = 2x

d)

y ' = x^2

14.

Write the slope of the line tangent to the graph of y=x^2-2 at the point x = 8.

a)

2

b)

-4

c)

-8

d)

-16

15.

This is the equation of the line tangent to the graph of y=x^2+3 at the point x = -1.

a)

y = -2x + 2

b)

y = 2x + 2

c)

-(1/2)x + (7/2)

d)

y = (1/2)x + (7/2)

16.

What is the derivative of a constant, C?

a)

C

b)

1

c)

0

17.

What is the derivative of ax?

a)

a

b)

x

c)

1

d)

0

18.

a)

b)

c)

d)

19.

a)

b)

c)

d)

20.

a)

b)

c)

d)

21.

a)

b)

c)

d)

22.

Find the derivative of the given equation
f(x) = 7

a)

7

b)

0

c)

7x

d)

14

23.

Find the derivative of the given equation
f(x) = x³ + x² + 3

a)

3x² + 2x

b)

3x + 2x

c)

3x + 2x + 3

d)

x³ + x²

24.

a)

DNE

b)

Infinity

c)

6

d)

12

25.

a)

b)

c)

d)

26.

How do you rewrite this in power form:

f\left(t\right)=\frac{1}{t}-\frac{3}{t^2}+\frac{5}{t^3}

a)

$f\left(t\right)=t^{-1}-3t^{-2}+5t^{-3}$

b)

$f\left(t\right)=-t^{-2}+6t^{-3}-15t^{-4}$

c)

$f\left(t\right)=-t^{-2}+t^{-3}-t^{-4}$

d)

$f\left(t\right)=\ t^{-2}-6t^{-3}+15t^{-4}$

27.

Find the second derivative of the polynomial...

a)

12x + 6

b)

6x² + 6x

c)

12x + 6x

d)

2x² + 6x

28.

Find the second derivative of the polynomial...

a)

6x + 6

b)

6x + 2

c)

6

d)

6x

29.

Find the second derivative of the polynomial...

a)

18x + 4

b)

9x² + 4x

c)

9x + 6

d)

12x + 4

30.

Find the second derivative of the polynomial...

a)

4

b)

4x

c)

4x + 3

d)

8

31.

Find the second derivative of the polynomial...

a)

2 + 6x

b)

2x + 3x²

c)

2x + 3x

d)

2 + 3x

32.

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

33.

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

34.

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

35.

a)

19.79898987

b)

16.97056275

c)

13.85640646

d)

28.28427125

36.

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

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.

Differentiate:
f(x) = x⁷ (5 + 8x)³

a)

f '(x) = x² (5 + 8x)⁶ (35 + 80x)

b)

f '(x) = x⁶ (5 + 8x)² (35 + 80x)

c)

f '(x) = 8x⁷ (5 + 8x)² (35 + 80x)

d)

f '(x) = x⁶ (5 + 8x)³ (35 + 80x)

39.

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

40.

a)

b)

c)

d)

41.

a)

b)

c)

d)

42.

Find $\frac{dv}{dt}\ if\ v=t+\frac{8}{t}$

a)

$1-\frac{8}{t^2}$

b)

$1-\frac{8}{t}$

c)

$t-\frac{8}{t^2}$

d)

$1+\frac{8}{t^2}$

43.

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

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)

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

44.

$\frac{dy}{dx}\ if\ y=\left(2x^2+4\right)^3$ Find

a)

$\left(12x+4\right)\left(2x^2+4\right)^2$

b)

$3\left(2x^2+4\right)^2$

c)

$12x\left(2x^2+4\right)^2$

d)

$12\left(2x^2+4\right)^2$

45.

Differentiate f(x) = (9 - 4x²)^-1

a)

8x/(9 - 4x²)^-2

b)

8x/(9 - 4x²)²

c)

-8x/(9 - 4x²)^-2

d)

-8x(9 - 4x²)²

46.

If f(x)=(6x²-5x)⁷
what is f'(x)?

a)

7(6x²-5x)⁶

b)

7(6x²-5x)⁶⋅(12x)

c)

7(6x²-5x)⁶⋅(12x-5)

d)

(12x-5)

47.

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

48.

Find the derivative of:
f(x) = 7(3x+ 4)⁵

a)

35(3x + 4)⁵

b)

35(3x + 4)⁴

c)

105(3x + 4)⁵

d)

105(3x + 4)⁴

49.

Over what interval(s) is f(x) increasing?

a)

(-∞, -3) ∪ (1, ∞)

b)

(-3, 1)

c)

(-5, 0) ∪ (2, ∞)

d)

(-5, ∞)

50.

Over what interval(s) is f(x) decreasing?

a)

(-3, 1)

b)

(-∞, -5) ∪ (0, 2)

c)

(-∞, -3) ∪ (1, ∞)

d)

(-5, 0) ∪ (2, ∞)

51.

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

52.

The slope of a function is described by its ____________.

a)

first derivative

b)

second derivative

c)

third derivative

d)

expression

53.

The slope of a function is described by its ____________.

a)

first derivative

b)

second derivative

c)

third derivative

d)

expression

54.

Over what intervals is f(x) decreasing?

a)

(-∞, -1) ∪ (1, ∞)

b)

(-∞, -√3) ∪ (0, √3)

c)

(-1, 1)

d)

(-√3, 0) ∪ (√3, ∞)

55.

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

56.

The concavity of a function is described by its _______________.

a)

first derivative

b)

second derivative

c)

third derivative

d)

expression

57.

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

58.

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

59.

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

60.

Which of the following could be the graph of f ' , the derivative of f ?

a)

b)

c)

d)

e)

61.

What is the relative minimum x value of y = x^3 + 6x^2?

a)

-4

b)

0

c)

12

d)

4

62.

What is the maximum value of f(x) = x³ - 3x² - 1 on the interval [-3, 2]?

a)

0

b)

-1

c)

2

d)

5

63.

Find the open interval(s) where the function is increasing:

f\left(x\right)=-x^3-10x^2-32x-32

a)

$\left(-\infty,-\frac{4}{3}\right),\left(-\frac{8}{9},\infty\right)$

b)

$\left(-4,-\frac{8}{3}\right)$

c)

$\left(-\infty,-4\right),\left(-\frac{8}{3},\infty\right)$

d)

$\left(-16,-\frac{32}{3}\right)$

64.

Find the open interval(s) where the function is decreasing.

y=-\frac{x^2}{2x+2}

a)

$\left(-2,-1\right),\left(-1,0\right)$

b)

$\left(-\infty,-8\right),\left(4,\infty\right)$

c)

$\left(-\infty,-2\right),\left(0,\infty\right)$

d)

$\left(-\frac{2}{3},-\frac{1}{3}\right),\left(-\frac{1}{3},\frac{1}{3}\right)$

65.

For the function given, identify the point(s) of relative maxima.

y=-\left(x+6\right)^{\frac{2}{3}}

a)

x = -24

b)

x = -6

c)

x = -2

d)

No relative maxima

66.

Where is the relative maximum of $f\left(x\right)=x^3-3x^2-1$ on the interval [-3, 2]?

a)

$x=0$

b)

$x=-1$

c)

$x=2$

d)

$x=5$

67.

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

a)

$\left(-\infty,-4\right)$

b)

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

c)

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

d)

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

68.

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)

$x=2$

69.

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

70.

Identify the interval from the derivative graph where the function is concave up.

a)

(-1,1) & (3,4)

b)

(-3,-2)

c)

(-2,-1)

d)

(-3,-1) & (1,3)

71.

Identify the interval from the derivative graph where the function is concave down

a)

(-1,1) & (3,4)

b)

(-2,4)

c)

(-3,-1) & (1,3)

d)

(-2,1) & (1,4)

72.

Which of the following statements must be true?

I. f has a relative Min at x=-3.

II. The graph of f has a point of inflection at x=2.

III. The graph of f is concave down for 0 < x < 4.

a)

I only

b)

II only

c)

III only

d)

I and II only

e)

I and III only

73.

You want to make a box to contain dirt and your pet earthworm. Using a 7 in by 10 in rectangle of cardboard, you cut congruent squares from the corners and fold up the sides.

Choose the equation would you use in order to do Calculus to find the maximum volume of dirt (including worm) the box can hold?

a)

$V=\left(7-2x\right)\left(10-2x\right)$

b)

$V=x\left(7-2x\right)\left(10-2x\right)$

c)

$V=x\left(7-x\right)\left(10-x\right)$

d)

$V=x\left(7+2x\right)\left(10+2x\right)$

74.

What can you conclude from this number line?

a)

there is a maximum for A at 12

b)

there is a minimum for A at 12

c)

you will have a profitable year

d)

the stars are aligning nicely

75.

Farmer Jo has 32 square feet of land in which to make an enclosure for bunnies, chicks, and penguins. (see picture)

Choose the equation that represents this information.

a)

$2x+4y=32$

b)

$2x+2y=32$

c)

$xy=32$

d)

$A=3xy$

76.

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

77.

A closed rectangular shipping box with square base is to be made from 120 square inches of cardboard. What dimensions should the box be for maximum volume?

Choose the constraint and optimization equations that represent the problem.

a)

$x^2=120$ and $V=x^3$

b)

$2x+y=120$ and $V=xy$

c)

$x^2y=120$ and $V=2x^2+4xy$

d)

$2x^2+4xy=120$ and $V=x^2y$

78.

A rectangle is bounded by the x-axis and the parabola $y=12-x^2$ . What length and width should the rectangle have so that its area is a maximum?

Given the constraint equation above and the optimization equation $A=2xy$ , choose the DERIVATIVE of the merged (combined) equation.

a)

$A'=2xy$

b)

$A'=\ \ 24-6x^2$

c)

$A'=2x\left(12-x^2\right)$

d)

$A'=12-2x$

79.

A geometry student wants to draw a rectangle inscribed in a semicircle of radius 7. If one side must be on the semicircle's diameter, what is the area of the largest rectangle that the

student can draw?

a)

49

b)

42

c)

14

d)

7√7

80.

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

81.

Oil spilling from a ruptured tanker spreads in a circle on the surface of the ocean. The radius of the spill increases at a rate of 5 m/min. How fast is the area of the spill increasing when the radius is 5 m?

a)

50π m²/min

b)

47π m²/min

c)

52π m²/min

d)

40π m²/min

82.

We want to construct a box whose base length is 3 times the base width. If the box must have a volume of 50 ft³, determine the dimensions that will minimize the amount of material used.

a)

w=2.027ft, h=4.055ft, l=6.082ft

b)

w=1.488ft, h=3.347ft, l=6.694ft

c)

w=2.231ft, h=3.347ft, l=6.694ft

d)

w=0.485ft, h=2.111ft, l=4.222ft

83.

A geometry student wants to draw a rectangle inscribed in a semicircle of radius 7. If one side must be on the semicircle's diameter, what is the area of the largest rectangle that the

student can draw?

a)

49

b)

42

c)

14

d)

7√7

84.

The radius r of a sphere is increasing at a rate of 2 inches per minute. Find the rate of change of the volume when r=6 inches.

a)

C=2πr

b)

A=πr²

c)

V=(4/3)πr³

d)

S=4πr²

85.

All edges of a cube are expanding at a rate of 3 centimeters per second. How fast is the volume changing each edge is 1 centimeter.

a)

A=l*w

b)

V=3s²

c)

A=s²

d)

V=s³

86.

In a cone, find the rate of change of the volume if dr/dt is 2 inches per minute and the height is 3r when the radius is 6.

a)

S=πr√(r^2+h^2)

b)

V=πr²h

c)

V=(πr²h)/3

d)

V=(4/3)πr³

87.

A construction worker pulls a five meter plank up the side of a building under construction by means of a rope tied to one end of the plank. Assume the opposite end of the plank follows a path perpendicular to the wall of the building and the worker pulls the rope at a rate of .15 meters per second. How fast is the end of the plank sliding along the ground when when it is 2.5 meters from the wall of the building?

a)

a²+b²=c²

b)

d=√(x²+y²)

c)

h=r⋅sinθ

d)

1=sin²θ+cos²θ

88.

A boat is being pulled into dock by means of a winch 12 feet above the deck of the boat. The winch pulls in rope at a rate of 4 feet per second. Determine the speed of the boat when there is 13 feet of rope out.

a)

h=tanθ

b)

r=cos(12)

c)

ax²+bx+c=0

d)

a²+b²=c²

89.

Find dy/dx by Implicit Differentiation
x³ +y³ = 36

a)

6 -x

b)

3x² +3y²

c)

−x²/y²

d)

0

90.

Find dy/dx
xy+y²=2

a)

-y/(x+2y)

b)

y/(x+2y)

c)

-3y/x

d)

-3x/y

91.

Find dy/dx at a given point.

a)

5/4

b)

4/5

c)

1

d)

-5

92.

A water tank, shaped like an inverted circular cone, has a base radius of 6 ft and a height of 9 ft. The tank is completely full and needs to be drained. The valve is opened and the water begins to decrease at a rate of 2 ft³/sec. How fast is the height of the water changing when the water is 2 ft deep?

a)

-9/(8pi) ft/sec

b)

9/(8pi) ft/sec

c)

-8/(9pi) ft/sec

d)

8/(9pi) f/tsec

93.

Louisa and Karis were each dropped off at the same bus stop. Louisa’s bus drops her off at 3:30 whereas Karis is dropped off ten minutes later. Louisa runs home at a constant rate of 6 mph and Karis runs home at 3 mph. Louisa lives north of the bus stop and Karis lives to the east. How fast is the distance between them changing at 4:00?

a)

6.512 mph

b)

7.115 mph

c)

6.708 mph

d)

6.641 mph

94.

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

95.

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

96.

9) Use Newton's Method to approximate the real zeros of each function rounded to 4 decimal places. You decide what the initial guess will be but continue the iterations until you see the value repeating.
$f\left(x\right)=x^3-13x^2+15x-81$

a)

A 7.1333

b)

B 11.0492

c)

C 3.6911

d)

D 5.4656

97.

10) Use Newton's Method to approximate the real zeros of each function rounded to 4 decimal places. You decide what the initial guess will be but continue the iterations until you see the value repeating.
$f\left(x\right)=-x^5+3x^3-2x+3$

a)

A 1.7078

b)

B 3.0000

c)

C 2.5942

d)

D -4.8591

H. PreCalc Ch 6

Total questions: 97

Worksheet time: 5hrs 51mins

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