Master Derivative Applications: Enhance Your Calculus II Skills

Question 1

A cylindrical can with no top is to be made to hold a volume of 1000 cm³. Find the radius and height of the can that will minimize the surface area.

Accepted Answer

r = 2.89 cm, h = 17.15 cm

Answer

r = 5.00 cm, h = 10.00 cm

Answer

r = 7.14 cm, h = 7.14 cm

Answer

r = 10.00 cm, h = 5.00 cm

Question 2

Find the maximum value of the function f(x) = x³ + 3x² - 9x + 1 on the interval [-2, 2].

Accepted Answer

f(-1) = 7

Answer

f(0) = 1

Answer

f(2) = 5

Answer

The function has no maximum value on the interval [-2, 2]

Question 3

Which of the following is NOT an application of the derivative?

Accepted Answer

Integration

Answer

Optimization

Answer

Related rates

Answer

Curve sketching

Question 4

Find the equation of the tangent line to the curve y = e^x at the point (1, e).

Accepted Answer

y = e + e(x - 1)

Answer

y = e - e(x - 1)

Answer

y = e² + e(x - 1)

Answer

y = e² - e(x - 1)

Question 5

Find the derivative of the function f(x) = (x² + 1)^5 * e^x.

Accepted Answer

5e^x(x² + 1)^4(2x + e^x)

Answer

10xe^x(x² + 1)^4

Answer

5(x² + 1)^4e^x + (x² + 1)^5e^x

Answer

(x² + 1)^5e^x + 5xe^x(x² + 1)^4

Question 6

Find the derivative of the function f(x) = e^(x^2 + 2x).

Accepted Answer

f'(x) = e^(x^2 + 2x) * (2x + 2)

Answer

f'(x) = (x^2 + 2x) * e^(x^2 + 2x)

Answer

f'(x) = 2xe^(x^2 + 2x)

Answer

f'(x) cannot be determined

Question 7

Find the critical points of the function f(x) = x^3 + 3x^2 - 9x + 5.

Accepted Answer

x = -1, x = 3

Answer

x = -3, x = 1

Answer

x = 0, x = 5

Answer

The function has no critical points

Question 8

Find the interval where the function f(x) = x⁴ - 2x² + 1 is increasing.

Accepted Answer

(-1, 0) ∪ (1, ∞)

Answer

(-∞, -1) ∪ (0, 1)

Answer

(0, ∞)

Answer

(-∞, ∞)

Question 9

Find the derivative of the function f(x) = sin(x² + 1).

Accepted Answer

2x cos(x² + 1)

Answer

cos(x² + 1)

Answer

sin(x² + 1)

Answer

x² cos(x² + 1)

Question 10

Use the Mean Value Theorem to find a number 'c' in the interval (0, 1) such that f'(c) = f(1) - f(0) for the function f(x) = x³ + 2x.

Accepted Answer

1/2

Answer

0

Answer

1

Answer

3/2

Question 11

Find the concavity of the function f(x) = x⁴ - 4x².

Accepted Answer

Concave up on (-∞, -2) and (2, ∞) and concave down on (-2, 2)

Answer

Concave up on (0, ∞) and concave down on (-∞, 0)

Answer

Concave up on (-∞, 0) and (0, ∞)

Answer

Concave down on (-∞, ∞)

Question 12

Consider the function f(x) = x^3 - 3x^2 + 2. Find the critical points of f(x).

Accepted Answer

x = 0, x = 2

Answer

x = 1, x = 2

Answer

x = 0, x = -2

Answer

x = 0, x = 1

Question 13

The second derivative of a function f(x) is given by f''(x) = 6x - 4. At what value(s) of x does the graph of f(x) have an inflection point?

Accepted Answer

x = 2/3

Answer

x = 0

Answer

x = -2/3

Answer

x = 1

Question 14

Consider the function f(x) = |x - 1| + |x + 1|. Determine the x-coordinate(s) of the cusp of the graph of f(x).

Accepted Answer

x = -1, x = 1

Answer

x = 1

Answer

x = -2, x = 2

Answer

x = 0

Question 15

A company's weekly profit (in dollars) from producing *x* units of a commodity is modeled by the function  *P(x) = -0.01x³ + 0.6x² + 20x - 100*.  How many units should the company produce to maximize its weekly profit?

Accepted Answer

x = 50

Answer

x = 75

Answer

x = 25

Answer

x = 100

Question 16

The velocity of a particle moving along a straight line is given by *v(t) = 3t² - 6t + 1*.  At what times is the particle at rest?

Accepted Answer

t = 0.5 and t = 1

Answer

t = 0.25 and t = 1.5

Answer

t = 0.75 and t = 1.25

Answer

t = 0.1 and t = 2

Question 17

A function *f(x)* has the following first and second derivatives: *f'(x) = 2x - 1* and *f''(x) = 2*. How many relative extrema does *f(x)* have?

Accepted Answer

1

Answer

3

Answer

0

Answer

2

Question 18

The graph of a function *f(x)* has a local minimum at *x = 2* and a local maximum at *x = 5*. Which of the following could be the derivative of *f(x)*?

Accepted Answer

f'(x) = (x - 2)(x - 5)

Answer

f'(x) = (x - 2)²(x - 5)²

Answer

f'(x) = (x + 2)(x + 5)

Answer

f'(x) = (x - 2)(x + 5)

Question 19

The graph of a function *f(x)* is continuous and differentiable on the interval [0, 3]. If *f'(x) > 0* for all *x* in [0, 3], which of the following statements is true?

Accepted Answer

f(x) is increasing on the interval [0, 3].

Answer

f(x) has a maximum at x = 0.

Answer

f(x) has a minimum at x = 3.

Answer

f(x) is decreasing on the interval [0, 3].

Question 20

A rectangular garden has a perimeter of 100 feet. What is the maximum possible area of the garden?

Accepted Answer

1250 square feet

Answer

1000 square feet

Answer

2000 square feet

Answer

625 square feet

Question 21

A spherical balloon is being inflated at a rate of 10 cubic feet per minute. What is the rate of change of the balloon's radius when its radius is 5 feet?

Accepted Answer

2/5 feet per minute

Answer

1 feet per minute

Answer

5/2 feet per minute

Answer

1/5 feet per minute

Question 22

Find the absolute maximum and minimum values of the function *f(x) = x³ - 3x² + 2* on the interval [0, 2].

Accepted Answer

max: (2, 0); min: (1, 0)

Answer

max: (0, 2); min: (2, 0)

Answer

max: (1, 0); min: (0, 2)

Answer

max: (2, 0); min: (0, 1)

Question 23

A boat is traveling in a straight line perpendicular to the shore.  When the boat is 1 mile from shore, its distance from shore is increasing at a rate of 10 mph.  When the boat is 3 miles from shore, its distance from shore is increasing at a rate of 6 mph. What is the average rate of change of the boat's distance from shore as it travels from 1 mile to 3 miles from shore?

Accepted Answer

8 mph

Answer

12 mph

Answer

4 mph

Answer

16 mph

Question 24

The velocity of a particle moving along a straight line is given by *v(t) = t² - 4t + 3*.  What is the particle's acceleration at *t = 1*?

Accepted Answer

0

Answer

1

Answer

-1

Answer

2

Question 25

A cone has a height of 12 inches and a base radius of 5 inches. What is the rate of change of the volume of the cone with respect to its height?

Accepted Answer

50π cubic inches per inch

Answer

75π cubic inches per inch

Answer

100π cubic inches per inch

Answer

25π cubic inches per inch

Question 26

A company's revenue function is given by R(x) = 500x - 2x^2, where x is the number of units sold. What is the marginal revenue when 25 units are sold?

Accepted Answer

$450

Answer

$500

Answer

$750

Answer

$250

Question 27

The position of an object moving along a straight line is given by s(t) = 2t^3 - 3t^2 - 12t + 5. What is the velocity of the object at t = 2?

Accepted Answer

11 mph

Answer

5 mph

Answer

15 mph

Answer

21 mph

Question 28

A farmer has 100 feet of fencing to enclose a rectangular plot of land. What is the maximum area that the farmer can enclose?

Accepted Answer

1250 square feet

Answer

1000 square feet

Answer

625 square feet

Answer

2000 square feet

Question 29

The graph of a function f(x) has a vertical tangent line at x = 2. What is the derivative of f(x) at x = 2?

Accepted Answer

Undefined

Answer

1

Answer

0

Answer

-1

Question 30

A rectangular garden is to be enclosed by a fence. If the length of the garden is 20 meters longer than its width, and the total area is 1000 square meters, find the dimensions of the garden.

Accepted Answer

Length = 50 meters, Width = 30 meters

Answer

Length = 40 meters, Width = 25 meters

Answer

Length = 30 meters, Width = 20 meters

Answer

Length = 60 meters, Width = 40 meters

Question 31

Find the critical points of the function f(x) = x^3 - 3x^2 + 2.

Accepted Answer

x = 0, x = 2

Answer

x = 1, x = -2

Answer

x = -1, x = 1

Answer

x = 0, x = -2

Question 32

The velocity of a particle moving along a straight line is given by v(t) = 6t - 4t^2. Find the particle's displacement at time t = 3.

Accepted Answer

6

Answer

-6

Answer

8

Answer

12

Question 33

A conical tank with a height of 10 feet and a radius of 5 feet is being filled with water at a rate of 1 cubic foot per minute. Find the rate at which the water level is rising when the water is 6 feet deep.

Accepted Answer

1/6 feet per minute

Answer

1/4 feet per minute

Answer

1/2 feet per minute

Answer

1/8 feet per minute

Question 34

Sketch the graph of the function f(x) = x^4 - 4x^2.

Accepted Answer

Has a local maximum at x = 0, local minima at x = ±2, concavity changes at x = ±2

Answer

Has a local minimum at x = 2, local maximum at x = 0, concavity changes at x = -2

Answer

Has a local minimum at x = 0, local maxima at x = ±2, concavity changes at x = 0

Question 35

Find the absolute maximum and minimum values of the function f(x) = sin(x) + cos(x) on the interval [0, π].

Accepted Answer

Absolute maximum: √2 at x = π/4, Absolute minimum: -√2 at x = 3π/4

Answer

Absolute maximum: 2 at x = π/4, Absolute minimum: -2 at x = 3π/4

Answer

Absolute maximum: √2 at x = 0, Absolute minimum: -√2 at x = π

Question 36

Find the derivative of the function f(x) = e^(2x) * ln(x).

Accepted Answer

2e^(2x)/x + e^(2x) * 1/x

Answer

2e^(2x) * ln(x) + e^(2x)

Answer

e^(2x) * ln(x)

Answer

2e^(2x)/x

Question 37

Find the second derivative of the function f(x) = x^3 - 2x^2 + 1.

Accepted Answer

6x - 4

Answer

x^3 - 4x + 1

Answer

6x^2 - 8x

Answer

3x^2 - 4x

Question 38

Which of the following functions has an inflection point at x = 0?

Accepted Answer

f(x) = x^3 - x

Answer

f(x) = sin(x)

Answer

f(x) = e^x

Answer

f(x) = x^4

Question 39

Find the derivative of the function f(x) = ln(x^2 + 1).

Accepted Answer

2x/(x^2 + 1)

Answer

x^2/(x^2 + 1)

Answer

1/(x^2 + 1)

Question 40

A spherical balloon is being inflated at a rate of 10 cubic centimeters per second. How fast is the radius increasing when the radius is 5 centimeters?

Accepted Answer

0.127 cm/s

Answer

1.018 cm/s

Answer

0.509 cm/s

Answer

0.254 cm/s

Question 41

Find the critical points of the function f(x) = x^3 - 3x^2 + 2.

Accepted Answer

x = 0, x = 2

Answer

x = 0, x = 1

Answer

x = 1, x = 2

Answer

x = -1, x = 2

Question 42

A rectangular garden is to be enclosed by 100 feet of fencing. What are the dimensions of the garden that maximize its area?

Accepted Answer

25 feet by 25 feet

Answer

15 feet by 35 feet

Answer

10 feet by 40 feet

Answer

20 feet by 30 feet

Question 43

Find the inflection point(s) of the function f(x) = x^4 - 4x^3 + 6x^2.

Accepted Answer

x = 1

Answer

x = 0, x = 1

Answer

x = 2

Answer

x = 0, x = 2

Question 44

What is the equation of the tangent line to the curve y = x^2 + 2x at the point (1, 3)?

Accepted Answer

y = 4x - 1

Answer

y = x + 2

Answer

y = 2x + 1

Answer

y = 3x

Question 45

A particle moves along a straight line with velocity v(t) = t^2 - 4t + 3. What is the particle's acceleration at time t = 2?

Accepted Answer

0

Answer

4

Answer

-2

Answer

2

Question 46

The function f(x) = x^3 + 3x^2 - 9x + 5 has a local maximum at what value of x?

Accepted Answer

x = -3

Answer

x = 1

Answer

x = 3

Answer

x = -1

Question 47

Find the absolute maximum and minimum values of the function f(x) = x^3 - 3x on the interval [-2, 2].

Accepted Answer

Maximum: 2, Minimum: -2

Answer

Maximum: 1, Minimum: -3

Answer

Maximum: 3, Minimum: -1

Answer

Maximum: 4, Minimum: -4

Question 48

A company's profit function is given by P(x) = -x^2 + 100x - 500, where x is the number of units produced. How many units should be produced to maximize profit?

Accepted Answer

50 units

Answer

75 units

Answer

100 units

Answer

25 units

Question 49

Which of the following statements is TRUE about the function f(x) = x^4 - 2x^2?

Accepted Answer

The function has a local minimum at x = 0.

Answer

The function is decreasing for all x.

Answer

The function has an inflection point at x = 0.

Answer

The function has a local maximum at x = 0.