Master Calculus Applications: Optimization, Rates, and Geometry

Question 1

Find the derivative of the function f(x) = e^(x^2 + 5x).

Accepted Answer

e^(x^2 + 5x) * (2x + 5)

Answer

e^(x^2 + 5x) * x

Answer

(x^2 + 5x) * e^(x^2 + 5x)

Answer

x * e^(x^2 + 5x)

Question 2

Evaluate the definite integral ∫(x^3 + 2x) dx from 0 to 2.

Accepted Answer

68/3

Answer

34/3

Answer

136/3

Answer

272/3

Question 3

Which of the following is not a valid method for solving optimization problems in calculus?

Accepted Answer

Using the midpoint method

Answer

Finding critical points

Answer

Using the second derivative test

Answer

Solving for the maximum or minimum value

Question 4

The antiderivative of the function f(x) = e^x is:

Accepted Answer

e^x + C

Answer

x e^x + C

Answer

e^(x^2) + C

Answer

x e^(x^2) + C

Question 5

A spherical balloon is being inflated at a constant rate of 3 cm³/s. Calculate the rate of change of its surface area when its radius is 5 cm.

Accepted Answer

5π cm²/s

Answer

3π cm²/s

Answer

30π cm²/s

Answer

50π cm²/s

Question 6

A cylindrical can is to be made to hold a volume of 1000 cubic centimeters. Find the dimensions of the can that will minimize the surface area.

Accepted Answer

Radius: 7.5 cm, Height: 13.33 cm

Answer

Radius: 5 cm, Height: 20 cm

Answer

Radius: 10 cm, Height: 10 cm

Answer

Radius: 13.33 cm, Height: 7.5 cm

Question 7

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

Accepted Answer

2 centimeters per second

Answer

1 centimeter per second

Answer

3 centimeters per second

Answer

4 centimeters per second

Question 8

To minimize the surface area of a rectangular box with a square base and a fixed volume of 64 cm³, what should be the dimensions of the box?

Accepted Answer

4 cm x 4 cm x 4 cm

Answer

2 cm x 8 cm x 4 cm

Answer

1 cm x 16 cm x 4 cm

Answer

6 cm x 3 cm x 4 cm

Question 9

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

Accepted Answer

5000 square meters

Answer

2500 square meters

Answer

10000 square meters

Answer

20000 square meters

Question 10

Which of the following integrals represents the volume of the solid generated by rotating the region bounded by the curves y = x^2 and y = 4 about the x-axis?

Accepted Answer

∫[0,2] π(4 - x^2)^2 dx

Answer

∫[0,2] 2πx(4 - x^2) dx

Answer

∫[0,2] πx^2(4 - x^2) dx

Answer

∫[0,2] 2π(4 - x^2) dx

Question 11

In optimization problems, the critical point where the first derivative changes sign is called the:

Accepted Answer

Extrema

Answer

Minimum

Answer

Maximum

Answer

Inflection point

Question 12

The formula for calculating the area under a curve between two points is:

Accepted Answer

Area = (int_{a}^{b} f(x) dx)

Answer

Area = (int_{a}^{b} |f(x)| dx)

Answer

Area = (rac{1}{2}int_{a}^{b} [f(x) - g(x)] dx)

Answer

Area = (b - a) * f(x)

Question 13

A definite integral differs from an indefinite integral by having:

Accepted Answer

Fixed limits of integration

Answer

A specific numerical value

Answer

A function as its result

Answer

No derivative

Question 14

The volume of a solid generated by rotating the region bounded by (y = f(x)), the (x)-axis, and the lines (x = a) and (x = b) about the (x)-axis is:

Accepted Answer

Volume = (piint_{a}^{b} [f(x)]^2 dx)

Answer

Volume = (int_{a}^{b} pi [f(x)]^2 dx)

Answer

Volume = (piint_{a}^{b} f(x) dx)

Answer

Volume = (int_{a}^{b} pi f(x) dx)

Question 15

The derivative of a logarithmic function is:

Accepted Answer

(rac{d}{dx} ln(x) = rac{1}{x})

Answer

(rac{d}{dx} ln(x) = x)

Answer

(rac{d}{dx} ln(x) = e^x)

Answer

(rac{d}{dx} ln(x) = ln(x))

Question 16

Implicit differentiation is a technique used to find the derivative of a function that is:

Accepted Answer

Implicitly defined

Answer

Explicitly defined

Answer

Defined by a graph

Answer

Defined by a table

Question 17

Which technique is used to find the extreme values (minimum or maximum) of a function?

Accepted Answer

Differentiation

Answer

Integration

Answer

Substitution

Answer

Logarithms

Question 18

Find the area beneath the curve of the function f(x) = x² between x = 0 and x = 2.

Accepted Answer

4/3

Answer

2/3

Answer

8/3

Answer

16/3

Question 19

Given the marginal cost function MC(x) = 2x + 10 with a fixed cost of £50, find the total cost of producing 10 units.

Accepted Answer

£200

Answer

£150

Answer

£175

Answer

£225

Question 20

Identify the correct integral to find the volume of the solid generated when the region bounded by y = x and y = 2x is rotated around the y-axis.

Accepted Answer

∫[0,2] π(2x - x²)² dx

Answer

∫[0,2] π(x - 2x²)² dx

Answer

∫[0,2] π(x² - 2x)³ dx

Answer

∫[0,2] π(2x³ - x⁴) dx

Question 21

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

Accepted Answer

2x cos(x^2)

Answer

x cos(x^2)

Answer

sin(2x)

Answer

cos(2x)

Question 22

Find the integral of the function f(x) = e^x.

Accepted Answer

e^x + C

Answer

x e^x + C

Answer

1/e^x + C

Answer

ln(e^x) + C

Question 23

Which of the following is not an area where calculus finds application?

Accepted Answer

Psychology

Answer

Engineering

Answer

Physics

Answer

Economics

Question 24

As the radius of a spherical balloon increases at a constant rate, which of the following statements describes the relationship between the surface area and the volume of the balloon?

Accepted Answer

The surface area increases at a faster rate than the volume.

Answer

The volume increases at a faster rate than the surface area.

Answer

The surface area and volume increase at the same rate.

Answer

The surface area and volume decrease at the same rate.

Question 25

The volume of a cone with a circular base of radius 'r' and height 'h' is given by which formula?

Accepted Answer

πr²h

Answer

2πr²h

Answer

πr³h

Answer

2πr³h

Question 26

Water is flowing into a spherical water tank with a radius of 2 meters at a rate of 2 cubic meters per minute. How quickly is the water level rising?

Accepted Answer

0.5 meters per minute

Answer

0.25 meters per minute

Answer

0.75 meters per minute

Answer

1 meter per minute

Question 27

A ladder 10 meters long is leaning against a wall. The bottom of the ladder is 6 meters from the wall. What is the height of the ladder on the wall?

Accepted Answer

8 meters

Answer

4 meters

Answer

6 meters

Answer

10 meters

Question 28

Find the volume of a cone with a height of 10 cm and a base radius of 5 cm.

Accepted Answer

100π cm³

Answer

50π cm³

Answer

150π cm³

Answer

200π cm³

Question 29

Find the derivative of the function f(x) = x^4 + 2x^2 - 1.

Accepted Answer

4x^3 + 4x - 0

Answer

3x^2 + 2 - 0

Answer

2x^3 + 4 - 0

Answer

x^4 + 4x^2 - 1

Question 30

The derivative of the function f(x) = x^3 + 2x^2 - 5x + 1 is:

Accepted Answer

3x^2 + 4x - 5

Answer

x^2 + 4x - 5

Answer

3x + 4

Answer

x + 4

Question 31

Find the area under the curve y = x² between x = 0 and x = 2 using the Riemann sum with n = 4.

Accepted Answer

1.667

Answer

1

Answer

2

Answer

2.667

Question 32

A company is considering investing in a new machine that costs $100,000. The machine is expected to generate revenue of $25,000 per year for the next 6 years. If the company uses a discount rate of 8%, calculate the net present value of the investment and determine if it is a wise investment.

Accepted Answer

- $7,921 (Investment is not recommended)

Answer

$100,000 (Investment is recommended)

Answer

$-100,000 (Investment is not recommended)

Answer

$12,155 (Investment is recommended)

Question 33

The velocity of a particle is given by the function v(t) = 2t^2 - 4t + 5. Calculate the total distance traveled by the particle from t = 0 to t = 3.

Accepted Answer

14.5 units

Answer

10.5 units

Answer

18.5 units

Question 34

The function f(x) = x^3 - 3x^2 + 2x + 1 has a critical point at x = 1. Determine whether this critical point corresponds to a local minimum or a local maximum.

Accepted Answer

Local minimum

Answer

Local maximum

Answer

Saddle point

Answer

Inflection point

Question 35

The function g(x) = sqrt(x^2 + 1) is not differentiable at:

Accepted Answer

x = 0

Answer

x = 1

Answer

x = 2

Answer

x = -1

Question 36

A rectangular garden is to be enclosed by a fence on three sides, with the house forming the fourth side. If 100 meters of fencing are available, what are the dimensions of the garden that maximize the enclosed area?

Accepted Answer

25 meters by 50 meters

Answer

20 meters by 60 meters

Answer

30 meters by 70 meters

Answer

10 meters by 90 meters

Question 37

A company wants to design a cylindrical can with a volume of 1000 cubic centimeters. What are the dimensions of the can (radius and height) that minimize the surface area?

Accepted Answer

Radius = 5.42 cm, Height = 10.84 cm

Answer

Radius = 7 cm, Height = 8 cm

Answer

Radius = 4 cm, Height = 20 cm

Answer

Radius = 6 cm, Height = 9 cm

Question 38

Calculate the area bounded by the curve y = x^2, the x-axis, and the lines x = 1 and x = 3.

Accepted Answer

8.67 square units

Answer

2.67 square units

Answer

9 square units

Question 39

A tank has a spherical shape with a radius of 5 meters. If it is filled to a depth of 3 meters, what is the volume of the water in the tank?

Accepted Answer

75π cubic meters

Answer

100π cubic meters

Answer

150π cubic meters

Answer

50π cubic meters

Question 40

A ladder 5 meters long is leaning against a vertical wall. The bottom of the ladder is pulled away from the wall at a rate of 0.5 meters per second. How fast is the top of the ladder moving down the wall when the bottom of the ladder is 3 meters away from the wall?

Accepted Answer

-0.33 meters per second

Answer

0.5 meters per second

Answer

-0.5 meters per second

Answer

0.33 meters per second

Question 41

A manufacturer wants to produce cylindrical cans with a volume of 250π cubic centimeters. If the radius of the can is fixed at 5 centimeters, what should be the height of the can to minimize the amount of material used?

Accepted Answer

10 centimeters

Answer

5 centimeters

Answer

15 centimeters

Answer

20 centimeters

Question 42

A spherical balloon is being inflated at a rate of 100 cubic centimeters per second. What is the rate of change of the balloon's radius when the radius is 5 centimeters?

Accepted Answer

0.1667 centimeters per second

Answer

0.5 centimeters per second

Answer

0.25 centimeters per second

Answer

0.3333 centimeters per second

Question 43

A conical tank has a radius of 2 meters and a height of 3 meters. Water flows into the tank at a rate of 5 cubic meters per minute.  How fast is the water level rising when the depth of the water is 1 meter?

Accepted Answer

0.25 meters per minute

Answer

0.3333 meters per minute

Answer

0.5 meters per minute

Answer

0.75 meters per minute

Question 44

A circular swimming pool has a diameter of 10 meters. If the water depth is increasing at a rate of 0.2 meters per hour, how fast is the volume of water in the pool increasing when the depth is 2 meters?

Accepted Answer

10π cubic meters per hour

Answer

20π cubic meters per hour

Answer

30π cubic meters per hour

Answer

40π cubic meters per hour

Question 45

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

Accepted Answer

1250 square meters

Answer

1500 square meters

Answer

2000 square meters

Answer

1000 square meters

Question 46

Water is leaking out of a conical tank at a rate of 10 cubic centimeters per second. The tank is 20 centimeters high and has a radius of 10 centimeters when full. How fast is the water level dropping when the water depth is 15 centimeters?

Accepted Answer

0.25 centimeters per second

Answer

1 centimeter per second

Answer

2 centimeters per second

Answer

0.5 centimeters per second

Question 47

A cylindrical can has a radius of 5 centimeters and a height of 10 centimeters. What is the volume of the can?

Accepted Answer

250π cubic centimeters

Answer

125π cubic centimeters

Answer

100π cubic centimeters

Answer

200π cubic centimeters

Question 48

The area under the curve y = x² + 1 from x = 0 to x = 2, as determined by integration, is:

Accepted Answer

14/3 square units

Answer

5 square units

Answer

11 square units

Answer

9 square units

Question 49

A particle moves along a straight line with a velocity of v(t) = 3t² - 2t + 1 meters per second. What is the distance traveled by the particle from t = 0 to t = 3 seconds?

Accepted Answer

20 meters

Answer

10 meters

Answer

25 meters

Answer

15 meters

Question 50

The derivative of the function f(x) = x³ + 2x² - 5 is:

Accepted Answer

f'(x) = 3x² + 4x

Answer

f'(x) = 3x² - 5

Answer

f'(x) = x³ - 5

Answer

f'(x) = x³ + 4x

Question 51

The chain rule for derivatives states that if f(x) = g(h(x)), then f'(x) is equal to:

Accepted Answer

f'(x) = g'(h(x)) * h'(x)

Answer

f'(x) = g'(x) * h'(x)

Answer

f'(x) = h'(x) / g'(x)

Answer

f'(x) = g'(x) / h'(x)

Question 52

The indefinite integral of sin(x) is:

Accepted Answer

-cos(x) + C

Answer

cos(x) + C

Answer

sin(x) + C

Question 53

The derivative of the function f(x) = e^x is:

Accepted Answer

f'(x) = e^x

Answer

f'(x) = 1/e^x

Answer

f'(x) = 0

Answer

f'(x) = x^(e-1)

Question 54

The limit of the function (x^2 - 4) / (x - 2) as x approaches 2 is:

Accepted Answer

4

Answer

0

Answer

2

Answer

Undefined