Applications of Calculus: Interactive Learning Quizzes for Community College Students

Question 1

Using the power rule, find the derivative of the function f(x) = x^3 + 2x^2 - 5x + 1:

Accepted Answer

3x^2 + 4x - 5

Answer

x^2 + 4x - 5

Answer

3x^2 - 4x - 5

Answer

x^2 - 4x - 5

Question 2

The population of a city is growing exponentially with a growth rate of 2% per year. If the current population is 100,000, what will be the population in 5 years?

Accepted Answer

110,408

Answer

112,000

Answer

102,000

Answer

115,000

Question 3

A projectile is launched vertically upward with an initial velocity of 100 feet per second. What is the maximum height reached by the projectile?

Accepted Answer

1250 feet

Answer

2000 feet

Answer

1000 feet

Answer

1500 feet

Question 4

The velocity of a moving object is given by v(t) = 2t^2 - 4t + 3 meters per second. What is the object's acceleration at time t = 2 seconds?

Accepted Answer

8 meters per second squared

Answer

12 meters per second squared

Answer

4 meters per second squared

Question 5

Which of the following is **not** an application of calculus in engineering?

Accepted Answer

Accounting

Answer

Dynamics

Answer

Statics

Answer

Thermodynamics

Question 6

The derivative of a function f(x) physically represents the:

Accepted Answer

Rate of change

Answer

Maximum value

Answer

Area under the curve

Question 7

Which of the following is a typical optimization problem solved using calculus?

Accepted Answer

Minimizing the cost of a manufacturing process

Answer

Estimating the population of a city

Answer

Predicting the weather

Answer

Analyzing the spread of a disease

Question 8

The integral of a function f(x) with respect to x physically represents the:

Accepted Answer

Area under the curve

Answer

Velocity of an object

Answer

Slope of the tangent line

Question 9

The limit of a function as x approaches infinity typically represents its:

Accepted Answer

Horizontal asymptote

Answer

Maximum value

Answer

Minimum value

Answer

Vertical asymptote

Question 10

Which of the following techniques is commonly used for integrating rational functions?

Accepted Answer

Partial fractions

Answer

Trigonometric substitution

Answer

Integration by parts

Answer

U-substitution

Question 11

The Divergence Theorem relates:

Accepted Answer

Surface integral to volume integral

Answer

Volume integral to line integral

Answer

Surface integral to line integral

Answer

Line integral to surface integral

Question 12

Which of the following is a common application of calculus in economics?

Accepted Answer

Maximizing revenue

Answer

Calculating the probability of an event

Answer

Estimating the size of a population

Answer

Predicting the weather

Question 13

The Taylor series expansion of a function about a point is typically expressed as:

Accepted Answer

An infinite sum of terms

Answer

A polynomial of finite degree

Answer

A rational function

Question 14

The rate of change of a bacteria population is modeled by the differential equation dP/dt = kP, where P represents the population size and k is a constant. What type of growth does this equation describe?

Accepted Answer

Exponential growth

Answer

Logistic growth

Answer

Linear growth

Question 15

A company aims to design a cylindrical can with a fixed volume of 1000 cm^3. What radius of the can minimizes the surface area?

Accepted Answer

5√(10/π) cm

Answer

5√(π/10) cm

Answer

10√(10/π) cm

Answer

√(10/π) cm

Question 16

The area under the curve of the function f(x) = x^2 from x = 0 to x = 2 is represented by which definite integral?

Accepted Answer

∫0^2 x^2 dx

Answer

∫2^0 x^2 dx

Answer

∫0^2 x dx

Answer

∫2^0 x dx

Question 17

A spherical balloon is being inflated at a rate of 10 cubic centimeters per second. What is the rate at which the surface area of the balloon is increasing when the radius is 5 centimeters?

Accepted Answer

20π

Answer

10π

Answer

40π

Answer

5π

Question 18

The function f(x) has the following properties: f'(x) > 0 for x < 2, f'(x) < 0 for x > 2, and f''(x) > 0 for all x. Which of the following describes the graph of f(x)?

Accepted Answer

Increasing on (-∞, 2), decreasing on (2, ∞), and concave up for all x.

Answer

Increasing on (-∞, 2), decreasing on (2, ∞), and concave down for all x.

Answer

Decreasing on (-∞, 2), increasing on (2, ∞), and concave up for all x.

Answer

Decreasing on (-∞, 2), increasing on (2, ∞), and concave down for all x.

Question 19

A spring-mass system has a mass of 2 kilograms and a spring constant of 100 newtons per meter. If the mass is pulled down 0.5 meters and released, what is the maximum height (in meters) reached by the mass above its starting point?

Accepted Answer

0.25

Answer

2

Answer

1

Answer

0.5

Question 20

A company sells a product for $20. The demand function for the product is q = 1000 - 5p, where q is the quantity demanded and p is the price. What is the revenue function (in terms of p) for the company?

Accepted Answer

R(p) = 20(1000 - 5p) = 20000 - 100p

Answer

R(p) = 1000 - 20p

Answer

R(p) = 1000 - 5p

Answer

R(p) = 20p - 1000

Question 21

A rectangular garden is to be enclosed by 100 meters of fencing. What is the maximum area (in square meters) of the garden?

Accepted Answer

625

Answer

1000

Answer

500

Answer

2500

Question 22

A company estimates that the demand for its product is given by D(p) = 100 - 2p, where p is the price. Find the price that maximizes revenue.

Accepted Answer

25

Answer

100

Answer

10

Answer

50

Question 23

A company's profit function is given by P(x) = -x^3 + 12x^2 - 36x, where x represents the number of units produced.  What is the maximum profit the company can achieve?

Accepted Answer

72

Answer

108

Answer

36

Answer

144

Question 24

The velocity of a particle is given by v(t) = 3t^2 - 6t.  What is the particle's acceleration at time t = 2?

Accepted Answer

6

Answer

0

Answer

-6

Answer

12

Question 25

A rectangular field is to be enclosed by 200 meters of fencing. What is the maximum area the field can have?

Accepted Answer

2500 square meters

Answer

1000 square meters

Answer

5000 square meters

Answer

10000 square meters

Question 26

The marginal cost of producing x units of a product is given by C'(x) = 2x + 5.  If the fixed cost is $10, what is the total cost function C(x)?

Accepted Answer

C(x) = x^2 + 5x + 10

Answer

C(x) = 2x + 15

Answer

C(x) = 2x + 5

Answer

C(x) = x^2 + 10

Question 27

A company is planning to build a cylindrical tank with a fixed volume of 100 cubic meters.  What is the radius of the tank that minimizes the surface area?

Accepted Answer

∛(50/π)

Answer

∛(10/π)

Answer

∛(25/π)

Answer

∛(100/π)

Question 28

The rate of change of a population of bacteria is given by dP/dt = 0.05P.  If the initial population is 1000, what is the population after 2 hours?

Accepted Answer

1000e

Answer

1000e^0.05

Answer

1000e^0.1

Answer

1000e^2

Question 29

Find the area bounded by the curves y = x^2 and y = 2x.

Accepted Answer

4/3

Answer

2/3

Answer

1/3

Answer

1

Question 30

A particle moves along a straight line with velocity v(t) = t^2 - 4t + 3.  Find the total distance traveled by the particle from t = 0 to t = 4.

Accepted Answer

4

Answer

8

Answer

6

Answer

2

Question 31

The volume of a solid generated by rotating the region bounded by the curve y = √x, the x-axis, and the line x = 4 about the x-axis is:

Accepted Answer

8π

Answer

16π

Answer

32π

Answer

4π

Question 32

The work done in stretching a spring from its natural length of 10 cm to 15 cm is 10 joules.  What is the spring constant k?

Accepted Answer

80 N/m

Answer

20 N/m

Answer

160 N/m

Answer

40 N/m

Question 33

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

Accepted Answer

π/80 cm/s

Answer

1/40π cm/s

Answer

1/80 cm/s

Answer

5/40π cm/s

Question 34

A farmer wants to fence a rectangular field using 1000 feet of fencing. One side of the field borders a river, so no fencing is needed there. What dimensions should the farmer use to maximize the area of the field?

Accepted Answer

250 ft x 500 ft

Answer

200 ft x 600 ft

Answer

300 ft x 400 ft

Answer

250 ft x 250 ft

Question 35

The velocity of a particle moving along a line is given by v(t) = 3t^2 - 6t + 2 meters per second. What is the total displacement of the particle from t = 0 to t = 3 seconds?

Accepted Answer

2 meters

Answer

9 meters

Answer

-2 meters

Answer

11 meters

Question 36

Find the area of the region bounded by the curves y = x^2 and y = 2x.

Accepted Answer

4/3 square units

Answer

2 square units

Answer

1/3 square units

Answer

8/3 square units

Question 37

A company's marginal cost function is given by MC(x) = 0.02x + 50 dollars per unit, where x is the number of units produced. Find the total cost of producing the first 100 units.

Accepted Answer

$6000

Answer

$5000

Answer

$5100

Answer

$1000

Question 38

The population of a city is growing at a rate proportional to its size. If the population doubles every 10 years, what is the growth constant k?

Accepted Answer

ln(2)/10

Answer

2

Answer

10

Answer

ln(10)/2

Question 39

Find the average value of the function f(x) = x^3 - 2x on the interval [0, 2].

Accepted Answer

0

Answer

1

Answer

2

Answer

-1

Question 40

The volume of a cube is increasing at a rate of 27 cubic centimeters per second. At what rate is the surface area increasing when the side length is 3 centimeters?

Accepted Answer

36 cm²/s

Answer

18 cm²/s

Answer

54 cm²/s

Answer

12 cm²/s

Question 41

A particle moves along a line with acceleration function a(t) = 6t - 4 meters per second squared. Its initial velocity is v(0) = -2 meters per second and its initial position is s(0) = 1 meter. Find the position function s(t).

Accepted Answer

s(t) = t³ - 2t² - 2t + 1

Answer

s(t) = 3t² - 4t - 2

Answer

s(t) = t³ - 2t² + 1

Question 42

Use the linear approximation of f(x) = √x at x = 4 to approximate √4.1.

Accepted Answer

2.025

Answer

2.05

Answer

2.02

Answer

2.1