Calculus Quizzes: Master Limits, Derivatives, and Integrals

Question 1

Find the indefinite integral of the function f(x) = 2x + 1.

Accepted Answer

x^2 + x + C

Answer

x^2 - x + C

Answer

2x^2 + x + C

Answer

2x^2 - x + C

Question 2

Which calculus concept is utilized to determine the area under a curve?

Accepted Answer

Integration

Answer

Optimization

Answer

Related rates

Answer

Limits

Question 3

Determine the derivative of the function f(x) = sin(x).

Accepted Answer

cos(x)

Answer

sin(x)

Answer

-sin(x)

Answer

-cos(x)

Question 4

What is the formula for the chain rule?

Accepted Answer

d/dx[f(g(x))] = f'(g(x)) * g'(x)

Answer

d/dx[f(g(x))] = f'(x) * g'(x)

Answer

d/dx[f(g(x))] = f'(g(x)) / g'(x)

Answer

d/dx[f(g(x))] = f'(x) / g'(x)

Question 5

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

Accepted Answer

e^x + C

Answer

e^x - C

Answer

x * e^x + C

Answer

x * e^x - C

Question 6

Determine the derivative of the function f(x) = ln(x).

Accepted Answer

1/x

Answer

-1/x

Answer

x

Answer

-x

Question 7

Calculate the derivative of the function f(x) = x³ + 2x² - 5x + 1.

Accepted Answer

3x² + 4x - 5

Answer

x³ + 4x² - 10x + 1

Answer

3x^4 + 4x^3 - 10x^2 + 1

Question 8

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^3 + 4x^2 - 5x + 1

Answer

x^3 + 2x^2 - 5x - 1

Question 9

Which of the following represents a second-order derivative?

Accepted Answer

d^2y/dx^2

Answer

d^3y/dx^3

Answer

dy/dx

Answer

d^4y/dx^4

Question 10

Find the integral of the function f(x) = sin(x) + cos(x).

Accepted Answer

-cos(x) + sin(x) + C

Answer

-sin(x) - cos(x) + C

Answer

-cos(x) - sin(x) + C

Answer

cos(x) - sin(x) + C

Question 11

Find the area under the curve y = x^2 between x = 1 and x = 3.

Accepted Answer

8

Answer

4

Answer

16

Answer

32

Question 12

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

Accepted Answer

0, 1, 2

Answer

0, 1

Answer

0, 2

Answer

1, 2

Question 13

Find the Taylor series expansion of the function f(x) = e^x about x = 0.

Accepted Answer

1 + x + x^2/2! + x^3/3! + ...

Answer

x + x^2/2! + x^3/3! + ...

Answer

1 + 2x + 3x^2 + 4x^3 + ...

Answer

x - x^2/2! + x^3/3! + ...

Question 14

Find the limit of (x^2 - 4) / (x - 2) as x approaches 2. Simplify your answer.

Accepted Answer

4

Answer

2

Answer

0

Answer

Undefined

Question 15

Find the derivative of f(x) = x^3 + 2x^2 - 5.

Accepted Answer

3x^2 + 4x

Answer

x^3 + 4x^2

Answer

3x + 2

Answer

x^2 - 5

Question 16

Calculate the indefinite integral of ∫(2x + 1) dx.

Accepted Answer

x^2 + x + C

Answer

2x - 1 + C

Answer

x^2 - x + C

Answer

2x^2 + 1 + C

Question 17

Determine the derivative of y = e^(2x).

Accepted Answer

2e^(2x)

Answer

e^(4x)

Answer

4e^(2x)

Answer

x * e^(2x)

Question 18

Find the indefinite integral of ∫(1 / x) dx.

Accepted Answer

ln(|x|) + C

Answer

x + C

Answer

(1 / x^2) + C

Question 19

Which of the following is the second derivative of the function f(x) = e^(2x)?

Accepted Answer

4e^(2x)

Answer

2e^(2x)

Answer

e^(2x)/2

Answer

e^(4x)

Question 20

What is the area under the curve of the function f(x) = sqrt(x) from x = 0 to x = 4?

Accepted Answer

(8/3)

Answer

4

Answer

8

Answer

2

Question 21

Which of the following differential equations is separable?

Accepted Answer

y' = y(1 + x)

Answer

y'' + y = e^x

Answer

y' = y^2 + x

Answer

y' + y^2 = x

Question 22

Calculate the derivative of the polynomial function f(x) = x³ + 2x² - 5x + 1.

Accepted Answer

3x² + 4x - 5

Answer

3x + 4

Answer

x² + 2x + 1

Answer

x³ + 4x - 5

Question 23

Determine the equation of the tangent line to the exponential curve y = e^x at the point (0, 1).

Accepted Answer

y = x + 1

Answer

y = 2x + 1

Answer

y = e^x

Answer

y = 1

Question 24

Find the inverse function of the linear function f(x) = 2x - 5.

Accepted Answer

f^(-1)(x) = (x + 5)/2

Answer

f^(-1)(x) does not exist

Answer

f^(-1)(x) = 1/(2x - 5)

Answer

f^(-1)(x) = 5 - 2x

Question 25

Obtain the Maclaurin series representation of the cosine function f(x) = cos(x).

Accepted Answer

1 - x²/2! + x⁴/4! - ...

Answer

1 + x²/2! + x⁴/4! + ...

Answer

x² - x⁴/2! + x⁶/3! - ...

Answer

x - x³/3! + x⁵/5! - ...

Question 26

Find the slope of the tangent line to the curve y = x³ at the point (1, 1).

Accepted Answer

3

Answer

2

Answer

1

Answer

4

Question 27

What is the integral of the function f(x) = 2x with respect to x?

Accepted Answer

x² + C

Answer

2x² + C

Answer

x + C

Answer

2x + C

Question 28

A particle moves along a straight line with a velocity function given by v(t) = t² - 4t. What is the particle's displacement from t = 0 to t = 3?

Accepted Answer

-3

Answer

3

Answer

0

Answer

9

Question 29

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

Accepted Answer

cos(x) - sin(x)

Answer

-cos(x) - sin(x)

Answer

cos(x) + sin(x)

Answer

sin(x) + cos(x)

Question 30

Which of the following functions is an antiderivative of f(x) = 1/x?

Accepted Answer

ln|x| + C

Answer

-1/x² + C

Answer

x + C

Answer

1/x² + C

Question 31

What is the limit of the function f(x) = (x² - 4) / (x - 2) as x approaches 2?

Accepted Answer

4

Answer

2

Answer

Does not exist

Answer

0

Question 32

Find the derivative of the function f(x) = 3x² + 2x - 1.

Accepted Answer

6x + 2

Answer

6x² + 2x

Answer

3x + 2

Question 33

What is the slope of the tangent line to the curve y = x³ at the point (1, 1)?

Accepted Answer

3

Answer

6

Answer

0

Answer

1

Question 34

Find the integral of the function f(x) = 2x + 1.

Accepted Answer

x² + x + C

Answer

x² + C

Answer

2x² + x + C

Answer

2x + C

Question 35

What is the area under the curve y = x² from x = 0 to x = 2?

Accepted Answer

8/3

Answer

4

Answer

16/3

Answer

2

Question 36

The velocity of a particle is given by v(t) = 2t + 1. Find the displacement of the particle from t = 0 to t = 2.

Accepted Answer

6

Answer

2

Answer

8

Answer

4

Question 37

What is the critical point of the function f(x) = x^3 - 3x^2 + 2?

Accepted Answer

x = 0 and x = 2

Answer

x = -1

Answer

x = 3

Answer

x = 1

Question 38

Find the volume of the solid generated by revolving the region bounded by the curve y = x^2, the x-axis, and the line x = 2 about the x-axis.

Accepted Answer

32π/5

Answer

64π/5

Answer

8π/3

Answer

16π/3

Question 39

What is the equation of the tangent line to the curve y = ln(x) at the point (1, 0)?

Accepted Answer

y = x - 1

Answer

y = -x + 1

Answer

y = -x - 1

Answer

y = x + 1

Question 40

Evaluate the following definite integral: ∫(0 to 1) (x^2 + 2x) dx.

Accepted Answer

5/3

Answer

4/3

Answer

7/3

Answer

2/3

Question 41

What is the limit of the function f(x) = (x^2 - 1) / (x - 1) as x approaches 1?

Accepted Answer

2

Answer

1

Answer

0

Answer

Does not exist

Question 42

What is the derivative of f(x) = 3x^2 + 2x - 1?

Accepted Answer

6x + 2

Answer

6x^2 + 2x

Answer

9x^2 + 2

Answer

3x + 2

Question 43

Find the equation of the tangent line to the curve f(x) = x^3 at the point (1, 1).

Accepted Answer

y = 3x - 2

Answer

y = x + 1

Answer

y = 3x + 1

Answer

y = x - 2

Question 44

The velocity of an object is given by v(t) = 2t + 1. What is the displacement of the object from t = 0 to t = 2?

Accepted Answer

6

Answer

2

Answer

8

Answer

4

Question 45

Find the area under the curve f(x) = x^2 from x = 0 to x = 2.

Accepted Answer

8/3

Answer

2

Answer

4

Question 46

Which of the following is a critical point of the function f(x) = x^3 - 3x^2?

Accepted Answer

x = 0 and x = 2

Answer

x = 0 and x = 1

Answer

x = 1 and x = 3

Question 47

If the second derivative of a function is negative at a point, what can you conclude about the function at that point?

Accepted Answer

It has a local maximum.

Answer

It has a local minimum.

Answer

It is concave down.

Answer

It has an inflection point.

Question 48

What is the indefinite integral of f(x) = cos(x)?

Accepted Answer

sin(x) + C

Answer

-cos(x) + C

Answer

-sin(x) + C

Answer

cos(x) + C

Question 49

Evaluate the definite integral of f(x) = x from x = 1 to x = 3.

Accepted Answer

4

Answer

2

Answer

6

Answer

8

Question 50

The volume of a solid generated by rotating the region bounded by the curve y = x^2, the x-axis, and the line x = 2 about the x-axis is given by which integral?

Accepted Answer

∫(from 0 to 2) π(x^2)^2 dx

Answer

∫(from 0 to 2) π(x^3) dx

Answer

∫(from 0 to 2) π(x^4) dx

Answer

∫(from 0 to 2) π(x^2) dx