Pre-Calculus Learning Quizzes: Master Calculus Prerequisites

Question 1

Calculate the derivative of f(x) = e^(x^2) by applying the chain rule.

Accepted Answer

2xe^(x^2)

Answer

x^2e^(x^2)

Answer

1/2xe^(x^2)

Question 2

Classify the function f(x) = x^4 - 2x^2 + 1. Is it even, odd, or neither?

Accepted Answer

Even

Answer

Odd

Answer

Neither

Question 3

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

Accepted Answer

0

Answer

2

Answer

4

Answer

The limit does not exist

Question 4

What is the graph of the function f(x) = |x + 2|?

Accepted Answer

A V-shape with vertex at (-2, 0)

Answer

A U-shape with vertex at (-2, -4)

Answer

A parabola opening up

Answer

A parabola opening down

Question 5

Determine the convergence or divergence of the following series: ∑[n=1, ∞] (n / (n^2 + 1)).

Accepted Answer

Converges

Answer

Diverges

Answer

Cannot be determined

Question 6

Which of the following is the equation of the line with a slope of -2 and passes through the point (1, 3)?

Accepted Answer

y = -2x + 3

Answer

y = -2x + 1

Answer

y = 2x - 1

Answer

y = 2x + 3

Question 7

Find the domain of the function f(x) = 1 / (x - 3), where x is a real number.

Accepted Answer

{x | x ≠ 3}

Answer

All real numbers

Answer

{x | x > 3}

Answer

{x | x < 3}

Question 8

Determine the range of the function f(x) = x^2 + 2, where x is an integer.

Accepted Answer

{y | y ≥ 2}

Answer

{y | y > 2}

Answer

{y | y ≤ 2}

Answer

{y | y < 2}

Question 9

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

Accepted Answer

2cos(2x)

Answer

cos(2x)

Answer

sin(4x)

Answer

2sin(2x)

Question 10

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

Accepted Answer

e^x + C

Answer

x * e^x + C

Answer

e^(x+1) + C

Answer

x * e^(x+1) + C

Question 11

Calculate the derivative of f(x) = x^3 + 2x^2 - 5x + 1.

Accepted Answer

3x^2 + 4x - 5

Answer

x^3 + 4x - 5

Answer

3x^2 + 2x - 5

Answer

x^3 + 2x + 1

Question 12

Determine whether the function f(x) = |x| is differentiable at x = 0.

Accepted Answer

No, because the function has a corner at x = 0.

Answer

Yes, because the function is differentiable at all points.

Answer

Yes, because the left and right derivatives exist and are equal.

Answer

No, because the left and right derivatives do not exist.

Question 13

Find the limit of the expression (sin(x) - x) / x^3 as x approaches 0.

Accepted Answer

1/6

Answer

0

Answer

-1/6

Answer

The limit does not exist.

Question 14

Establish whether the function f(x) = x^4 - 4x^3 + 3x^2 is continuous at x = 0.

Accepted Answer

Continuous

Answer

Discontinuous

Answer

Cannot be determined

Question 15

Solve the differential equation dy/dx = cos(x) using the Fundamental Theorem of Calculus.

Accepted Answer

y = sin(x) + C

Answer

y = cos(x) + C

Answer

y = x + C

Answer

y = C

Question 16

Determine the derivative of f(x) = x^3 - 2x^2 + 5x - 1.

Accepted Answer

3x^2 - 4x + 5

Answer

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

Answer

x^3 - 4x + 5

Answer

3x - 4 + 5

Question 17

Calculate the area under the curve of the function f(x) = x^2 between x = 0 and x = 2.

Accepted Answer

4/3

Answer

2/3

Answer

8/3

Answer

16/3

Question 18

Find the derivative of f(x) = sin(x)cos(x) using the product rule.

Accepted Answer

cos(2x)

Answer

cos(x)sin(x)

Answer

sin(2x)

Answer

-cos(x)sin(x)

Question 19

Which of the following integrals represents the area under the curve y = x^2 from x = 0 to x = 2?

Accepted Answer

∫(x^2)dx from 0 to 2

Answer

∫(2x)dx from 0 to 2

Answer

∫(x)dx from 0 to 4

Answer

∫(x^2)dx from 2 to 0

Question 20

Find the integral of ∫(sqrt(x))dx.

Accepted Answer

(2/3)x^(3/2) + C

Answer

x^(1/2) + C

Answer

2x^(1/2) + C

Answer

(3/2)x^(1/2) + C

Question 21

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

Accepted Answer

π∫(x - x^2)^2 dx from 0 to 1

Answer

∫(x - x^2)^2 dx from 0 to 1

Answer

∫(x^2 - x)^2 dx from 0 to 1

Answer

π∫(x^2 - x)^2 dx from 0 to 1

Question 22

Calculate the derivative of the function g(x) = x^3 - 2x^2 + 5x - 1.

Accepted Answer

3x^2 - 4x + 5

Answer

x^3 - 4x + 5

Answer

3x^2 + 4x - 5

Answer

x^3 + 4x + 5

Question 23

Which of the following is the equation of the tangent line to the graph of f(x) = x^2 - 3x + 2 at the point (1, 0)?

Accepted Answer

y = 2x - 1

Answer

y = -x - 1

Answer

y = x - 1

Answer

y = -2x + 1

Question 24

Evaluate the limit of the sequence a_n = (n^2 - 1) / (n^2 + 1) as n approaches infinity. (Simplify before evaluating.)

Accepted Answer

1

Answer

0

Answer

∞

Answer

-1

Question 25

Classify the function f(x) = x^2 * e^(-x) as even, odd, or neither.

Accepted Answer

Neither

Answer

Even

Answer

Odd

Question 26

Evaluate the following limit: lim_(x->0) (x^2 - 1) / (x - 1)

Accepted Answer

1

Answer

0

Answer

2

Answer

Undefined

Question 27

Find the antiderivative of the function f(x) = x^3 - 3x^2 + 2

Accepted Answer

(x^4) / 4 - x^3 + 2x + C

Answer

x^3 - 3x^2 + 2 + C

Answer

x^4 - 6x^3 + 4x + C

Answer

No antiderivative exists

Question 28

Find the domain of the function f(x) = sqrt(x^2 - 4)

Accepted Answer

(-∞, -2] U [2, ∞)

Answer

[-2, 2]

Answer

[2, ∞)

Answer

(-∞, -2) U (2, ∞)

Question 29

Which of the following represents the graph of f(x) = x^3 - 2x^2 + x?

Accepted Answer

A cubic function with a maximum and a minimum

Answer

A parabola opening upward

Answer

A parabola opening downward

Answer

A straight line

Question 30

Evaluate ∫[0,1] x^3 dx

Accepted Answer

0.25

Answer

0.5

Answer

1

Answer

2

Question 31

Find the derivative of the function f(x) = x^3 + 2x^2 - 5x + 1.

Accepted Answer

3x^2 + 4x - 5

Answer

9x^2 + 4x - 5

Answer

x^3 + 4x^2 - 10x + 5

Answer

x^3 + 4x - 15

Question 32

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

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 33

Identify the graph that represents the function f(x) = 1 / x.

Accepted Answer

Hyperbola

Answer

Parabola

Answer

Circle

Answer

Line

Question 34

Calculate the derivative of the function:
f(x) = x^4 + 3x^3 - 2x^2 + 5

Accepted Answer

4x^3 + 9x^2 - 4x

Answer

2x^3 + 9x^2 - 4x

Answer

x^3 + 3x^2 - 2x

Answer

4x^3 - 9x^2 - 4x

Question 35

Find the domain and range of the function:
f(x) = √(16 - x^2)

Accepted Answer

Domain: [-4, 4]; Range: [0, 4]

Answer

Domain: (16, ∞); Range: (4, ∞)

Answer

Domain: (4, ∞); Range: (0, ∞)

Answer

Domain: (-4, 4]; Range: [0, 4]

Question 36

Simplify the trigonometric expression:
cos(π/4) + sin(π/4)

Accepted Answer

√2/2

Answer

1

Answer

0

Question 37

Find the derivative of the function f(x) = 2x^3 - 5x^2 + 3x - 1.

Accepted Answer

6x^2 - 10x + 3

Answer

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

Answer

6x^3 - 5x^2

Answer

3x^2 - 10x + 3

Question 38

Identify the critical point of the function f(x) = |x - 2|.

Accepted Answer

x = 2

Answer

x = 0

Answer

x = -2

Answer

The function has no critical points

Question 39

Determine which of the following sequences converges: an = 1/n^2

Accepted Answer

an = 1/n^2

Answer

an = 2n + 1

Answer

an = n^2

Answer

an = (-1)^n

Question 40

What is the average rate of change of the function *f(x) = x² - 3x + 2* over the interval from *x = 1* to *x = 4*?

Accepted Answer

(f(4) - f(1)) / (4 - 1)

Answer

(4 - 1) / (f(4) - f(1))

Answer

f(4) - f(1)

Answer

(f(4) + f(1)) / (4 - 1)

Question 41

At which point is the function *h(x) = |x|* continuous but not differentiable?

Accepted Answer

x = 0

Answer

x = 1

Answer

h(x) is differentiable everywhere

Answer

x = -1

Question 42

Which of the following is an antiderivative of the function *f(x) = 2x + cos(x)*?

Accepted Answer

x² + sin(x) + C

Answer

x² - sin(x) + C

Answer

2x² - sin(x) + C

Answer

2x² + sin(x) + C

Question 43

What is the derivative of the function *g(x) = e^(2x - 1)*?

Accepted Answer

2e^(2x - 1)

Answer

e^(2x - 1)

Answer

(2x - 1)e^(2x - 2)

Answer

2e^(2x)

Question 44

Which of the following is the derivative of the function f(x) = x^3 + 2x^2 - 5x + 1?

Accepted Answer

3x^2 + 4x - 5

Answer

x^3 + 4x + 1

Answer

3x^3 - 2x^2 - 5

Answer

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

Question 45

Find the integral of the function f(x) = 2x + 3 with respect to x.

Accepted Answer

x^2 + 3x + C

Answer

x^2 - 3x + C

Answer

2x - 3x + C

Answer

2x^2 + 3x + C

Question 46

Evaluate the following indefinite integral: ∫(x^2 + 1) dx

Accepted Answer

(x^3)/3 + x + C

Answer

x^2 + x + C

Answer

(x^3)/2 + x + C

Answer

x^3 + C

Question 47

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

Accepted Answer

2e^(2x + 1)

Answer

e^(2x + 1)/2

Answer

e^(x + 1)

Answer

2xe^(2x + 1)

Question 48

Which of the following is the integral of the function f(x) = 1/(x^2 + 1)?

Accepted Answer

arctan(x) + C

Answer

x/(x^2 + 1) + C

Answer

1/x + C

Answer

ln(x^2 + 1) + C

Question 49

Which of the following is the derivative of the function f(x) = ln(x^2 + 1)?

Accepted Answer

2x/(x^2 + 1)

Answer

ln(x) + 1

Answer

1/(x^2 + 1)

Answer

x^2/(x^2 + 1)

Question 50

Find the integral of the function f(x) = cos(2x) with respect to x.

Accepted Answer

(1/2)sin(2x) + C

Answer

2cos(2x) + C

Answer

sin(2x) + C

Answer

cos(x) + C

Question 51

Find the limit of the function f(x) = (x^2 - 4) / (x - 2) as x approaches 2.

Accepted Answer

4

Answer

The limit does not exist

Answer

2

Answer

0

Question 52

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

Accepted Answer

y = 3x - 2

Answer

y = 2x - 1

Answer

y = x + 1

Answer

y = x - 1

Question 53

Find the area under the curve y = x^2 from x = 0 to x = 2.

Accepted Answer

8/3

Answer

4

Answer

2

Question 54

What is the domain of the function f(x) = √(x - 2)?

Accepted Answer

[2, ∞)

Answer

(-∞, 2]

Answer

(-∞, ∞)

Answer

[0, ∞)

Question 55

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

Accepted Answer

2

Answer

The limit does not exist

Answer

1

Answer

0

Question 56

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

Accepted Answer

6x + 2

Answer

3x + 2

Answer

6x - 1

Question 57

Which of the following is the derivative of sin(2x)?

Accepted Answer

2cos(2x)

Answer

cos(2x)

Answer

-2cos(2x)

Answer

-cos(2x)

Question 58

Evaluate the definite integral of the function f(x) = 2x from x = 0 to x = 2.

Accepted Answer

4

Answer

0

Answer

8

Answer

2

Question 59

If f(x) = x^2 and g(x) = x + 1, what is the composition f(g(x))?

Accepted Answer

(x + 1)^2

Answer

x^3 + x^2

Answer

x^2 + 1

Answer

x + 2

Question 60

Determine the domain of the function f(x) = sqrt(x - 4).

Accepted Answer

x ≥ 4

Answer

x ≤ 4

Answer

All real numbers

Answer

x > 4

Question 61

What is the horizontal asymptote of the function  *f(x) = (2x + 1) / (x - 1)*?

Accepted Answer

y = 2

Answer

y = 1

Answer

x = 1

Answer

No horizontal asymptote

Question 62

Which of the following is equivalent to *log(a) + log(b)*?

Accepted Answer

log(a * b)

Answer

log(a) * log(b)

Answer

log(a / b)

Answer

log(a - b)

Question 63

What is the limit of the expression *(x² - 4) / (x - 2)* as *x* approaches 2?

Accepted Answer

4

Answer

Undefined

Answer

0

Answer

2

Question 64

What is the derivative of the function *f(x) = x³ - 2x² + 5*?

Accepted Answer

3x² - 4x

Answer

3x² - 5

Answer

x³ - 4x + 5

Answer

x² - 4x + 5