Pre-Calculus Quizzes: Master Calculus Foundations

Question 1

Does the sequence a_n = n / (n^2 + 1) converge or diverge? If it converges, what is its limit?

Accepted Answer

Converges to 0

Answer

Diverges to infinity

Answer

Converges to 1

Answer

Diverges to -1

Question 2

To evaluate the integral ∫ sqrt(1 - x^2) dx, which trigonometric substitution should be used?

Accepted Answer

x = sin(θ)

Answer

x = cos(θ)

Answer

x = tan(θ)

Answer

x = cot(θ)

Question 3

Which of the following is the correct definition of a limit?

Accepted Answer

The value that a function approaches as its input approaches a certain value

Answer

The value that a function takes at a specific input

Answer

The slope of a function at a specific point

Answer

The area under a curve between two points

Question 4

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

Accepted Answer

2x cos(x^2)

Answer

cos(2x)

Answer

x^2 cos(x)

Question 5

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

Accepted Answer

3x^2 + 4x - 5

Answer

3x^2 + 2x

Answer

3x + 4

Answer

x^3 + 2x^2

Question 6

Determine the domain of the function f(x) = sqrt(x - 2).

Accepted Answer

(2, ∞)

Answer

[-2, ∞)

Answer

(∞, 2)

Answer

[2, ∞)

Question 7

Classify the function f(x) = x^3 - 3x as odd, even, or neither.

Accepted Answer

Odd

Answer

Even

Answer

Neither

Answer

Both even and odd

Question 8

Find the equation of the vertical asymptote of the function g(x) = (x^2 - 9) / (x + 3).

Accepted Answer

x = 3

Answer

x = -3

Answer

x = 0

Answer

The function has no vertical asymptotes.

Question 9

Determine whether the alternating series converges or diverges:

1 - 1/2 + 1/3 - 1/4 + ...

using the Alternating Series Test.

Accepted Answer

Converges

Answer

Diverges

Answer

Cannot be determined

Answer

Absolutely convergent

Question 10

Evaluate the convergence or divergence of the improper integral ∫_1^∞ (x - 1) / x^3 dx using an appropriate test. Justify your answer.

Accepted Answer

Convergent, by the Limit Comparison Test with 1/x^2

Answer

Divergent, by the Limit Comparison Test with 1/x^3

Question 11

Using the epsilon-delta definition, prove that the function **f(x, y) = e^(x + y)** is continuous at any point **(a, b)** in the **xy**-plane.

Accepted Answer

Answer

The function is continuous because it is differentiable at every point in the **xy**-plane.

Answer

The function is continuous due to its smooth surface graph.

Answer

The function is continuous because it is defined at each point in the **xy**-plane.

Question 12

Which of the following is NOT a valid technique used to evaluate limits?

Accepted Answer

Using a graphing calculator

Answer

L'Hopital's rule

Answer

Factoring

Answer

Substitution

Question 13

Find the derivative of f(x) = x^3 + 2x^2 - 5x + 1 using the power rule and sum/difference rule.

Accepted Answer

3x^2 + 4x - 5

Answer

3x^2 - 4x + 5

Answer

x^2 + 4x - 5

Answer

2x^2 + 4x - 5

Question 14

Find the derivative of f(x) = (x^2 + 1)^5 using the chain rule.

Accepted Answer

5(x^2 + 1)^4(2x)

Answer

5(x^2 + 1)^4

Answer

10x(x^2 + 1)^4

Answer

(x^2 + 1)^5

Question 15

Calculate the second derivative of f(x) = ln x.

Accepted Answer

-1/x^2

Answer

1/x

Answer

-1/x

Answer

1/x^2

Question 16

Find the derivative of the function y = x^3 - 2x:

Accepted Answer

3x^2 - 4

Answer

3x^2 - 2

Answer

x^3 - 2

Answer

x^2 - 4

Question 17

Determine the equation of the tangent line to the curve y = sin(x) at x = π/4:

Accepted Answer

y = √(2)/2 x

Answer

y = 0

Answer

y = x

Answer

y = x - π/4

Question 18

Determine if the sequence $a_n = rac{n^2 + 1}{n^3 - 4}$ converges or diverges. If it converges, find its limit.

Accepted Answer

Converges to 0

Answer

Converges to 1

Answer

Diverges

Answer

Converges to $rac{1}{4}$

Question 19

Determine whether the following improper integral converges or diverges: ∫[0, ∞] e^(-x) dx

Accepted Answer

Converges

Answer

Diverges

Answer

Cannot be determined

Question 20

Define the limit of a function.

Accepted Answer

The value that the function approaches as the input approaches a specific value

Answer

The average value of the function over a given interval

Answer

The maximum or minimum value of the function

Answer

The derivative of the function

Question 21

State the Fundamental Theorem of Calculus.

Accepted Answer

The integral of a derivative is the original function

Answer

The derivative of an integral is the original function

Answer

The derivative of a function is equal to its integral

Answer

The integral of a function is equal to its derivative

Question 22

What is the domain of the function f(x) = √(x - 3)?

Accepted Answer

[3, ∞)

Answer

(-∞, ∞)

Answer

[-3, ∞)

Answer

(-∞, 3]

Question 23

Find the inverse function of f(x) = 2x + 3.

Accepted Answer

f^-1(x) = (x - 3) / 2

Answer

f^-1(x) = 3x + 2

Answer

f^-1(x) = 3 - 2x

Answer

f^-1(x) = 2x - 3

Question 24

Which function has a vertical asymptote at x = 3?

Accepted Answer

f(x) = (x - 3)^-1

Answer

f(x) = x^2 - 3x + 2

Answer

f(x) = |x - 3|

Question 25

Find the equation of the tangent line to the function f(x) = x^2 - 2x + 1 at the point (1, 0).

Accepted Answer

y = 2x - 1

Answer

y = x^2 + 1

Answer

y = x - 1

Answer

y = x^2 - 2x

Question 26

Identify the logarithmic equation:

Accepted Answer

log(x + 2) = 3

Answer

e^x = 5

Answer

sin(x) = 0.5

Answer

x^2 + 2x = 0

Question 27

Find the inverse function of f(x) = 2x + 3.

Accepted Answer

f^-1(x) = (x - 3) / 2

Answer

f^-1(x) = 2x - 3

Answer

f^-1(x) = 3x - 2

Question 28

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

Accepted Answer

[2, ∞)

Answer

[-2, ∞)

Answer

(2, ∞)

Answer

[0, ∞)

Question 29

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

Accepted Answer

f'(x) = 3x² + 4x - 5

Answer

f'(x) = 3x³ + 4x - 5

Answer

f'(x) = x² + 4x + 5

Answer

f'(x) = 3x² - 4x + 5

Question 30

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

Accepted Answer

6x - 2

Answer

6x^2 - 2

Answer

9x^2 - 2x + 1

Answer

3x - 2

Question 31

The slope of the tangent line to the curve y = x^3 at the point (1, 1) is:

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 32

The average value of the function f(x) = 2x on the interval [0, 2] is:

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 33

What is the derivative of the function *f(x) = sin(x²)*?

Accepted Answer

2x cos(x²)

Answer

sin(2x)

Answer

cos(x²)

Answer

2 cos(x²)

Question 34

Find the indefinite integral of the function *f(x) = e^(2x)*.

Accepted Answer

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

Answer

e^(x) + C

Answer

2e^(2x) + C

Answer

e^(2x) + C

Question 35

The function *f(x) = x³ - 3x² + 2x* has a local maximum at which x-value?

Accepted Answer

x = 1

Answer

x = 2

Answer

The function has no local maximum.

Answer

x = 0

Question 36

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

Accepted Answer

[2, ∞)

Answer

(2, ∞)

Answer

(-∞, 2]

Answer

(-∞, ∞)

Question 37

What is the derivative of the function *f(x) = |x|*?

Accepted Answer

1 if x > 0, -1 if x < 0, and undefined if x = 0

Answer

0 for all x

Answer

-1 for all x

Answer

1 for all x

Question 38

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

Accepted Answer

4

Answer

0

Answer

Does not exist

Answer

2

Question 39

What is the derivative of the function f(x) = 3x² + 2x - 1?

Accepted Answer

6x + 2

Answer

3x² + 2

Answer

6x² + 2x

Answer

3x + 2

Question 40

What is the slope of the tangent line to the curve y = x³ - 2x at the point (1, -1)?

Accepted Answer

1

Answer

-1

Answer

3

Answer

-3

Question 41

What is the indefinite integral of the function f(x) = 2x + 1?

Accepted Answer

x² + x + C

Answer

2x + C

Answer

x² + C

Answer

2x² + x + C

Question 42

What is the definite integral of the function f(x) = x² from 0 to 2?

Accepted Answer

8/3

Answer

2

Answer

4

Answer

16/3

Question 43

The function f(x) = |x| is:

Accepted Answer

Not differentiable at x = 0

Answer

Differentiable only for x > 0

Answer

Differentiable only for x < 0

Answer

Differentiable everywhere

Question 44

What are the critical points of the function f(x) = x³ - 3x² + 2?

Accepted Answer

x = 0, x = 2

Answer

x = 1, x = 2

Answer

x = -1, x = 2

Answer

x = 0, x = 1

Question 45

What is the area under the curve of the function f(x) = 2x from x = 1 to x = 3?

Accepted Answer

8

Answer

4

Answer

6

Answer

12

Question 46

What is the equation of the line tangent to the curve y = x² at the point (2, 4)?

Accepted Answer

y = 4x - 4

Answer

y = 2x + 4

Answer

y = 2x

Answer

y = 4x + 4

Question 47

The function f(x) = sin(x) is:

Accepted Answer

Periodic

Answer

Constant

Answer

Exponential

Answer

Linear

Question 48

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

Accepted Answer

2

Answer

0

Answer

Does not exist

Answer

1

Question 49

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

Accepted Answer

6x + 2

Answer

3x + 2

Answer

6x^2 + 2x

Answer

6x - 1

Question 50

The slope of the tangent line to the curve y = x^3 - 2x at the point (1, -1) is:

Accepted Answer

1

Answer

-1

Answer

-3

Answer

3

Question 51

The integral of the function f(x) = 2x + 1 from x = 0 to x = 2 is:

Accepted Answer

6

Answer

8

Answer

2

Answer

4

Question 52

The area under the curve of the function f(x) = x^2 from x = 0 to x = 2 is:

Accepted Answer

8/3

Answer

16/3

Answer

4

Answer

2

Question 53

The limit of the function f(x) = (sin(x))/x as x approaches 0 is:

Accepted Answer

1

Answer

-1

Answer

Does not exist

Answer

0

Question 54

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

Accepted Answer

e^x

Answer

e^(x+1)

Answer

1/e^x

Answer

x*e^(x-1)

Question 55

The integral of the function f(x) = cos(x) from x = 0 to x = pi/2 is:

Accepted Answer

1

Answer

-1

Answer

pi/2

Answer

0

Question 56

A particle moves along a straight line with a velocity function v(t) = 2t + 1. The displacement of the particle from t = 0 to t = 2 is:

Accepted Answer

6

Answer

8

Answer

2

Answer

4

Question 57

The equation of the line tangent to the curve y = x^2 at the point (2, 4) is:

Accepted Answer

y = 4x - 4

Answer

y = x + 2

Answer

y = 2x

Question 58

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

Accepted Answer

4

Answer

Does Not Exist

Answer

2

Answer

0

Question 59

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

Accepted Answer

6x + 5

Answer

6x^2 + 5x

Answer

3x + 5

Answer

6x - 2

Question 60

Find the equation of the tangent line to the curve y = x^3 - 2x + 1 at the point (1, 0).

Accepted Answer

y = x - 1

Answer

y = -x + 1

Answer

y = x + 1

Answer

y = 3x - 3

Question 61

What is the integral of the function f(x) = 2x + 1?

Accepted Answer

x^2 + x + C

Answer

2x^2 + x + C

Answer

x^2 + 1 + C

Answer

2x + C

Question 62

The graph of f(x) = sin(x) has a period of:

Accepted Answer

2π

Answer

π

Answer

4π

Answer

π/2

Question 63

Find the value of the expression cos(π/4) * sin(π/3).

Accepted Answer

√6 / 4

Answer

√3 / 2

Answer

√2 / 2

Answer

1 / 2

Question 64

The graph of the function f(x) = (x - 2)^2 is:

Accepted Answer

A parabola opening upwards with vertex at (2, 0)

Answer

A parabola opening downwards with vertex at (2, 0)

Answer

A line with a slope of 2