Calculus: Master Limits, Derivatives, and Integrals

Question 1

Determine the derivatives of the parametric equations x(t) = t^2 and y(t) = t^3.

Accepted Answer

dx/dt = 2t, dy/dt = 3t^2

Answer

dx/dt = t, dy/dt = t^2

Answer

dx/dt = 2t, dy/dt = 2t

Answer

dx/dt = t^2, dy/dt = t^3

Question 2

Determine the antiderivative of the function f(x) = e^(2x) + cos(3x).

Accepted Answer

½ e^(2x) + 1/3 sin(3x) + C

Answer

2 e^(2x) - 3 sin(3x) + C

Answer

½ e^(2x) - 1/3 sin(3x) + C

Answer

e^(2x) + sin(3x) + C

Question 3

Determine whether the sequence {a_n} = (1 + 1/n)^n converges. If it converges, find its limit.

Accepted Answer

Converges to e

Answer

Converges to 0

Answer

Diverges to infinity

Question 4

Find the integral of the function f(x) = 2x + 1 from x = 0 to x = 2.

Accepted Answer

8

Answer

6

Answer

10

Answer

12

Question 5

Define a limit of a function.

Accepted Answer

The value a function approaches as its input gets infinitely close to a certain value

Answer

The rate at which a function changes at a given point

Answer

The area under a curve between two points

Answer

The maximum or minimum value of a function

Question 6

Which of the following is NOT an application of Calculus?

Accepted Answer

Predicting the weather using mathematical models

Answer

Finding the velocity of a moving object

Answer

Optimizing the shape of a can

Answer

Calculating the volume of a solid of revolution

Question 7

Calculate the derivative of f(x) = e^(x^2 + 1) using the chain rule. Provide a clear step-by-step explanation of your process.

Accepted Answer

2xe^(x^2 + 1)

Answer

e^(x^2 + 1)

Answer

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

Answer

2e^(x^2 + 1)

Question 8

Determine the equation of the tangent plane to the surface z = xy^2 + x^2 at the point (1, 1, 3).

Accepted Answer

z = 2x - y + 1

Answer

z = x^2 + 2y

Answer

z = y^2 + 3

Question 9

Evaluate the improper integral ∫[0, 1] (sin x) / x dx. Determine if it converges or diverges, and provide a brief explanation.

Accepted Answer

The integral diverges.

Answer

The integral converges to 1.

Answer

The integral converges to 0.

Answer

The integral converges to π / 2.

Question 10

Define the formal definition of a limit.

Accepted Answer

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

Answer

The exact value a function reaches when the input variable equals a specific value

Answer

The rate of change of a function as the input variable changes

Answer

The value a function has when the input variable is zero

Question 11

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

Accepted Answer

2x + 2

Answer

2x - 1

Answer

x^2 + 2x

Answer

x + 1

Question 12

Find the surface area of the surface generated by rotating the curve y = √x from x = 0 to x = 1 about the x-axis.

Accepted Answer

π(1 + √2)

Answer

2π

Answer

π(1 - √2)

Answer

4π

Question 13

A sequence is defined as a_n = (1 + 1/n)^n. Does this sequence converge or diverge? If it converges, what is its limit?

Accepted Answer

The sequence converges to e

Answer

The sequence diverges to infinity

Answer

The sequence converges to 1

Question 14

Determine the convergence or divergence of the series Σ(n=1,∞) (1/n) and Σ(n=1,∞) (1/n^2) using appropriate convergence tests. Explain your reasoning clearly.

Accepted Answer

Σ(1/n) diverges and Σ(1/n^2) converges by the Limit Comparison Test.

Answer

Both series converge by the Ratio Test.

Answer

Both series diverge by the Ratio Test.

Answer

Σ(1/n) converges and Σ(1/n^2) diverges by the Limit Comparison Test.

Question 15

What is the limit of the function (x² + 2x - 1) / (x² - 3x + 2) as x approaches infinity?

Accepted Answer

1

Answer

0

Answer

Infinity

Answer

-1

Question 16

Determine the behavior of the sequence {n^2 / (n^2 + 1)}. Does it converge or diverge?

Accepted Answer

Converges to 0

Answer

Converges to 1

Answer

Diverges to infinity

Answer

Oscillates

Question 17

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

Accepted Answer

3x^2 + 4x - 5

Answer

3x^2 + 2x - 5

Answer

x^3 + 4x^2 - 5x

Answer

x^2 + 2x - 5

Question 18

Find the integral of the function f(x) = e^x from x = 0 to x = 1.

Accepted Answer

e - 1

Answer

e^2 - 1

Answer

e^2 + 1

Answer

e - 2

Question 19

Specify the correct formula for the derivative of a logarithmic function.

Accepted Answer

d/dx ln(x) = 1/x

Answer

d/dx ln(x) = x

Answer

d/dx ln(x) = ln(x)

Answer

d/dx ln(x) = -1/x

Question 20

Define a limit of a function.

Accepted Answer

The value a function approaches as the input gets closer to a specific value.

Answer

The value a function takes at a particular input.

Answer

The rate of change of a function at a specific input.

Answer

The area under a curve.

Question 21

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

Accepted Answer

2x + 3

Answer

x^2 + 3

Answer

2x

Answer

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^2 - 5

Answer

3x^2 + 2x - 5

Answer

x^3 + 2x - 5

Question 23

Which of the following methods is NOT used to evaluate limits?

Accepted Answer

Graphing the function

Answer

Factoring and cancellation

Answer

Substitution

Answer

L'Hopital's rule

Question 24

How is the derivative of a function related to its graph?

Accepted Answer

The derivative represents the slope of the tangent line to the graph at any given point.

Answer

The derivative represents the area under the curve of the graph over a given interval.

Answer

The derivative represents the integral of the function with respect to x.

Answer

There is no relationship between the derivative and the graph.

Question 25

What does the Fundamental Theorem of Calculus state?

Accepted Answer

The integral of a derivative is the original function.

Answer

The derivative of an integral is the original function.

Answer

The derivative and the integral are inverse operations.

Answer

The area under the curve of a function is equal to the derivative of the function.

Question 26

Calculate the definite integral ∫[0,1] x^2 * e^(x^3) dx using the Fundamental Theorem of Calculus.

Accepted Answer

(1 / 3) [e^(1^3) - e^(0^3)]

Answer

e^(1^3) - e^(0^3)

Answer

(1 / 3)x^3 * e^(x^3)

Answer

0

Question 27

Calculate the derivative of the function f(x) = x^3 + 2x.

Accepted Answer

3x^2 + 2

Answer

3x

Answer

x^2 + 2

Answer

x^3 + 1

Question 28

Determine if the improper integral ∫[0,∞) e^(-x) dx is convergent or divergent.

Accepted Answer

Convergent

Answer

Semi-convergent

Answer

Divergent

Answer

Oscillatory

Question 29

Which of the following techniques is used to integrate rational functions with quadratic denominators?

Accepted Answer

Partial Fraction Decomposition

Answer

U-Substitution

Answer

Integration by Parts

Answer

Trig Substitution

Question 30

Which of the following is the formal definition of a limit? Select the option that best aligns with the concept in Calculus.

Accepted Answer

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

Answer

The minimum value that a function can attain.

Answer

The maximum value that a function can attain.

Question 31

Determine the derivative of the function f(x) = x^2. Select the correct option that represents the derivative of this function.

Accepted Answer

2x

Answer

2

Answer

x

Answer

x^3

Question 32

Determine the derivative of the function f(x) = e^x. Select the option that correctly represents the derivative of this exponential function.

Accepted Answer

e^x

Answer

ln(e^x)

Answer

e^(x^2)

Answer

x*e^x

Question 33

Select the option that correctly represents the formula for calculating the area under the curve of a function f(x) from a to b.

Accepted Answer

∫[a,b] f(x) dx

Answer

f(x) * (b - a)

Answer

(b - a) * f(x)

Answer

∫[b,a] f(x) dx

Question 34

Calculate the derivative of the function f(x) = x² + 3x - 5.

Accepted Answer

2x + 3

Answer

x² - 3x + 5

Answer

x³ + 3x² - 5x

Answer

3x - 5

Question 35

Which of the following functions is continuous at x = 0?

Accepted Answer

f(x) = |x|

Answer

f(x) = 1/x

Answer

f(x) = sqrt(x)

Answer

f(x) = tan(x)

Question 36

What is the derivative of the logarithmic function f(x) = ln(x² + 1)?

Accepted Answer

2x/(x² + 1)

Answer

(x² + 1)/x

Answer

1/(x² + 1)

Answer

x² + 1

Question 37

Find the antiderivative of the function f(x) = e^(2x).

Accepted Answer

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

Answer

1/e^(2x) + C

Answer

e^(2x)

Answer

2e^(2x) + C

Question 38

Which of the following is NOT an application of calculus?

Accepted Answer

Predicting the stock market

Answer

Calculating the area under a curve

Answer

Solving optimization problems

Answer

Finding the velocity of a moving object

Question 39

Determine the area of the region bounded by the curve y = x^2 and the line y = 4.

Accepted Answer

8/3

Answer

2/3

Answer

4/3

Answer

16/3

Question 40

Identify the antiderivative of the function f(x) = 1/(x^2 + 1).

Accepted Answer

arctan(x) + C

Answer

1/x + C

Answer

x/2 + C

Answer

tan(x) + C

Question 41

Given the function f(x) = x^2 - 4x + 3, evaluate the limit of f(x) as x approaches positive infinity.

Accepted Answer

Infinity

Answer

3

Answer

Negative infinity

Answer

0

Question 42

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

Accepted Answer

2x * cos(x^2)

Answer

-2x * cos(x^2)

Answer

cos(x^2)

Answer

x * cos(x^2)

Question 43

Calculate the indefinite integral of the function f(x) = e^x + 2x.

Accepted Answer

e^x + x^2 + C

Answer

e^x - x^2 + C

Answer

e^x / x^2 + C

Answer

e^x + 2x + C

Question 44

Find the area under the curve of the function f(x) = sqrt(x) between x = 0 and x = 4 using integrals.

Accepted Answer

8/3

Answer

2

Answer

4

Answer

1

Question 45

Determine if the function f(x) = x^3 + 2x^2 - 1 is continuous at x = 0.

Accepted Answer

Yes

Answer

Cannot be determined

Answer

No

Answer

Depends on the value of the function at x = 0

Question 46

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

Accepted Answer

2x / (x^2 + 1)

Answer

1 / (x^2 + 1)

Answer

(x^2 + 1) / 2x

Answer

-2x / (x^2 + 1)

Question 47

Find the indefinite integral of the function f(x) = cos(2x) using integration by substitution.

Accepted Answer

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

Answer

sin(2x) + C

Answer

-sin(2x) + C

Answer

-cos(2x) + C

Question 48

Find the antiderivative of f(x) = cos(x).

Accepted Answer

sin(x) + C

Answer

-sin(x) + C

Answer

cos(x) + C

Answer

-cos(x) + C

Question 49

If the derivative of a function is positive on an interval, the function is...

Accepted Answer

Increasing on that interval

Answer

Undefined on that interval

Answer

Decreasing on that interval

Answer

Constant on that interval

Question 50

To find the volume of a solid generated by revolving the region bounded by y = x^2, the x-axis, and x = 2 about the x-axis, which integration method should be used?

Accepted Answer

Disk method

Answer

Shell method

Answer

None of the above

Answer

Washer method

Question 51

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

Accepted Answer

6x + 2

Answer

9x + 2

Answer

6x^2 + 2x

Answer

3x + 2

Question 52

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

Accepted Answer

x = 0, x = 2

Answer

x = 1, x = 2

Answer

x = 0, x = 1

Question 53

On which interval is the function f(x) = x^2 increasing?

Accepted Answer

(0, ∞)

Answer

(-∞, ∞)

Answer

(-∞, 0)

Answer

(-1, 1)

Question 54

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

Accepted Answer

e^x + C

Answer

x * e^x + C

Answer

ln(x) + C

Answer

e^(x+1) + C

Question 55

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

Accepted Answer

3

Answer

2

Answer

4

Answer

1

Question 56

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

Accepted Answer

6x + 2

Answer

6x^2 + 2

Answer

3x + 2

Answer

6x - 1

Question 57

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

Accepted Answer

3

Answer

1

Answer

0

Answer

6

Question 58

The function f(x) = x^2 + 2x - 3 is increasing on the interval:

Accepted Answer

(-1, ∞)

Answer

(-∞, 1)

Answer

(-∞, -1)

Answer

(-∞, ∞)

Question 59

Find the area under the curve y = x^2 from x = 0 to x = 3.

Accepted Answer

9

Answer

3

Answer

12

Answer

6

Question 60

Evaluate the indefinite integral ∫(1/x) dx.

Accepted Answer

ln|x| + C

Answer

1/x^2 + C

Answer

-1/x^2 + C

Answer

x + C