Calculus: Limits, Derivatives, Integrals, and Applications

Question 1

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

Accepted Answer

3x² + 4x - 5

Answer

x³ + 4x² - 10x + 1

Answer

3x⁴ + 4x³ - 5x² + 1

Answer

3x² - 4x + 5

Question 2

Find the derivative of the function f(x) = x³.

Accepted Answer

3x²

Answer

x

Answer

x⁴

Answer

2x

Question 3

Find the derivative of the function f(x) = e^(x) * ln(x).

Accepted Answer

(1 + ln(x)) * e^(x)

Answer

x * e^(x) * ln(x)

Answer

ln(x) * e^(x)

Answer

e^(x) / ln(x)

Question 4

Find the equation of the tangent line to the curve y = x² at the point (1, 1).

Accepted Answer

y = 2x - 1

Answer

y = -x + 1

Answer

y = x²

Answer

y = x + 1

Question 5

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

Accepted Answer

3x^2 + 4x - 5

Answer

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

Answer

3x^2 - 4x + 5

Answer

x^3 + 2x^2 - 10x

Question 6

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

Accepted Answer

(x^2 * cos(x)) + (2x * sin(x))

Answer

x^2 * cos(x)

Answer

2x * sin(x)

Answer

x * sin(x)

Question 7

Calculate the limit of the function f(x) = (x^2 - 1) / (x - 1) as x tends to 1.

Accepted Answer

2

Answer

-2

Answer

1

Answer

0

Question 8

Determine the area under the curve of the function f(x) = x^2 on the interval [0, 2].

Accepted Answer

8/3

Answer

16/3

Answer

4

Answer

8

Question 9

Determine the critical point(s) of the function f(x) = x^4 - 4x^2 + 3.

Accepted Answer

x = ±1

Answer

x = -3

Answer

x = 0

Answer

x = 2

Question 10

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

Accepted Answer

3x^2 + 4x - 5

Answer

x^2 + 2x - 5

Answer

x^3 + 2x^2

Answer

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

Question 11

Which of the following is the antiderivative of the function f(x) = sin(x)?

Accepted Answer

-cos(x) + C

Answer

tan(x) + C

Answer

cos(x) + C

Answer

sin(x) + C

Question 12

Determine the critical points of the function f(x) = x^3 - 3x^2 + 2x + 1.

Accepted Answer

x = 0, x = 1, x = 2

Answer

x = 0, x = 2

Answer

x = 1, x = 2

Answer

x = 0, x = -1, x = 2

Question 13

Which of the following is the formal definition of the limit of a function f(x) as x approaches a?

Accepted Answer

lim_(x->a) f(x) = L if for any positive number ε, there exists a positive number δ such that if 0 < |x - a| < δ, then |f(x) - L| < ε

Answer

lim_(x->a) f(x) = L if for any x, there exists an L such that |f(x) - L| < ε

Question 14

Use the limit definition of the derivative to find the derivative of the function f(x) = x^3 - 2x^2 + 1.

Accepted Answer

f'(x) = 3x^2 - 4x

Answer

f'(x) = 3x^2 + 4x

Answer

f'(x) = x^3 - 4x

Answer

f'(x) = x^2 - 4x

Question 15

State the chain rule for differentiation.

Accepted Answer

If f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x)

Answer

If f(x) = g(h(x)), then f'(x) = g'(x) + h'(x)

Answer

If f(x) = g(h(x)), then f'(x) = g'(h(x)) + h'(x)

Answer

If f(x) = g(h(x)), then f'(x) = g'(x) * h'(x)

Question 16

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

Accepted Answer

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

Answer

cos(x) + sin(x) + C

Answer

sin(x) - cos(x) + C

Answer

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

Question 17

State the fundamental theorem of calculus.

Accepted Answer

If f(x) is continuous on [a, b], then the net change in the area under the curve y = f(x) from x = a to x = b is equal to the definite integral of f(x) from a to b

Answer

If f(x) is continuous on [a, b], then the area under the curve y = f(x) from x = a to x = b is equal to the indefinite integral of f(x) from a to b

Question 18

Derive 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

Question 19

Find the indefinite integral of (x^2 + 1) dx.

Accepted Answer

(x^3)/3 + x + C

Answer

(x^2)/2 + C

Answer

x^3 + C

Answer

3x^2 + C

Question 20

Evaluate the limit of the expression (e^x - 1) / x as x approaches 0.

Accepted Answer

1

Answer

infinity

Answer

0

Answer

e

Question 21

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

Accepted Answer

1/x

Answer

ln(x)

Answer

x

Answer

-1/x

Question 22

Determine the integral of sin(x) + cos(x) dx.

Accepted Answer

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

Answer

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

Answer

sin(x) + cos(x)

Answer

cos(x) - sin(x) + C

Question 23

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

Accepted Answer

2xe^(x^2)

Answer

e^(x^2)

Answer

2x^2e^(x^2)

Answer

x^2e^(x)