Calculus: Master Differential and Integral Concepts | Online Learning

Question 1

Which application of calculus involves finding the area of a region under a curve?

Accepted Answer

Definite integration

Answer

Optimization

Answer

Related rates

Answer

Newton's method

Question 2

Describe the Chain Rule in calculus.

Accepted Answer

A rule for finding the derivative of a composite function

Answer

A rule for finding the integral of a composite function

Answer

A rule for finding the antiderivative of a composite function

Answer

A rule for finding the slope of the tangent line to a composite function

Question 3

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

Accepted Answer

f'(x) = 3x² + 4x - 5

Answer

f'(x) = 3x³ + 4x² - 5x

Answer

f'(x) = x² + 4x - 5

Answer

f'(x) = 3x² - 4x + 5

Question 4

Calculate the volume of the solid generated by rotating the region bounded by the curves y = x and y = x² about the x-axis.

Accepted Answer

π/5

Answer

2π/5

Answer

π/2

Answer

π/10

Question 5

Determine if the function f(x) = x³ - 3x² + 2x - 1 is increasing or decreasing on the interval (-∞, 1).

Accepted Answer

Decreasing

Answer

Cannot be determined

Answer

Constant

Answer

Increasing

Question 6

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

Accepted Answer

3x² + 4x - 5

Answer

x³ - 5

Answer

x² + 2x - 5

Answer

x³ + 4x² - 5x - 1

Question 7

Identify the critical point(s) of the function f(x) = x⁴ - 2x² + 1.

Accepted Answer

x = 0, ±1

Answer

x = 1, -1

Answer

x = -1, 1

Answer

x = 0, 2

Question 8

Determine the volume of the solid generated by rotating the region bounded by y = x and y = x² about the x-axis.

Accepted Answer

π/6

Answer

π/4

Answer

π/8

Answer

π/2

Question 9

Determine the equation of the tangent line to the curve y = x³ - 2x at the point (1, -1).

Accepted Answer

y = 3x - 4

Answer

y = x - 2

Answer

y = -x + 2

Answer

y = 3x + 4

Question 10

What is the definition of the derivative of a function *f(x)*?

Accepted Answer

The limit of the slope of secant lines as the change in *x* approaches zero.

Answer

The integral of the function with respect to *x*.

Answer

The rate of change of the function with respect to *x*.

Question 11

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

Accepted Answer

2*x* + 2

Answer

x² + 4*x* - 1

Answer

2*x² - 1

Answer

x² + 4

Question 12

Which of the following statements represents the Fundamental Theorem of Calculus, Part I?

Accepted Answer

If *f(x)* is continuous on the interval [*a*, *b*], then the definite integral of *f(x)* from *a* to *b* is equal to the net area between the graph of *f(x)* and the *x*-axis on [*a*, *b*].

Answer

The derivative of an integral is equal to the original function.

Question 13

What is the Chain Rule for derivatives?

Accepted Answer

If *f(x)* is differentiable at *x* and *g(x)* is differentiable at *f(x)*, then the composite function *g(f(x))* is differentiable at *x* and its derivative is given by *g'(f(x))* * f'(x).

Answer

The Product Rule.

Question 14

Which of the following statements describes the Mean Value Theorem for Derivatives?

Accepted Answer

If *f(x)* is continuous on [*a*, *b*] and differentiable on (*a*, *b*), then there exists a number *c* in (*a*, *b*) such that *f'(c)* = (*f(b)* - *f(a)*)/(*b* - *a*).

Answer

If *f(x)* has a maximum on [*a*, *b*], then *f'(c)* = 0 for some *c* in (*a*, *b*).

Question 15

What represents the instantaneous rate of change of a function at a point?

Accepted Answer

The derivative of the function at that point.

Answer

The slope of the secant line through that point.

Answer

The area under the curve of the function.

Answer

The average rate of change over a small interval.

Question 16

What does the definite integral of *f(x)* from *a* to *b* represent?

Accepted Answer

The area under the curve of *f(x)* between *x* = *a* and *x* = *b*.

Answer

The slope of the tangent line to *f(x)* at *x* = *a*.

Answer

The derivative of *f(x)* at *x* = *b*.

Question 17

What is the area enclosed between the curve *y = x²*, the *x*-axis, and the lines *x* = 1 and *x* = 2?

Accepted Answer

7/3

Answer

8/3

Answer

1/3

Answer

4/3

Question 18

What is the derivative of *sin(x)*?

Accepted Answer

cos(x)

Answer

-cos(x)

Answer

-sin(x)

Answer

sin(x)

Question 19

What does the fundamental theorem of calculus state?

Accepted Answer

The definite integral of a function can be calculated using its antiderivative.

Answer

The derivative of a function is always equal to its antiderivative.

Answer

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

Question 20

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

Accepted Answer

3x^2 - 4x

Answer

x^3 - 4x

Answer

3x^2 + 4x

Answer

x^2 - 4x

Question 21

Determine the area under the curve y = sin(x) from 0 to π.

Accepted Answer

2

Answer

π

Answer

1

Answer

4

Question 22

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

Accepted Answer

sin(x) + C

Answer

-cos(x) + C

Answer

-sin(x) + C

Answer

cos(x) + C

Question 23

Which of the following is 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

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

Answer

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

Answer

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

Question 24

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

Accepted Answer

8/3

Answer

2

Answer

4

Answer

16/3