Master the Derivative: Interactive Quizzes for Calculus I

Question 1

Which of the following is the formal definition of the derivative of a function y = f(x)?

Accepted Answer

The instantaneous rate of change of y with respect to x, defined as the limit as h approaches 0 of (f(x+h) - f(x)) / h

Answer

The slope of the tangent line to the graph of f(x) at any point (x, f(x))

Answer

The area under the curve of f(x) from x to x=0

Answer

The y-intercept of the graph of f(x)

Question 2

What is the derivative of f(x) = sin(x) using the chain rule of differentiation?

Accepted Answer

cos(x)

Answer

sin(x)/cos(x)

Answer

cos(x)/sin(x)

Answer

sin(x)cos(x)

Question 3

Which of the following is an application of the derivative in optimization?

Accepted Answer

Finding the maximum and minimum values of a function

Answer

Solving differential equations

Answer

Calculating the area under a curve

Answer

Computing the volume of a solid of revolution

Question 4

What is the derivative of the inverse function of f(x) = x^3 - 1?

Accepted Answer

1/(3x^2)

Answer

3x^2

Answer

x^2 + 1

Answer

There is no inverse function for this function.

Question 5

What is the derivative of f(x) = arctan(x) using the inverse trigonometric function rule?

Accepted Answer

1/(1+x^2)

Answer

x/(1+x^2)

Answer

-(1+x^2)/x

Answer

x - 1

Question 6

Find the derivative of an exponential function with base 'a' and exponent 'x'.

Accepted Answer

a^x * ln(a)

Answer

a^(x+1)

Answer

x*a^(x-1)

Answer

(1/a) * a^x

Question 7

Calculate 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

6x^2 - 4x + 5

Answer

2x^2 - 4x + 5

Question 8

Which of the following represents the derivative of the function f(x) = e^x?

Accepted Answer

e^x

Answer

xe^x

Answer

1/e^x

Answer

x

Question 9

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

Accepted Answer

1/x

Answer

x

Answer

-1/x

Answer

ln(x)

Question 10

Which of the following is the derivative of the function f(x) = 1/x?

Accepted Answer

-1/x^2

Answer

1/x^2

Answer

1/x

Answer

x

Question 11

Calculate the derivative of the function f(x) = sqrt(x).

Accepted Answer

1/(2sqrt(x))

Answer

1/sqrt(x)

Answer

2sqrt(x)

Answer

sqrt(x)

Question 12

Determine the derivative of the function implicitly defined by the equation x^3 + y^2 = 2xy.

Accepted Answer

2y + x(dy/dx) = 3x^2 + 2x(dy/dx)

Answer

x^2 + y = 2xy

Answer

dy/dx = 3x^2 + 2y

Question 13

The derivative of the constant function f(x) = c for any real number c, is:

Accepted Answer

0

Answer

1

Answer

c

Answer

x

Question 14

Define the derivative of a function f(x).

Accepted Answer

The limit of the difference quotient as the change in x approaches zero

Answer

The instantaneous rate of change of the function

Answer

The slope of the tangent line to the graph of the function at a specific point

Answer

The area under the curve of the function

Question 15

Apply the power rule to find the derivative of f(x) = x^3 + 2x - 5.

Accepted Answer

3x^2 + 2

Answer

x^2 + 2

Answer

3x - 5

Answer

x + 2

Question 16

Identify the function that is NOT differentiable at x = 0.

Accepted Answer

f(x) = |x|

Answer

f(x) = x^2

Answer

f(x) = sin(x)

Answer

f(x) = e^x

Question 17

Employ the chain rule to find the derivative of f(x) = e^(2x).

Accepted Answer

e^(2x) * ln(2)

Answer

2e^(2x)

Answer

e^(2x)/2

Answer

e^(x)

Question 18

Calculate the derivative of f(x) = ln(x^2 + 1) using logarithmic differentiation.

Accepted Answer

(2x)/(x^2 + 1) * ln(x^2 + 1)

Answer

2x/(x^2 + 1)

Answer

(1/x) * ln(x^2 + 1)

Answer

1/(x^2 + 1)

Question 19

Identify the critical points of the function f(x) = x^4 - 4x^2.

Accepted Answer

x = 0, x = ±2

Answer

x = 0, x = ±3

Answer

x = 0, x = ±1

Answer

x = 0 only

Question 20

Apply Rolle's Theorem to determine if f(x) = x^3 - 3x^2 + 2 is continuous and differentiable on the interval [-1, 2].

Accepted Answer

Continuous and differentiable

Answer

Continuous but not differentiable

Answer

Differentiable but not continuous

Answer

Neither continuous nor differentiable

Question 21

The derivative of a function f(x) is defined as:

Accepted Answer

The instantaneous rate of change of f(x) with respect to x

Answer

The slope of the secant line passing through two points on the graph of f(x)

Answer

The area under the curve of f(x) between two points x and x+h

Answer

The integral of f(x) with respect to x

Question 22

Calculate the derivative of f(x) = x^3 - 2x^2 + 1 using the limit definition of the derivative:

Accepted Answer

3x^2 - 2

Answer

x^2 - 2

Answer

3x - 2

Answer

2x - 1

Question 23

Which of the following is a rule for finding the derivative of a constant function?

Accepted Answer

f'(x) = 0

Answer

f'(x) = 1

Answer

f'(x) = x

Answer

f'(x) = e^x

Question 24

Find the critical points of the function f(x) = x^3 - 3x^2 + 2, if any exist.

Accepted Answer

x = 0, x = 2

Answer

x = 0, x = 1

Answer

x = 1, x = 2

Answer

No critical points exist

Question 25

For the function f(x) = x^3 - 6x^2 + 9x + 2, determine the intervals of concavity and the x-coordinate of the inflection point.

Accepted Answer

Concave up on (-∞, 2) and concave down on (2, ∞), inflection point at x = 2

Answer

Concave down on the entire real line

Answer

Concave up on (-∞, 2) and concave down on (2, ∞), inflection point at x = -2

Answer

Concave up on the entire real line

Question 26

The derivative of a function f(x) is given as follows:
f'(x) = {x < 0 : positive, x > 0 : negative}
Describe the behavior of f(x) in terms of its increasing/decreasing nature and local extrema.

Accepted Answer

Increases on (-∞, 0) and decreases on (0, ∞), with a local maximum at x = 0

Answer

Decreases on (-∞, 0) and increases on (0, ∞), with a local minimum at x = 0

Answer

Increases on the entire real line, with no local extrema

Answer

Decreases on the entire real line, with no local extrema

Question 27

Given the function f(x) = x³ - 3x² + 2 defined on the closed interval [0, 2], identify the value(s) of c in the open interval (0, 2) that satisfy Rolle's Theorem.

Accepted Answer

1

Answer

-1

Answer

0

Answer

2

Question 28

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

Accepted Answer

3x^2 - 4x

Answer

3x^2 + 4x

Answer

x^3 - 4x

Answer

3x - 4

Question 29

What is the derivative of f(x) = sin(x) with respect to x?

Accepted Answer

cos(x)

Answer

-sin(x)

Answer

sin(x)

Question 30

What is the slope of the tangent line to the graph of f(x) = 2x^2 + 1 at x = 2 ?

Accepted Answer

8

Answer

4

Answer

2

Answer

16

Question 31

Find the derivative of the function f(x) = (x^2 + 1) /(x-1) using the quotient rule.

Accepted Answer

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

Answer

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

Answer

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

Answer

(2x)/(x-1) - x

Question 32

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

Accepted Answer

3x^2 + 4x - 5

Answer

x^2 + 4x - 5

Answer

x^3 + 4x - 5

Answer

3x^2 + 2x - 5

Question 33

Find the derivative of f(x) = sin(x) + cos(x).

Accepted Answer

cos(x) - sin(x)

Answer

sin(x) - cos(x)

Answer

-sin(x) + cos(x)

Answer

-cos(x) - sin(x)

Question 34

Calculate the derivative of y = e^(3x).

Accepted Answer

3e^(3x)

Answer

3x * e^(3x)

Answer

x * e^(3x)

Answer

e^(3x)

Question 35

Find the maximum and minimum values of the function f(x) = -x^2 + 4x - 3 on the interval [0, 3].

Accepted Answer

Maximum: 5 at x = 2; Minimum: -3 at x = 0

Answer

Maximum: -3 at x = 3; Minimum: 5 at x = 0

Answer

Maximum: 7 at x = 3; Minimum: -3 at x = 0

Answer

Maximum: 5 at x = 0; Minimum: -3 at x = 2

Question 36

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

Accepted Answer

6x + 2

Answer

9x + 2

Answer

3x + 2

Answer

6x^2 + 2

Question 37

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

Accepted Answer

1

Answer

-3

Answer

3

Answer

-1

Question 38

If f(x) = sin(x), what is f'(x)?

Accepted Answer

cos(x)

Answer

-sin(x)

Answer

-cos(x)

Answer

sin(x)

Question 39

Which of the following functions has a derivative of 0 for all x?

Accepted Answer

f(x) = 5

Answer

f(x) = sin(x)

Answer

f(x) = x^2

Answer

f(x) = x

Question 40

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

Accepted Answer

6x(x^2 + 1)^2

Answer

3(x^2 + 1)^2

Answer

6x(x^2 + 1)

Answer

3x(x^2 + 1)^2

Question 41

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

Accepted Answer

x = 0 and x = 2

Answer

x = 0 and x = 1

Answer

x = 1 and x = 2

Answer

x = 1 and x = 3

Question 42

A particle moves along a straight line with its position given by s(t) = t^2 - 4t + 3. What is the particle's velocity at time t = 2?

Accepted Answer

0

Answer

4

Answer

-2

Answer

2

Question 43

The function f(x) = |x| is not differentiable at x = 0 because:

Accepted Answer

The left and right-hand derivatives do not agree at x = 0

Answer

The function is not continuous at x = 0

Answer

The function is not defined at x = 0

Question 44

What is the second derivative of the function f(x) = e^x?

Accepted Answer

e^x

Answer

x*e^x

Answer

e^(x^2)

Answer

-e^x

Question 45

The derivative of the product of two functions, u(x) and v(x), is given by:

Accepted Answer

u'(x)v(x) + u(x)v'(x)

Answer

u'(x)v'(x)

Answer

u(x)v'(x) - u'(x)v(x)

Question 46

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

Accepted Answer

2x - 3

Answer

2x

Answer

-3x + 2

Answer

x^2 - 3

Question 47

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

Accepted Answer

cos(x)

Answer

-sin(x)

Answer

sin(x)

Question 48

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

Accepted Answer

e^x

Answer

x * e^x

Answer

e

Answer

0

Question 49

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

Accepted Answer

1/x

Answer

ln(x)

Answer

-1/x

Answer

0

Question 50

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

Accepted Answer

1 for x > 0, -1 for x < 0

Answer

-x for x < 0

Answer

0

Answer

x for x > 0

Question 51

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

Accepted Answer

6x(x^2 + 2x)^2

Answer

(x^2 + 2x)^3

Answer

6x^2

Answer

0

Question 52

What is the derivative of the function f(x) = x^2 * sin(x)?

Accepted Answer

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

Answer

x^2 + sin(x)

Answer

2x * cos(x)

Answer

0

Question 53

Find the derivative of f(x) = 1 / (x^3 + 1).

Accepted Answer

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

Answer

0

Answer

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

Answer

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

Question 54

What is the derivative of the function f(x) = sqrt(x^2 - 1)?

Accepted Answer

x / sqrt(x^2 - 1)

Answer

0

Answer

2x / sqrt(x^2 - 1)

Answer

-x / sqrt(x^2 - 1)

Question 55

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

Accepted Answer

(x^2 - 4x + 2) / (x^2 - 1)^2

Answer

0

Answer

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

Answer

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