Master Differentiation Applications: Enhance Your Calculus Skills

Question 1

Find the derivative of the function f(x) = x^3 using the power rule of differentiation.

Accepted Answer

3x²

Answer

x²

Answer

3x

Answer

1

Question 2

Use integration to find the area under the curve y = x² from x = 0 to x = 2.

Accepted Answer

4/3

Answer

2/3

Answer

8/3

Answer

16/3

Question 3

Analyze the function f(x) = x^3 - 3x² + 2x + 1 to determine its global maximum and global minimum.

Accepted Answer

The global maximum is at x = 1 and the global minimum is at x = 0.

Answer

The global maximum is at x = 0 and the global minimum is at x = 1.

Answer

The function has no global maximum or minimum.

Answer

The global maximum is at x = 2 and the global minimum is at x = -1.

Question 4

Determine the intervals where the function f(x) = x^4 + 2x^3 - 3x + 2 is concave upward or concave downward.

Accepted Answer

Concave upward for x < -1 and x > 1, concave downward for -1 < x < 1

Answer

Concave upward on the entire domain

Answer

Concave downward on the entire domain

Answer

Concave downward for x < -1 and x > 1, concave upward for -1 < x < 1

Question 5

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

Accepted Answer

x = -1, 0, 1

Answer

x = 0, 1, 2

Answer

x = -2, -1, 0

Answer

x = 1, 2, 3

Question 6

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

Accepted Answer

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

Answer

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

Answer

2x/(x + 1)^2

Answer

-2x/(x + 1)^2

Question 7

Which condition is necessary for a function to have a relative maximum or minimum?

Accepted Answer

The first derivative is zero at that point.

Answer

The second derivative is positive at that point.

Answer

The second derivative is negative at that point.

Answer

The function is continuous at that point.

Question 8

Which of the following is NOT a practical application of differentiation?

Accepted Answer

Predicting stock market trends

Answer

Determining the slope of a tangent line to a curve

Answer

Calculating the velocity of a moving object

Answer

Finding the area under a curve

Question 9

Which of the following is the equation of the normal line to the curve y = x^3 - 2x at the point (1, -1)?

Accepted Answer

y = -x + 0

Answer

y = x - 2

Answer

y = x^3 - 2x

Answer

y = 3x - 4

Question 10

A rectangular garden is to be fenced on three sides, with the fourth side bounded by a wall. If the garden is to have an area of 100 square feet, what dimensions will minimize the amount of fencing required?

Accepted Answer

Width = 10 feet; Length = 20 feet

Answer

Width = 25 feet; Length = 15 feet

Answer

Width = 20 feet; Length = 10 feet

Answer

Width = 5 feet; Length = 40 feet

Question 11

The velocity of a particle moving along a straight line is given by v(t) = 3t^2 - 4t + 2 m/s, where t is the time in seconds. Find the position function s(t) such that s(0) = 1.

Accepted Answer

s(t) = t^3 - 2t^2 + 2t + 1

Answer

s(t) = t^3 - 4t^2 + 2t + 1

Answer

s(t) = 3t^2 - 4t + 1

Answer

s(t) = 3t^3 - 4t^2 + 2t + 1

Question 12

Which of the following is NOT a direct application of differentiation?

Accepted Answer

Integration

Answer

Determining concavity of a function

Answer

Calculating the instantaneous velocity of an object

Answer

Finding the slope of a tangent line

Question 13

The derivative of the position function of an object with respect to time represents its:

Accepted Answer

Velocity

Answer

Displacement

Answer

Acceleration

Answer

Speed

Question 14

What is the slope of the line tangent to the curve y = x^2 at the point where x = 2?

Accepted Answer

4

Answer

1

Answer

2

Answer

8

Question 15

The derivative of the function y = e^x is:

Accepted Answer

e^x

Answer

1/e^x

Answer

e^x/x

Answer

x*e^x

Question 16

What is the second derivative of the function y = x^3 + 2x^2 - 1?

Accepted Answer

6x + 4

Answer

x^2 + 4x

Answer

3x^2 + 4x

Answer

x^3 + 4x^2

Question 17

A farmer wants to enclose a rectangular field using 500 feet of fencing. What dimensions will maximize the area of the field?

Accepted Answer

125 ft by 125 ft

Answer

100 ft by 150 ft

Answer

50 ft by 200 ft

Answer

250 ft by 250 ft

Question 18

The position of a particle moving along a straight line is given by s(t) = t³ - 6t² + 9t + 2, where t is measured in seconds. What is the particle's velocity at t = 2 seconds?

Accepted Answer

-3 units/second

Answer

3 units/second

Answer

0 units/second

Answer

9 units/second

Question 19

Find the equation of the tangent line to the curve y = x² - 4x + 3 at the point (2, -1).

Accepted Answer

y = 0x - 1

Answer

y = 4x - 9

Answer

y = -2x + 3

Answer

y = 2x - 5

Question 20

A spherical balloon is being inflated at a rate of 10 cubic centimeters per second. How fast is the radius increasing when the radius is 5 centimeters?

Accepted Answer

1/(5π) cm/sec

Answer

1/π cm/sec

Answer

5π cm/sec

Answer

50π cm/sec

Question 21

Determine the critical points of the function f(x) = x³ - 3x².

Accepted Answer

x = 0, x = 2

Answer

x = 0 only

Answer

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

Answer

x = 2 only

Question 22

What is the absolute maximum value of the function f(x) = -x² + 4x on the interval [0, 3]?

Accepted Answer

4

Answer

3

Answer

0

Question 23

Which of the following is NOT a direct application of differentiation?

Accepted Answer

Finding the sum of an infinite geometric series

Answer

Optimizing the dimensions of a rectangular box

Answer

Calculating the instantaneous velocity of an object

Answer

Determining the maximum height of a projectile

Question 24

The function f(x) = x³ - 6x² + 9x is concave up on which interval?

Accepted Answer

(2, ∞)

Answer

(1, 2)

Answer

(-∞, 1)

Answer

(-∞, ∞)

Question 25

Approximate the value of √17 using linear approximation, starting with the point x = 16.

Accepted Answer

4.125

Answer

4.25

Answer

4

Answer

4.5

Question 26

A rectangular piece of cardboard with dimensions 10 inches by 15 inches is used to make an open box by cutting out squares of equal size from each corner and folding up the sides. What size squares should be cut out to maximize the volume of the box?

Accepted Answer

5/3 inches by 5/3 inches

Answer

5 inches by 5 inches

Answer

2 inches by 2 inches

Answer

3 inches by 3 inches