Differential Calculus for Business: Master Limits, Derivatives, and Applications

Question 1

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

Accepted Answer

The rate of change of the function with respect to x

Answer

The second derivative of the function

Answer

The integral of the derivative

Answer

The area under the curve of the function

Question 2

Which of the following is not typically a step in finding the limit of a function?

Accepted Answer

Applying the Chain Rule

Answer

Substituting the value of x into the function

Answer

Simplifying the function algebraically

Answer

Factoring the function

Question 3

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

Accepted Answer

1/x

Answer

x

Answer

ln(x)

Answer

0

Question 4

What is the second derivative of the function f(x) = e^(2x)?

Accepted Answer

4e^(2x)

Answer

2e^(2x)

Answer

e^(2x)

Answer

0

Question 5

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

Accepted Answer

2e^(2x)

Answer

e^(2x)

Answer

x*e^(2x)

Answer

2xe^(2x)

Question 6

Determine the antiderivative of the function ∫(2x + 1) dx.

Accepted Answer

x^2 + x + C

Answer

2x^2 + x + C

Answer

x^2 + C

Answer

2x + C

Question 7

Calculate the area under the curve y = x^2 between the points x = 0 and x = 2.

Accepted Answer

4/3

Answer

2/3

Answer

8/3

Answer

16/3

Question 8

Determine the concavity of the function f(x) = -x^4 + 2x^2.

Accepted Answer

Concave down for all x

Answer

Concave up for all x

Answer

Concave up for x > 0 and concave down for x < 0

Answer

Concave down for x > 0 and concave up for x < 0

Question 9

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

Accepted Answer

3x² - 4x

Answer

x² - 4x + 5

Answer

x³ - 4x

Answer

3x² + 4x

Question 10

Identify the formula for the slope of the tangent line to a curve at a specific point (x₀, y₀).

Accepted Answer

f'(x₀)

Answer

f(x₀) / f'(x₀)

Answer

f(x₀) - f'(x₀)

Answer

f'(x₀) - f(x₀)

Question 11

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

Accepted Answer

2e^(2x)

Answer

e^(2x)

Answer

-e^(2x)

Answer

-2e^(2x)

Question 12

Given the function f(x) = x⁴ - 3x² + 2, find its critical points.

Accepted Answer

x = 0 and x = ±√3

Answer

x = 0

Answer

x = ±√3

Answer

No critical points

Question 13

Identify the type of discontinuity that occurs when the limit of a function exists but is not equal to the value of the function at that point.

Accepted Answer

Jump discontinuity

Answer

Removable discontinuity

Answer

Infinite discontinuity

Answer

Continuous discontinuity

Question 14

Find the vertical asymptote(s) of the function f(x) = (x - 2) / (x² - 4).

Accepted Answer

x = ±2

Answer

x = 2

Answer

x = -2

Answer

No vertical asymptotes

Question 15

Which method is appropriate for solving optimization problems using derivatives?

Accepted Answer

Any of the above

Answer

Graphical method

Answer

Algebraic method

Answer

Numerical method

Question 16

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

Accepted Answer

3x^2 + 4x

Answer

x^3 + 4x^2

Answer

x^2 + 4x - 1

Answer

3x - 2

Question 17

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

Accepted Answer

e^x

Answer

e^(x^2)

Answer

1/e^x

Answer

x*e^x

Question 18

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

Accepted Answer

1/x

Answer

x

Answer

ln(x^2)

Answer

1/ln(x)

Question 19

What is the derivative of the trigonometric function f(x) = sin(x)?

Accepted Answer

cos(x)

Answer

-sin(x)

Answer

tan(x)

Answer

sin(x)

Question 20

Find the derivative of the product function f(x) = x^2 * e^x

Accepted Answer

x^2 * e^x + 2x * e^x

Answer

2x * e^x

Answer

x^3 * e^x

Question 21

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

Accepted Answer

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

Answer

1 / (x - 1)^2

Answer

2x / (x - 1)

Answer

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

Question 22

Find the derivative of the absolute value function f(x) = |x|

Accepted Answer

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

Answer

1 for all x

Answer

0 for all x

Answer

-1 for all x

Question 23

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

Accepted Answer

1 / (1 + x^2)

Answer

-1 / (1 + x^2)

Answer

(1 + x^2) / x

Answer

arctan(x)