Question 1

Which of the following best describes the meaning of the derivative $$f'(x) of a $$function $$f(x) at a $$point $$x$$?

Accepted Answer

It gives the slope of the tangent line to the graph at $$x.$$

Question 2

Given $$f(x) = 5x + 3$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = 5$$

Question 3

If $$f(x) = x^4$, what is f$$'(x)$$?

Accepted Answer

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

Question 4

Which of the following is NOT a correct notation for the derivative of $$y = f(x)$$?

Accepted Answer

$$f(x)'$$

Question 5

Suppose $$f(x) = x$$^{-2}$$. What is f$$'(x)$$?

Accepted Answer

$$f'(x) = -2x$$^{-3}$$

Question 6

A company models its profit by $$P(x) = 2x^2$ + 3x + 5$$, where $$x is $$the number of units sold. What is the instantaneous rate of change of profit with respect to units sold?

Accepted Answer

$$P'(x) = 4x + 3$$

Question 7

Find the equation of the tangent line to $$f(x) = x^2$ at x = 3$.

Accepted Answer

$$y = 6x - 9$$

Question 8

If $$f(x) = \sqrt{x}$$, what is the slope of the tangent line at $$x = 4$$?

Accepted Answer

$$\frac{1}{4}$$

Question 9

Given $$f(x) = \frac{1}{x}$$, what is the equation of the tangent line at $$x = 2$$?

Accepted Answer

$$y = -\frac{1}{4}x + 1$$

Question 10

If $$f(x) = x^r$$, where $$r is $$any real number, what is the general formula for $$f'(x)$$?

Accepted Answer

$$f'(x) = r x$$^{r-1}$$

Question 11

A medication's concentration in the bloodstream is modeled by $$C(t) = 8 - 2t$$, where $$t is $$time in hours. What is the rate at which the concentration is changing?

Accepted Answer

$$C'(t) = -2$$

Question 12

If $$f(x) = 7$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = 0$$

Question 13

Which step is necessary to find the equation of the tangent line to $$y = f(x) at$$ $$x = a$$?

Accepted Answer

Evaluate $$f'(a) to $$find the slope.

Question 14

If $$f(x) = x$$^{1/2}$$, what is f$$'(x)$$?

Accepted Answer

$$f'(x) = \frac{1}{2}x$$^{-1/2}$$

Question 15

A population grows according to $$P(t) = 100e$$^{0.05t}$$. What is the instantaneous rate of change of the population at time t$?

Accepted Answer

Question 16

If $$f(x) = x^3$, what is the equation of the tangent line at x = 2$?

Accepted Answer

$$y = 12x - 32$$

Question 17

Which of the following statements is true about the derivative of a constant function $$f(x) = b$$?

Accepted Answer

The derivative is always $$0.$$

Question 18

If $$f(x) = \frac{5}{x^3$}$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = -15x$$^{-4}$$

Question 19

A car's position is given by $$s(t) = 4t^2$ + 2t. $$What is the car's instantaneous velocity at time $$t$$?

Accepted Answer

$$s'(t) = 8t + 2$$

Business Calculus: The Derivative and Tangent Lines

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