Question 1

Given the function $$f(x) = x^2$ + 4$$, what is the average rate of change of $$f(x)$$ between $$x = 2$$ and $$x = 5$$?

Accepted Answer

$$9$$

Question 2

For the function $$f(x) = \dfrac{3}{x-1}$$, what is the average rate of change between $$x = 2$$ and $$x = 5$$?

Accepted Answer

$$-\dfrac{1}{3}$$

Question 3

Find the instantaneous rate of change of $$f(x) = \dfrac{4}{x+5} at$$ $$x = -1.$$

Accepted Answer

$$-\dfrac{2}{3}$$

Question 4

The distance in feet of an object from a starting point is given by $$s(t) = 2t^2$ - 5t + 40$$, where $$t is in $$seconds. What is the average velocity of the object from $$t = 2$$ seconds to $$t = 4$$ seconds?

Accepted Answer

$$9 ft/sec$$

Question 5

Given $$s(t) = 2t^2$ - 5t + 40$$, what is the instantaneous velocity at $$t = 4$$ seconds?

Accepted Answer

$$11 ft/sec$$

Question 6

A company’s total cost for manufacturing $$x$$ cases of t-shirts is $$C(x) = 50 + 75x - x^2$ for 0$$ \leq x \leq 10$$. What is the average change in total cost per case if production increases from 1 to 5 cases?

Accepted Answer

$$51$$

Question 7

For $$C(x) = 50 + 75x - x^2$, what is the additional cost when production increases from 1 to 2 cases?

Accepted Answer

$$72$$

Question 8

Given $$C(x) = 50 + 75x - x^2$, what is the instantaneous rate of change of cost with respect to the number of cases produced when just one case is produced?

Accepted Answer

$$73$$

Question 9

Suppose the cost function for producing t-shirts is $$C(x) = 50 + 75x - x^2$. If the company wants to minimize the cost per case, at what number of cases is the instantaneous rate of change of cost zero?

Accepted Answer

$$x = 37.5 ($$not in domain),$$x = 37.5 is $$outside the allowed range, so the minimum occurs at $$x = 10$$

Question 10

A company’s cost function is $$C(x) = 50 + 75x - x^2$. If the company produces 3 cases, what is the marginal cost (instantaneous rate of change of cost) at x = 3$?

Accepted Answer

$$69$$

Question 11

Given $$f(x) = x^2$ + 4$$, which of the following best describes the meaning of the average rate of change between $$x = a$$ and $$x = b$$?

Accepted Answer

The slope of the secant line connecting $$(a, f(a))$$ and $$(b, f(b))$$

Question 12

If $$f(x) = \dfrac{4}{x+5}$$, what is the derivative $$f'(x)$$?

Accepted Answer

$$-\dfrac{4}{(x+5)^2$}$$

Question 13

A company’s cost function is $$C(x) = 50 + 75x - x^2$. If the company increases production from x = 4$ to x = 5$, what is the additional cost?

Accepted Answer

$$66$$

Question 14

Given $$s(t) = 2t^2$ - 5t + 40$$, what is the formula for the instantaneous velocity at any time $$t$$?

Accepted Answer

$$4t - 5$$

Question 15

Suppose $$f(x) = x^2$ + 4. If $$the average rate of change from $$x = a to$$ $$x = b is$$ $$12$$, and $$a = 1$$, what is $$b$$?

Accepted Answer

$$4$$

Question 16

A company’s cost function is $$C(x) = 50 + 75x - x^2$. If the company produces 0 cases, what is the marginal cost at x = 0$?

Accepted Answer

$$75$$

Question 17

Given $$f(x) = \dfrac{3}{x-1}$$, what is the derivative $$f'(x)$$?

Accepted Answer

$$-\dfrac{3}{(x-1)^2$}$$

Question 18

A company’s cost function is $$C(x) = 50 + 75x - x^2$. If the company produces 10 cases, what is the marginal cost at x = 10$?

Accepted Answer

$$-25$$

Question 19

Suppose $$s(t) = 2t^2$ - 5t + 40$$ models the position of a car in feet after $$t$$ seconds. If the car starts at $$t = 0$$, how far has it traveled after 3 seconds?

Accepted Answer

$$49 ft$$

Business Calculus Practice: Rates of Change, Average and Instantaneous Velocity, and Cost Analysis

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