Question 1

Suppose the population of a city grows from 1.2 million in 2020 to 1.5 million in 2023. What is the average rate of change of the population (in millions per year) between 2020 and 2023?

Accepted Answer

$$0.1$$

Question 2

Given the function $$f(x) = x^2$ - 4x + 7$$, what is the average rate of change of $$f(x)$$ from $$x = 1 to$$ $$x = 5$$?

Accepted Answer

$$\frac{f(5) - f(1)}{5 - 1} = \frac{12 - 4}{4} = 2$$

Question 3

A company’s revenue is modeled by $$R(t) = 100(1.05)^t (in $$millions of dollars), where $$t is $$years since 2020. What is the average rate of change in revenue from 2020 to 2023?

Accepted Answer

$$\frac{R(3) - R(0)}{3} = \frac{100(1.05)^3$ - 100}{3} \approx \frac{115.76 - 100}{3} \approx 5.25$$

Question 4

The position of a car is given by $$s(t) = 4t^2$ - 3t + 2 (in $$meters), where $$t is in $$seconds. What is the instantaneous velocity at $$t = 2$$ seconds?

Accepted Answer

$$s'(2) = 8t - 3 at$$ $$t=2$$ gives $$8(2) - 3 = 13 m/s$$

Question 5

If the cost to produce $$x$$ units is $$C(x) = 50x + 200$$, what is the marginal cost when $$x = 10$$?

Accepted Answer

$$C'(x) = 50$$, so marginal cost is $$50$$

Question 6

Suppose the profit from selling $$x$$ units is $$P(x) = 10x - x^2$. What is the marginal profit when x = 3$?

Accepted Answer

$$P'(x) = 10 - 2x$$, so at $$x=3$$, $$P'(3) = 10 - 6 = 4$$

Question 7

A medication’s concentration in the bloodstream is modeled by $$C(t) = 20e$$^{-0.2t}$$ (mg/L), where t$ is hours after administration. What is the average rate of change in concentration from t=0$ to t=5$?

Accepted Answer

$$\frac{C(5) - C(0)}{5} = \frac{20e$$^{-1}$$ - 20}{5} \approx \frac{7.36 - 20}{5} = -2.53$$

Question 8

The cost function for producing $$x$$ smartphones is $$C(x) = 1000 + 150x + 0.5x^2$. What is the marginal cost when x = 20$?

Accepted Answer

$$C'(x) = 150 + x$$, so at $$x=20$$, $$C'(20) = 170$$

Question 9

A company’s spending is modeled by $$S(t) = 80(1.037)^t (in $$billions of dollars), where $$t=0 is 2018. $$What is the average rate of change in spending from 2018 to 2023?

Accepted Answer

$$\frac{S(5) - S(0)}{5} = \frac{95.94 - 80}{5} = 3.19$$

Question 10

Suppose the position of an object is $$s(t) = 5t^2$ - 2t - 7 (in $$feet), where $$t is in $$seconds. What is the instantaneous velocity at $$t = 3$$?

Accepted Answer

$$s'(t) = 10t - 2$$, so at $$t=3$$, $$s'(3) = 28 ft/s$$

Question 11

If the profit function is $$P(x) = 20x - 0.5x^2$, what is the marginal profit when x = 10$?

Accepted Answer

$$P'(x) = 20 - x$$, so at $$x=10$$, $$P'(10) = 10$$

Question 12

A tech market grew from $$134$$ billion in 2023 to $$184$$ billion in 2024. What is the average rate of change in billions of dollars per year?

Accepted Answer

$$\frac{184 - 134}{1} = 50$$

Question 13

The total cost to produce $$x$$ units is $$C(x) = 500 + 20x. $$What does the marginal cost represent in this context?

Accepted Answer

The additional cost to produce one more unit

Question 14

If $$f(x) = 3x^2$ + 2x$$, what is the difference quotient $$\frac{f(a+h) - f(a)}{h}$$?

Accepted Answer

$$\frac{3(a+h)^2$ + 2(a+h) - [3a^2$ + 2a]}{h}$$

Question 15

A company’s profit from selling $$x$$ units is $$P(x) = 5x^2$ - 3x + 10. $$What is the marginal profit when $$x = 4$$?

Accepted Answer

$$P'(x) = 10x - 3$$, so at $$x=4$$, $$P'(4) = 37$$

Question 16

Which of the following best describes the difference between average and instantaneous rate of change?

Accepted Answer

Average rate of change measures change over an interval; instantaneous rate of change measures change at a single point.

Question 17

Suppose $$f(x) = x^3$. What is the instantaneous rate of change at x = 2$?

Accepted Answer

$$f'(x) = 3x^2$, so at x=2$, f$$'(2) = 12$$

Rates of Change: Average and Instantaneous (Section 3.3 Study Notes)

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