Question 1

Which formula correctly represents the average rate of change of a function $$f(x)$$ over the interval $$[x_1$, x_2$]$$?

Accepted Answer

$$\frac{f(x_2$) - f(x_1$)}{x_2$ - x_1$}$$

Question 2

Usain Bolt ran 100 meters in 9.58 seconds. What was his average velocity during the sprint?

Accepted Answer

$$10.44 m/s$$

Question 3

A car's distance function is $$f(t) = \frac{1}{480}t^2$, where t$ is in seconds and f(t$$)$$ is in miles. What is the average rate of change of f$ from t=0$ to t=4$?

Accepted Answer

$$\frac{1}{120}$$ miles/sec

Question 4

Given $$f(x) = x^2$, what is the average rate of change between x = 3$ and x = 4$?

Accepted Answer

$$7$$

Question 5

For $$f(x) = x^2$, estimate the instantaneous rate of change at x = 3$ using the average rate of change between x = 3$ and x = 3.01$.

Accepted Answer

$$6.01$$

Question 6

Which of the following best describes the instantaneous rate of change of a function at a point?

Accepted Answer

The slope of the tangent line to the function at that point.

Question 7

Given $$f(x) = 3x^2$, find the average rate of change between x = 1$ and x = 1 + h$.

Accepted Answer

$$6 + 3h$$

Question 8

For $$g(x) = \sqrt{x}$$, what is the instantaneous rate of change at $$x = 1$$?

Accepted Answer

$$\frac{1}{2}$$

Question 9

A chemical reaction's concentration $$C(t)$$ decreases according to the graph shown. If $$C(0.5) = 6$$ and $$C(2.5) = 1$$, what is the average rate of change of concentration between $$t = 0.5$$ and $$t = 2.5$$?

Accepted Answer

$$-2.5$$ units/time

Question 10

For $$h(x) = \frac{1}{x}$$, what is the instantaneous rate of change at $$x = 1$$?

Accepted Answer

$$-1$$

Question 11

A car accelerates from $$t = 1 to$$ $$t = 2$$ seconds following $$f(t) = \frac{1}{480}t^2$. What is the average rate of change of f$ over this interval?

Accepted Answer

$$\frac{3}{480}$$ miles/sec

Question 12

If $$f(x) = x^2$, what is the average rate of change between x = 3$ and x = 2$?

Accepted Answer

$$5$$

Question 13

Suppose $$f(x)$$ models the height of a ball thrown upward. What does the instantaneous rate of change of $$f at $$time $$t$$ represent?

Accepted Answer

The velocity of the ball at time $$t.$$

Question 14

Given $$f(x) = x^2$, what is the average rate of change between x = 3$ and x = 3.1$?

Accepted Answer

$$6.1$$

Question 15

For $$f(x) = x^2$, as h$$ 	o 0$$, what does the average rate of change formula $$\frac{f(1+h) - f(1)}{h}$$ approach?

Accepted Answer

$$2$$

Question 16

If the average rate of change of a function $$f$$ over $$[a, b] is $$negative, what does this indicate about $$f on $$that interval?

Accepted Answer

The function is decreasing on average over $$[a, b].$$

Question 17

A medication's concentration in the bloodstream is modeled by $$C(t). If $$the instantaneous rate of change at $$t = 2$$ hours is $$-0.5 mg/hr$$, what does this mean?

Accepted Answer

The concentration is decreasing at a rate of $$0.5 mg/hr at$$ $$t = 2$$ hours.

Question 18

Given $$f(x) = x^2$, what is the average rate of change between x = 3$ and x = 2.99$?

Accepted Answer

$$5.99$$

Question 19

For $$f(x) = x^2$, what is the instantaneous rate of change at x = 3$?

Accepted Answer

$$6$$

Question 20

If $$f(x) is a $$differentiable function, which of the following statements is true about the instantaneous rate of change at $$x = a$$?

Accepted Answer

It is equal to $$f'(a)$$, the derivative of $$f at$$ $$x = a.$$

Average and Instantaneous Rates of Change (Calculus Section 2.1 Study Notes)

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