Question 1

Suppose the position of a car moving along a straight road is given by $s(t) = 3t^2 + 2t + 1$, where $s$ is in meters and $t$ is in seconds. What is the average rate of change of the car's position from $t = 2$ to $t = 5$?

Accepted Answer

$23$

Question 2

Given the function $f(x) = x^3 - 6x^2 + 9x + 2$, what is the instantaneous rate of change of $f$ at $x = 2$?

Accepted Answer

$-2$

Question 3

A biologist is tracking the population of a bacteria culture, modeled by $P(t) = 100e^{0.05t}$, where $t$ is in hours. What is the average rate of change of the population from $t = 0$ to $t = 10$?

Accepted Answer

$100(e^{0.5} - 1)$

Question 4

Which of the following best describes the difference between average rate of change and instantaneous rate of change for a function $f(x)$?

Accepted Answer

Average rate of change is the slope of the secant line, while instantaneous rate of change is the slope of the tangent line.

Question 5

If the average rate of change of $g(x)$ from $x = 1$ to $x = 4$ is $5$, and $g(1) = 2$, what is $g(4)$?

Accepted Answer

$17$

Question 6

A chemical reaction's temperature in degrees Celsius is modeled by $T(t) = 20 + 4t - t^2$, where $t$ is in minutes. What is the instantaneous rate of change of temperature at $t = 3$?

Accepted Answer

$-2$

Question 7

Given $h(x) = \sin(x)$, what is the average rate of change of $h$ from $x = 0$ to $x = \pi$?

Accepted Answer

$0$

Question 8

A company’s profit in thousands of dollars is given by $P(t) = 5t^2 - 20t + 30$, where $t$ is the number of years since founding. What is the average rate of change of profit from year $t = 1$ to $t = 4$?

Accepted Answer

$-5$

Question 9

Which of the following statements is TRUE about the instantaneous rate of change of a function at a point?

Accepted Answer

It is the slope of the tangent line to the function at that point.

Question 10

A medication’s concentration in the bloodstream is modeled by $C(t) = 50e^{-0.1t}$, where $t$ is in hours. What is the average rate of change of concentration from $t = 0$ to $t = 5$?

Accepted Answer

$\frac{50(e^{-0.5} - 1)}{5}$

Question 11

If $f(x)$ is a function and the average rate of change from $x = a$ to $x = b$ is $k$, which equation must be true?

Accepted Answer

$f(b) - f(a) = k(b - a)$

Question 12

A runner’s distance from the starting line is given by $d(t) = 12t - t^2$, where $t$ is in seconds. What is the instantaneous velocity at $t = 6$ seconds?

Accepted Answer

$-12$

Question 13

Which of the following real-life scenarios best illustrates the concept of instantaneous rate of change?

Accepted Answer

Measuring the speed of a car at exactly 1 hour into the trip using a speedometer.

Question 14

Given $f(x) = 2x^2 - 3x + 1$, what is the average rate of change from $x = 1$ to $x = 4$?

Accepted Answer

$7$

Question 15

A function $f(x)$ has an average rate of change of $8$ from $x = 2$ to $x = 6$. If $f(2) = 5$, what is $f(6)$?

Accepted Answer

$37$

Question 16

If the graph of a function $y = f(x)$ is a straight line, what can be said about its average and instantaneous rates of change?

Accepted Answer

They are always equal at every point.

Rates of Change in Calculus: Average and Instantaneous Rate of Change

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