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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