Q: Find the average rate of change of $$f(x) = \sqrt{3x + 4} on $$the interval $$[-1, 4].$$

$$\frac{\sqrt{3 \times 4 + 4} - \sqrt{3 \times (-1) + 4}}{4 - (-1)} = \frac{\sqrt{16} - \sqrt{1}}{5} = \frac{4 - 1}{5} = 0.6$$

Question 1

Given the graph of $$f(x)$$, is $$f$$ increasing or decreasing at $$x = -1$$? How do you know?

Accepted Answer

Decreasing, because the slope of the tangent line at $$x = -1 is $$negative.

Question 2

For the function $$f(x)$$ shown in the first graph, on which interval is the graph concave down?

Accepted Answer

$$(-2, 0)$$

Question 3

Estimate the $$x-$$value of the point of inflection for $$f(x) in $$the first graph. What does this point tell you about how $$f is $$changing?

Accepted Answer

$$x \approx 1$$, where the concavity changes from down to up, indicating the rate of change of $$f$$ switches from decreasing to increasing.

Question 4

Given the table for $$H(t)$$, the temperature of tea cooling, what is the average rate of change of $$H$$ from $$t = 0 to$$ $$t = 10$$? Include units.

Accepted Answer

$$\frac{43 - 66}{10 - 0} = -2.3$$ degrees Celsius per minute

Question 5

For $$f(x) = \sqrt{3x + 4}$$, which is greater: the instantaneous rate of change at $$x = 0 or at$$ $$x = 4$$?

Accepted Answer

At $$x = 0$$, because the derivative $$f'(x) = \frac{3}{2\sqrt{3x+4}}$$ decreases as $$x$$ increases.

Question 6

Find the average rate of change of $$f(x) = \sqrt{3x + 4} on $$the interval $$[-1, 4].$$

Accepted Answer

$$\frac{\sqrt{3 	imes 4 + 4} - \sqrt{3 	imes (-1) + 4}}{4 - (-1)} = \frac{\sqrt{16} - \sqrt{1}}{5} = \frac{4 - 1}{5} = 0.6$$

Question 7

Given the graph of $$g(x)$$, what is the average rate of change on the interval $$[0, 2]$$?

Accepted Answer

$$\frac{g(2) - g(0)}{2 - 0} = \frac{1 - 8}{2} = -3.5$$

Question 8

Order the following from least to greatest for $$g(x)$$: (1) Average rate of change on $$[0, 3.5]$$, (2) Instantaneous rate of change at $$x = 1$$, (3) Average rate of change on $$[1, 3]$$, (4) Instantaneous rate of change at $$x = 0.5.$$

Accepted Answer

1, 3, 2, 4

Question 9

Given the graph with points $$(-3, 2)$$ and $$(-1, -2)$$, what is the average rate of change of $$f on$$ $$[-3, -1]$$?

Accepted Answer

$$\frac{-2 - 2}{-1 - (-3)} = \frac{-4}{2} = -2$$

Question 10

Given the table for $$H(t)$$, what is the average rate of change from $$t = 2 to$$ $$t = 9$$? Explain your answer in context.

Accepted Answer

$$\frac{44 - 60}{9 - 2} = -2.29$$ degrees Celsius per minute; the tea cools by about $$2.29$$ degrees per minute on average during this interval.

Question 11

For the function $$f(x) = \sqrt{3x + 4}$$, write the equation of the secant line on the interval $$[-1, 4].$$

Accepted Answer

$$y - 1 = 0.6(x + 1)$$

Question 12

Using the secant line from the previous question, approximate $$f(-0.9).$$

Accepted Answer

$$y = 1 + 0.6(-0.9 + 1) = 1 + 0.6(0.1) = 1.06$$

Question 13

Is the approximation for $$f(-0.9)$$ using the secant line on $$[-1, 4] an $$overestimate or underestimate? Explain using concavity.

Accepted Answer

Overestimate, because $$f(x) is $$concave down on $$[-1, 4].$$

Question 14

Given the table for $$g(x)$$, find the average rate of change on $$[0, 4] if$$ $$g(0) = 2$$ and $$g(4) = 6.$$

Accepted Answer

$$\frac{6 - 2}{4 - 0} = 1$$

Question 15

Given the table for $$h(x)$$, if the average rate of change on $$[-1, 1] is $$constant, what type of function is $$h(x)$$? Justify your answer.

Accepted Answer

Linear, because a constant average rate of change indicates a straight line.

Question 16

Given the table for $$j(x)$$, if the average rate of change on $$[7, 14] is$$ $$-0.5$$ and $$j(x)$$ represents the depth of a hole being dug, what does this rate mean in context?

Accepted Answer

The depth of the hole decreases by $$0.5$$ feet per minute on average between $$7$$ and $$14$$ minutes.

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