Question

1. find the average rate of change for each of the following graphs over the given interval.

a. -2,0
b. 2,6
c. 0, - 2
d. -2,2

1. find the average rate of change for each of the functions over the given interval.

a. f(x)=x² - 8x on 1,6
b. g(x)=5 - x/4 on 1,4
c. h(x)=x² - 2 on -4,-1
d. s(t)=(t + 3)² - 4 on -3,5

1. the rate at which people enter an art museum, in people per hours, is modeled by the function r. the graph of y = r(t) is shown below, where t is measured in hours after the museum opens.

a. identify the interval(s) on which r is increasing. what does this mean in context of the problem?
b. identify the interval(s) on which r is constant. what does this mean in context of the problem?
c. identify the maximum value of r and interpret its meaning in the context of this problem.
d. when is r(t)=0? what does this mean in the context of this problem?

Explanation:

2A Step1: Calculate f(6)

$f(6) = 6^3 - 8 \times 6 = 216 - 48 = 168$

2A Step2: Calculate f(1)

$f(1) = 1^3 - 8 \times 1 = 1 - 8 = -7$

2A Step3: Compute average rate

$\frac{f(6)-f(1)}{6-1} = \frac{168 - (-7)}{5} = 35$

2B Step1: Calculate g(4)

$g(4) = 5 - \frac{4^2}{4} = 5 - 4 = 1$

2B Step2: Calculate g(1)

$g(1) = 5 - \frac{1^2}{4} = 5 - 0.25 = 4.75$

2B Step3: Compute average rate

$\frac{g(4)-g(1)}{4-1} = \frac{1 - 4.75}{3} = -1.25$

2C Step1: Calculate h(-1)

$h(-1) = (-1)^2 - 2 = 1 - 2 = -1$

2C Step2: Calculate h(-4)

$h(-4) = (-4)^2 - 2 = 16 - 2 = 14$

2C Step3: Compute average rate

$\frac{h(-1)-h(-4)}{-1 - (-4)} = \frac{-1 - 14}{3} = -5$

2D Step1: Calculate s(5)

$s(5) = (5+3)^2 - 4 = 64 - 4 = 60$

2D Step2: Calculate s(-3)

$s(-3) = (-3+3)^2 - 4 = 0 - 4 = -4$

2D Step3: Compute average rate

$\frac{s(5)-s(-3)}{5 - (-3)} = \frac{60 - (-4)}{8} = 8$

Answer:

2A. 35, 2B. -1.25, 2C. -5, 2D. 8