# Lesson 10

Looking at Rates of Change

Let's calculate average rates of change for exponential functions.

### Problem 1

A store receives 2,000 decks of popular trading cards. The number of decks of cards is a function, $$d$$, of the number of days, $$t$$, since the shipment arrived. Here is a table showing some values of $$d$$.

$$t$$ $$d(t)$$
0 2,000
5 1,283
10 823
15 528
20 338

Calculate the average rate of change for the following intervals:

1. day 0 to day 5
2. day 15 to day 20

### Problem 2

A study was conducted to analyze the effects on deer population in a particular area. Let $$f$$ be an exponential function that gives the population of deer $$t$$ years after the study began.

If $$f(t)=a \boldcdot b^t$$ and the population is increasing, select all statements that must be true.

A:

$$b>1$$

B:

$$b<1$$

C:

The average rate of change from year 0 to year 5 is less than the average rate of change from year 10 to year 15.

D:

The average rate of change from year 0 to year 5 is greater than the average rate of change from year 10 to year 15.

E:

$$a > 0$$

### Problem 3

Function $$f$$ models the population, in thousands, of a city $$t$$ years after 1930.

The average rate of change of $$f$$ from $$t=0$$ to $$t=70$$ is approximately 14 thousand people per year.

Is this value a good way to describe the population change of the city over that time period? Explain or show your reasoning.

### Problem 4

The function, $$f$$, gives the number of copies a book has sold $$w$$ weeks after it was published. The equation $$f(w) = 500 \boldcdot 2^w$$ defines this function.

Select all domains for which the average rate of change could be a good measure for the number of books sold.

A:

$$0 \leq w \leq 2$$

B:

$$0 \leq w \leq 7$$

C:

$$5 \leq w \leq 7$$

D:

$$5 \leq w \leq 10$$

E:

$$0 \leq w \leq 10$$

### Problem 5

The graph shows a bacteria population decreasing exponentially over time.

The equation $$p = 100,\!000,\!000 \boldcdot \left(\frac{2}{3}\right)^h$$ gives the size of a second population of bacteria, where $$h$$ is the number of hours since it was measured at 100 million.

Which bacterial population decays more quickly? Explain how you know.

​​​​​​

(From Unit 5, Lesson 6.)

### Problem 6

Technology required. A moth population, $$p$$, is modeled by the equation $$p = 500,\!000 \boldcdot \left(\frac{1}{2}\right)^w$$, where $$w$$ is the number of weeks since the population was first measured.

1. What was the moth population when it was first measured?
2. What was the moth population after 1 week? What about 1.5 weeks?
3. Use technology to graph the population and find out when it falls below 10,000.
(From Unit 5, Lesson 9.)

### Problem 7

Give a value for $$r$$ that would indicate that a line of best fit has a positive slope and models the data well.

(From Unit 3, Lesson 7.)

### Problem 8

The size of a district and the number of parks in it have a weak positive relationship.

Explain what it means to have a weak positive relationship in this context.

(From Unit 3, Lesson 8.)

### Problem 9

Here is a graph of Han’s distance from home as he drives.

Identify the intercepts of the graph and explain what they mean in terms of Han’s distance from home.