# 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:

- day 0 to day 5
- 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.

\(b>1\)

\(b<1\)

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.

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.

\(a > 0\)

### 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.

\(0 \leq w \leq 2\)

\(0 \leq w \leq 7\)

\(5 \leq w \leq 7\)

\(5 \leq w \leq 10\)

\(0 \leq w \leq 10\)

### 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.

### 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.

- What was the moth population when it was first measured?
- What was the moth population after 1 week? What about 1.5 weeks?
- Use technology to graph the population and find out when it falls below 10,000.

### 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.

### 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.

### 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.