Unit 5

Looking at Rates of Change

Intro to Exponential Functions

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Lesson 10

HSF-IF.B.6: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Expressions and Equations

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Equivalent or Not?

Unit 5 ● Lesson 10

p(t) = 80(¾)ᵗ

-7.549

Notice: Exponential functions have different rates of change for different input intervals.

Warm-up

Page 254

Page 353

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Unit 5 ● Lesson 10

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

We can calculate the average rate of change of a function over a specified period of time so that we can make sense of and describe the relationship at certain intervals.

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Where were we? Where are we? Where are we going?

Unit 5 ● Lesson 10

Agenda Review

You are successful today when...,

● You can calculate the average rate of change of a function over a specified period of time.

● You know how the average rate of change of an exponential function differs from that of a linear function.

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

½

now better than 50%?!?

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

10.2 Activity: Coffee Shops

You can calculate the average rate of change of a function over a specified period of time.

15 mins Total

4 mins individual - 4 mins group - 3 mins class share

pg 354-5

1) Find the average rate of change for each period of time. Show your reasoning.

1. 1987 and 1990
2. 1987 and 1993
3. 1987 and 1997

2) Make some observations about the rates of change you calculated. What do these average rates tell us about how the company was growing during this time period?

3) Use the graph to support your answers to these questions. How well do the average rates of change describe the growth of the company in:

1. the first 3 years?
2. the first 6 years?
3. the entire 10 years?

4) Let 𝒇 be the 𝒇(t) function so that represents the number of stores t years since 1987. The value of 𝒇(20) is 15,011. Find [𝒇(20) - 𝒇(10)] / (20 - 10) and say what it tells us about the change in the number of stores.

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

10.2 Activity: Coffee Shops

You can calculate the average rate of change of a function over a specified period of time.

• The average rate of change of the first 6 years is almost double that of the first 3 years.
• The average rate of change of the 10 year time period is almost 7 times greater than that of the first 3 years and more than 3 times greater than the first 6 years.
• The company was growing at an increasing rate over the 10 year period.

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

10.2 Activity: Coffee Shops

You can calculate the average rate of change of a function over a specified period of time.

a. An average rate of change of about 22⅓. A poor prediction of growth much beyond 1992.

b. An average rate of change of about 42.5. It overestimates early on and greatly underestimates the growth in years after 1994.

c. An average rate of change of about 139.5 does not accurately predict the actual growth of the company between 1987 and 1997.

The average rate of growth from 1997 to 2007 is 1,359.9 stores per year. This is 60 times greater than the average rate of change over the first 3 years and almost 10 times greater than the average rate of change over the first 10 years of the time period. The number of stores continued to grow at an increasing rate from 1997 to 2007.

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

10.2 Activity: Coffee Shops

You can calculate the average rate of change of a function over a specified period of time.

• When we calculated the average rate of change for each of the three time periods, we were in effect finding the slope of the line that goes through two points that represent the starting year and the ending year.
• The line that connects the points for 1987 and 1990 fit the data for that period fairly well, so the slope of that line describes the growth in those three years fairly accurately.
• In contrast, the line that connects the points for 1987 and 1997, does not at all fit the data, so the slope of that line does not paint an accurate picture of how the company was growing that decade.
• Is there a single period of time whose average rate of change would well summarize how the company was growing from 1987 to 1997?

Since 1987

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

10.3 Activity: Revisiting Cost of Solar Cells

You can calculate the average rate of change of a function over a specified period of time.

10 mins Total

2 mins individual - 4 mins group - 3 mins class share

pg 356

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

10.3 Activity: Revisiting Cost of Solar Cells

You can calculate the average rate of change of a function over a specified period of time.

1) Clare said, "In the first five years, between 1977 and 1982, the cost fell by about $12 per year. But in the second five years, between 1983 and 1988, the cost fell only by about $2 a year." Show that Clare is correct.

2) If the trend continues, will the average decrease in price be more or less than $2 per year between 1987 and 1992? Explain your reasoning.

From the first 5 years to the next 5 years, the rate of decrease slowed down, so if the trend continues the decrease in price from 1987 to 1992 should be less than $2 per year.

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Looking at Rates of Change

You can calculate the average rate of change of a function over a specified period of time.

A decrease in value of about $300/yr.

No. The value decreases by almost $1,000 in the 1st year and very little in the fifth.

Yes, from year 1 to year 2 it is a reasonable estimate

Lesson Synthesis

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

An Average Rate of Change

You can calculate the average rate of change of a function over a specified period of time.

The average rate of change does not accurately describe how the mold changes over the 6 day period. The first day it only grows by 1 square millimeter while on the fifth day it grows by 32 square millimeters.

8

Cool-down

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Unit 5 ● Lesson 10

● I can calculate the average rate of change of a function over a specified period of time.

● I know how the average rate of change of an exponential function differs from that of a linear function.

Learning

Targets

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Glossary

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.

Slides are CC BY NC Kendall Hunt Publishing. Curriculum excerpts are CC BY Open Up Resources, with adaptations CC BY Illustrative Mathematics.