AP Precalc 1.2 Rates of Change 1.
2 Notes
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and thoughts here!
Average Rate of Change
𝐜𝐡𝐚𝐧𝐠𝐞 𝐢𝐧 𝐨𝐮𝐭𝐩𝐮𝐭 𝐯𝐚𝐥𝐮𝐞𝐬 𝚫𝒚 𝒚𝟐 𝒚𝟏
𝐒𝐋𝐎𝐏𝐄
𝐜𝐡𝐚𝐧𝐠𝐞 𝐢𝐧 𝐢𝐧𝐩𝐮𝐭 𝐯𝐚𝐥𝐮𝐞𝐬 𝚫𝐱 𝒙𝟐 𝒙𝟏
1 1
𝑦 𝑥 3𝑥
1. Find the average rate of change between 1, 2 and 3, 4 . y 2 2
x
Instead of a graph, let’s say we have a table of values, and you want to calculate the average rate of
change on an interval. Be careful that the output values are always in the numerator.
2. What is the average rate of change on the interval from 13 to 25 seconds?
𝒕
10 13 19 25
time (seconds)
𝒅 𝒕
80 76 43 30
distance (meters)
The word per is a key indicator that you are dealing with a rate of change. Miles per gallon. Students per
classroom. Online gamers per server. It also tells you which variable is dependent or independent, as the
dependent is always listed first.
Rate of Change at a Point
What about the rate of change of a function at a point? This tells us about the rate at which the output
values would change were the input values to change at that point. We can approximate the rate of
change at a point by using the average rates of change over small intervals that contain the point.
3. Estimate the rate of change at 𝑥 1 for the function 𝑓 𝑥 𝑥 3𝑥 .
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and thoughts here! 4. Try it on your own! Find an approximation for the rate of change of 𝑔 at 𝑥 2 if 𝑔 𝑥 𝑒 .
Positive and Negative Rates of Change
A positive rate of change indicates that as one quantity increases or decreases, the
other quantity _________________.
A negative rate of change indicates that as one quantity increases, the other
__________________.
State whether the situation represents a positive or negative rate of change.
5. As the years increase, a high school student 6. As Mr. Bean’s weight decreases, his running
body increases. distances increase.
1.2 Rates of Change
AP Precalculus
1.2 Practice
Find the average rate of change of the function on the given interval.
1. ℎ 𝑡 3𝑥 𝑥 over the interval 2. 𝑏 𝑤 𝑤 2 over the interval 3. 𝑓 𝑥 ln 3𝑥 over the interval
2 𝑥 5. 1, 2 . 1 𝑥 4.
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Use the information in each table to find the average rate of change on the given interval.
4.
𝒕
3 10 21 43 55
Minutes
𝒅 𝒕
5 102 135 140 143
meters
a. 3 𝑡 55 b. 10 𝑡 43 c. 21 𝑡 55
5.
𝒕
6 12 24 48 72
months
𝒅 𝒕
20,000 19,800 15,000 10,000 7,500
hair follicles
a. 6 𝑡 48 b. 12 𝑡 72 c. 6 𝑡 72
Estimate the rate of change of each function at the given point.
6. 𝑓 𝑥 at 𝑥 4 7. 𝑓 𝑥 2𝑥 1 at 𝑥 2
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8. 𝑓 𝑥 7√𝑥 at 𝑥 2 9. 𝑓 𝑥 ln 2𝑥 at 𝑥 3
State whether the situation represents a positive or negative rate of change.
10. A candy company uses pints of chocolate to make 11. The amount of money is Josh’s savings account
candy. The more chocolate they use, the more boxes decreases for each semester he attends college.
of candy are produced.
12. As the number of cats Mr. Sullivan owns increases, 13. As the amount of water Mr. Brust drinks decreases,
the number of mice in his barn decreases. the fewer trips to the restroom he needs to make.
1.2 Rates of Change 1.2 Test Prep
14. A continuous function 𝑓 is defined on the closed interval y
5 𝑥 6 and is shown in the graph below. For how
many values of 𝑏, 5 𝑏 6, is the average rate of
change of 𝑓 on the interval 𝑏, 5 equal to 0? Give a reason
for your answer.
x
𝑓 𝑥
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