Lesson Plan #6
Lesson Plan #34
Class: AP Calculus Date: Thursday December 1st, 2011
Topic: Estimating the derivative of a function from a table using the difference quotient.
Aim: How can we estimate a derivative of a function from a table using the difference quotient?
Objectives:
1) Students will be able to estimate the derivative of a function from a table using the difference quotient
HW# 34: [pic]Show your work
Do Now:
1) Evaluate [pic]
2) Given the table below, find the average rate of change of [pic]over the interval [pic]
|[pic] |1 |2 |3 |4 |5 |
|[pic] |3 |7 |1 |8 |6 |
Procedure:
Write the Aim and Do Now
Get students working!
Take attendance
Give back work
Go over the HW
Collect HW
Go over the Do Now
Sketch a line segment joining the points on the curve at [pic]and
[pic].
The graph of the function [pic]on [pic]is shown at right. Find the
slope of the line segment joining the points at [pic]and [pic].
For what can we use this slope as an approximation?
We can approximate the slope of the tangent line at [pic]by using the difference quotient.
Instead of using the line segment between 2 and 4, we could have chose the line segment between 2 and 3 or the line segment between 3 and 4 to approximate [pic]
Recall the definition of the derivative:
Sometimes we are given a table of values instead of a function. Without the function, we can’t calculate an exact value of the derivative, but we can get an estimate of the derivative using the difference quotient[pic]
For example, take a look at the table below.
Andrews’s height on the last day of summer for consecutive years when his age was 9 through 17
|Age |Height (in inches) |
|9 |46 |
|10 |47 |
|11 |48 |
|12 |50 |
|13 |53 |
|14 |58 |
|15 |62 |
|16 |67 |
|17 |69 |
Give an approximation for the rate of Andrew’s growth when he was 15?
Solution:
We can approximate the derivative using values that are close to 15, so we could do
[pic]or [pic] or the average of those two difference quotients called the symmetric difference quotient or
[pic]
Additional questions on the above table.
1) What is the average rate of Andrew’s growth between the ages of 9 and 17?
Example 1:
1) From the values of [pic]shown, estimate [pic].
|[pic] |1.92 |1.94 |1.96 |1.98 |2.00 |
|[pic] |6.00 |5.00 |4.40 |4.10 |4.00 |
Let [pic]be a continuous and differentiable function. Selected values of [pic]are shown below. Find the approximate value of [pic]at [pic].
|[pic] |-2 |-1 |0 |1 |2 |3 |
|[pic] |1 |0 |1 |1.59 |2.08 |2.52 |
World AIDS Day Activity:
Sample Test Questions:
1) A car driven on a test track have positive velocity [pic]. At time [pic]seconds, the car is at a starting line denoted [pic]. The velocity of the car, in feet per minute, is recorded for selected values of [pic]over the interval [pic]minutes as shown in the table. Find the average acceleration of the car over the interval [pic]minutes. Indicated unites of measure.
[pic]0102030405060708090[pic]80012002500320040003100250020001500900
3)
-----------------------
| |0 |1 |4 |9 |16 |
| | | | | | |
|[pic] | | | | | |
| | | | | | |
|[pic] | | | | | |
| | | | | | |
|[pic] | | | | | |
| | | | | | |
|Question: | | | | | |
|Approximate| | | | | |
|the rate of| | | | | |
|change of | | | | | |
|AIDS | | | | | |
|diagnoses | | | | | |
|in 2009 in | | | | | |
|New York | | | | | |
|City. | | | | | |
| | | | | | |
| | | | | | |
|[pic] | | | | | |
| | | | | | |
|[pic] | | | | | |
| | | | | | |
| | | | | | |
| | | | | | |
| | | | | | |
| | | | | | |
| | | | | | |
| | | | | | |
|2) The | | | | | |
|number of | | | | | |
|passengers | | | | | |
|in a subway| | | | | |
|car [pic] | | | | | |
|minutes | | | | | |
|after it | | | | | |
|comes into | | | | | |
|service is | | | | | |
|modeled by | | | | | |
|the | | | | | |
|function | | | | | |
|[pic]for | | | | | |
|[pic]. | | | | | |
|Values of | | | | | |
|[pic]are | | | | | |
|given in | | | | | |
|the table | | | | | |
|below. | | | | | |
|Estimate | | | | | |
|the rate at| | | | | |
|which the | | | | | |
|number of | | | | | |
|passengers | | | | | |
|is changing| | | | | |
|at | | | | | |
|[pic]minute| | | | | |
|s. SHOW the| | | | | |
|calculation| | | | | |
|s that lead| | | | | |
|to your | | | | | |
|answer and | | | | | |
|indicate | | | | | |
|the units. | | | | | |
|[pic] | | | | | |
|[pic] |1 |5 |11 |1 |0 |
Question:
Approximate the rate of people living with AIDS in 2009 in New York City.
1) Given the table below approximate [pic].
|[pic] |[pic] |
|2 |1.3 |
|3 |1.6 |
|5 |2.9 |
|6 |2.8 |
Questions:
Approximate the rate of change of deaths to persons with AIDS in 2009 in New York City.
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