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.

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download