4.4: Linear Approximation
This activity is intended to supplement Calculus, Chapter 3, Lesson 8.
ID: 9470
Time required: 45 minutes
Activity Overview
In this activity, students will graph the functions, construct the tangent line at a point, and find an estimate from a linear approximation. They will then determine whether the estimate is an overestimate or an underestimate, and find an interval for a desired accuracy.
Topic: Application of Derivatives
 Calculate the equation of the tangent line to a graph at any given point.

Construct a tangent line to the graph of a differentiable function at
x=a to approximate its value nearx=a . 
Construct a tangent line to the graph of a differentiable function at
x=0 to approximate its value nearx=0 .
Teacher Preparation and Notes

This investigation offers an opportunity for students to develop an understanding of how a tangent line to a curve can be used to approximate the values of a function near the point of tangency. Linear approximations are often used in scientific applications, including formulas used in physics where
sinθ is replaced with its linear approximation,θ .  This activity is designed to be studentcentered with the teacher acting as a facilitator while students work cooperatively. The student worksheet is intended to guide students through the main ideas of the activity and provide a place for them to record their observations.
 The students will need to be able to enter the functions and use the commands on their own.

Before starting this activity, students should go to the home screen and select
F6 :Clean Up > 2:NewProb, then press ENTER. This will clear any stored variables, turn off any functions and plots, and clear the drawing and home screens.
Associated Materials
 Student Worksheet: Linear Approximation http://www.ck12.org/flexr/chapter/9728, scroll down to the fourth activity.
Part 1 – Introduction
From top to bottom, the points where the horizontal lines cross the
Since
To get the horizontal line, students can use graph >
To get the vertical line, they can use graph >
Part 2 – Investigating linear approximation
Students are to graph the function of
For the
The error is the length
The true value,
linear approx. of 
real value of 
error  underestimation/overestimation  


9.6  6.272  3.328  overestimation 

7.5  6.125  1.375  overestimation 

6.8  5.904  0.896  overestimation 

2.6  2.352  0.248  overestimation 
As you get close to the point of tangency, the graph of the function and the graph of the tangent line appear to be the same.
They are called local linearization because the graph acts like a straight line at the point of tangency and that line is the tangent line.
Students are to find the derivative of
Part 3 – Underestimates versus overestimates
Students are to graph
If
If
The point
Students should see that in general the linear approximation will overestimate if the curve is concave down and it will underestimate when the curve is concave up.
Part 4 – Finding intervals of accuracy
Students are posed with the question: How close to –1 must
When students look at the graph, they want the tangent line to be within the bound of
Since the linear approximation overestimates in this region, they want to compare
Students can zoom in graphically or solve algebraically. The interval is (–1.1799, –0.81453) using the solve command.
Because the tangent line overestimates to the left of
Since the linear approximation overestimates in the region to the left of
They can zoom in graphically or solve algebraically. The interval is (0.415196, 1) using the solve command.
Since the linear approximation underestimates in the region to the right of \begin{align*}x = 1\end{align*}, students want to compare \begin{align*}L(x)\end{align*} and \begin{align*}f1(x)  0.2\end{align*} there.
They can zoom in graphically or solve algebraically. The interval is (1, 1.5848) using the solve command.
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