MATH 163, Spring 2001 Due Date: Name(s):
Project 3.10: Linear Approximation and Differentials
Objective
To illustrate the connection between the linear approximation of a function and its differential.
Narrative
If you have not already done so, read Section 3.10 of the text.
Linear approximations and differentials are important since they allow us to at least estimate the values
of functions that otherwise might be difficult or even impossible to evaluate. In this project we illustrate
this by using differentials to approximate the value of f(x) =

2x/(1 - x) when x = 0.7.
Task
a) Type the command lines in the left-hand column below into Maple in the order in which they are listed.
> # Project 3.10: Linear Approximation and Differentials
> restart; Clear Maple's memory.
> f := x -> sqrt(2*x)/(1-x); Let f(x) =

2x
1 - x
.
> f(0.5); Evaluate f(x) when x = 0.5.
> df := (x,dx) -> (D(f)(x))*dx; Let df be the differential of f.
> df(0.5,0.2); Evaluate df(x, dx) when x = 0.5 and dx = 0.2.
> with(student): Use the student package to ...
> showtangent(f(x),x=0.5,x=0.1..0.9,y=0..5); Plot the graph of f(x) near x = 0.5, and the
tangent to the graph of f at P(0.5, f(0.5)).
At this point, make a hard-copy of your typed input and Maple's responses. Then ...
b) draw by hand the horizontal line whose equation is y = 2, and the vertical lines whose equations are
x = 0.5 and x = 0.7, and label each with its equation,
c) clearly identify and label by hand the segments whose lengths are y(0.5, 0.2) and dy(0.5, 0.2), and
d) using the values of f(0.5) and dy(0.5, 0.2) you computed above, estimate the value of f(0.7).
Your lab report will be a hard copy of your typed input and Maple's responses (both text and hand-labeled
graphics), and your approximation.
Comments
The actual value of f(0.7) to 4 decimal places is 3.9441. While there is certainly some error in the approximation
you made in this project, your approximation should still be fairly good considering the computations
you had to perform (vs. the complexity of computing f(0.7) =

1.4/0.3 without using any computational
devices other than a pencil and piece of paper). We will say considerably more about approximations later
in MATH 164.