MATH 163, Spring 2001 Due Date: Name(s):
Project 2.1c: Tangential Approximation
Objective
In this project we illustrate one reason the problem of finding the equation of the tangent line to the graph
of a function at a point is important.
Narrative
One of the important practical problems faced in the compilation of tables and in the programming of
computational devices (such as hand-calculators and computers) is how to find the value of functions such
as

x, sin x, ex
, and ln x for specific choices of x. Tables and computational devices are so ubiquitous that it
might not be (immediately) apparent that this is an issue, but someone has to program the devices we use,
and evaluating functions such as these is an issue for anyone involved in writing low-level computer code.
Knowing the equation of the tangent line to the graph of a function y = f(x) at a given point is important
since the tangent line is the graph of a linear function y = mx + b, and linear functions are easy to evaluate.
For example, later we will see that if angles are measured in radians then the tangent line to the graph of
y = f(x) = sin x at x = 0 is the graph of y = x. Thus for "small" angles x, sin x  x. So if we're interested
in computing sin 3
, for example, we might convert 3
to radians and then use this value to approximate
sin 3
.
Task
a) Type the command lines below into Maple in the order in which they are listed. These command lines
produce graphs of y = sin x and y = x at two different scales. Note that the natural unit for angle
measurement (in the computation of trig functions such as sin) in Maple is radians. Also observe how close
the graphs of y = sin x and y = x are to one another!
> # Project 2.1c: Tangential Approximation
> # Task a
> restart:
> plot({x,sin(x)},x=-Pi/2..Pi/2);
> plot({x,sin(x)},x=-Pi/6..Pi/6);
b) Continue by typing the following command lines into Maple. They produce numerical information about
the values of the functions f(x) = sin x and f(x) = x for small values of x (in degrees and in radians) and
store them in a matrix (a rectangular array of numbers). (Remember that 1
= /180 radians.)
> # Task b
> OneDeg := evalf(Pi/180);
> M := matrix(32,4,(Row,Col) -> 0):
> M[1,1] := "x (in degrees)": M[1,2] := "x (in radians)":
M[1,3] := "sin(x)": M[1,4] := "error":
> for i from 0 to 30 do
M[i+2,1] := 2*i: M[i+2,2] := evalf(2*i*OneDeg):
M[i+2,3] := evalf(sin(2*i*OneDeg)):
M[i+2,4] := evalf(M[i+2,2]-M[i+2,3]): od:
> eval(M);
At this time make a hard-copy of your typed input and Maple's responses. Then, ...
c) By hand, use the data in the chart you produced to find an approximation to sin 3
to 4 decimal places
of accuracy. Roughly how much error is there in this approximation? (Make sure it is clear how you draw
your conclusions.)
d) Later we will see that the tangent line to the graph of y =

x at x = 1 is the graph of y = 1
2 x + 1
2 . Using
this fact, approximate

1.1. Based on the fact that Maple says

1.1 = 1.048808848, roughly how much
error is there in your approximation?
Comments
1. We will return to this topic later in MATH 163 in our discussion of differentials.
2. Evaluating functions such as

x, sin x, ex
, and ln x is only one reason finding the equation of the tangent
line to the graph of a function at a point is important. There are many others which we will discuss
later in MATH 163.
3. In MATH 164 we will discuss how small "small" is, and how to handle "large" angles when evaluating
functions such as sin.