The purpose of this lab is to use Maple to become more familiar with the linear approximation to a differentiable function.

Suppose that is a differentiable function and that **a** is some
fixed number in the domain of **f**.
We define the linear approximation to at **x=a**, by the
equation

In this
equation, the parameter **a** is called the base point, and **x** is the
independent variable. It might help you understand the definition
better if you
keep in mind that **a** stands for a fixed number.

For example if , then
would be the straight line that is tangent to the graph of
at **x=1**. We can use the definition to find this tangent line
by evaluating and and then plugging these
numbers into the definition to obtain

The
straight line has the two properties that and
. That is, intersects the graph of
at **x=a** and has the same slope as at
**x=a**. To see this, first set **x=a** in the definition, giving

Then differentiate the definition with respect to **x** to get the slope
of the line, which is

Unfortunately, Maple does not provide the linear approximation
directly, so a procedure called `tangentline` has been written as
part of the `CalcP` package. Its syntax is similar to that of the
`secantline` command introduced in the previous lab. The Maple
commands below show
how to load the `CalcP` package into your Maple session and provide
several examples of how to use the `tangentline` function.

> with(CalcP);

> f := x -> x^5+4*x^2+1;

> tangentline(f,x=-1);

> plot(f(x),tangentline(f,x=-1),x=-2..2);

> plot(f(x),tangentline(f,x=-1),tangentline(f,x=1),x=-2..2);

The `tangentline` procedure produces an expression, which can be
manipulated using standard Maple commands. In the next example, we
show how to use the Maple `unapply` command to turn the result of
`tangentline` into a function.

> f_T := unapply(tangentline(f,x=-1),x);

> f_T(x);

As shown in the next example, the `animate` command from the `
plots` package can be used to
see how the tangent line changes as the base point is changed.

> with(plots):

> animate(x^2,tangentline(x^2,x=t),x=-2..2,t=-1..1);

The next two examples show how to apply `tangentline` to arbitrary
functions and . Note that Maple uses the notation `
D(g)` to stand for the
derivative of **g**, and the notation `D(g)(a)` to stand for the
derivative of **g** evaluated at **x=a**.

> tangentline(g(x),x=a);

> tangentline(g(x)+h(x),x=a);

> tangentline(g(x),x=a) + tangentline(h(x),x=a);

The last Maple command in the previous example shows that if is
the sum of two functions, , then , which means that is tangent to
at **x=a**. To check that this is true, you need to show
that the values and are equal at **x=a**
and that the derivatives of and are
equal at **x=a**. Maple commands that do this are shown below.

> t1 := tangentline(g(x),x=a) + tangentline(h(x),x=a);

> subs(x=a,t1);

> subs(x=a,diff(t1,x));

- Try out
`tangentline`on the following examples. This part of the lab is only to help you become familiar with the concept of the tangent line to a function at a point**x=a**. Some suggested values of**a**are given, but feel free to try other values of**a**and/or other functions.

- with
**a = 2**and**a = -3**. - with
**a = 0**and . - with
**a = 2**and**a = -1**.

- with
- More generally, the graphs of two functions and are
tangent at
**x=a**if and . For example, the functions and are tangent at**x=0**, but the functions and are not tangent at**x=0**, even though they have the same slope, because and so the graphs of**p**and**q**don't intersect at**x=0**.Show that is tangent to at

**x=a**under this more general definition. - Suppose that and . Determine if the
following statements are true or not.
- is tangent to at
**x=-1**. - is tangent to at
**x=1**. - is tangent to at
**x=1**.

- is tangent to at

Mon Jun 26 13:37:30 EDT 1995