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));

Wed Sep 20 11:01:11 EDT 1995