Subsections

# Linear Approximation and Tangent Functions

## Linear Approximation

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

In this equation, the parameter is called the base point, and is the independent variable. You may recognize the equation as the equation of the tangent line at the point . It is this line that will be used to make the linear approximation. For example if , then would be the line tangent to the parabola at
> f:=x->x^2;
> fT1:=D(f)(1)*(x-1)+f(1);
> plot({f(x),fT1},x=-1..3);

Obviously the two things the function and the tangentline have in common at are their y-value and their slope. Looking at the plot, the line will approximate the function exactly at the base point a and the approximation will be close if you stay close to that base point. To see how far off the approximation becomes on the given interval of the plot it is easy to see that the largest errors occur at the ends of the interval, i.e. and . Simply subtract the y-values to calculate the error.
> abs(f(-1)-subs(x=-1,fT1));
> abs(f(3)-subs(x=3,fT1));

Notice that the error grows to four at each end of the interval. How could you find an interval such that the greatest error will be a specific value? Think about this as it will appear in the exercises.

## Tangent Functions

Two functions are said to be tangent at a point if their y-values are equal, and if their tangent slopes are equal . For example, the functions and are tangent at . Check their y-values and their derivatives at the given point.
> f:=x=>sin(x);g:=x->x;
> f(0);g(0);
> D(f)(0);D(g)(0);

However, and aren't tangent at because even though they have the same slope they don't have the same y-value.
> h:=x->x+1;
> D(f)(0);D(h)(0);
> f(0);h(0);


## Exercises

1. Given:

1. Graph the three functions on the same plot using the intervals and
2. Find the point where the functions intersect.
3. Are any of the functions tangent to each other at the intersection point? If so plot them with a small enough domain to show the tangent functions clearly.