One way to find an equation for a line tangent to the graph of a function is to assume that the function and the tangent line have the same formula.

Suppose f(x) = sin(2x) + 3cos(x). (Define f this way on your calculator.)

Suppose we want to find an equation for the line tangent to the graph of f at (0, f(0)).

First observe that any line has an equation like y = ax+b.

If the function and its tangent line have the same formula, we have ax+b = f(x).
(Enter this equation on your calculator and store it in some convenient place, such as z. Don't forget the multiplication symbol between the a and the x.)

To find b, set x = 0. (Store the resulting value in b and look at the equation again.)

Since the graph of f really isn't a straight line, we can't expect to solve for a and get a number. The next best idea is to solve for a and set x=0. (We really only expect good results near the point of tangency.) Alas, that doesn't work either with this problem. But if we take derivatives first and then set x = 0, everything works fine.
 
 
 

One way to find an equation for the best quadratic approximation to a function is to assume the function and the parabola have the same formula.

Start a new problem, and define f(x) = sin(2x) + 3cos(x). We want to find a good quadratic approximation to f near the point (0, f(0)).

Any quadratic equation has a formula of the form ax² + bx + c.
If the quadratic approximation is perfect, we'll have ax² + bx + c = f(x).
(Enter this equation on your calculator and store it in some convenient place, such as z. Don't forget the multiplication symbols after the a and the b.)

To find c, set x = 0. (Store the resulting value in c and look at the equation again.)

To find b, differentiate first, and then set x = 0. (Store the resulting value in b and look at the equation again.)

How can you find a?
 
 

Exercises

• Graphf and its quadratic approximation and see how they compare.
• Find a cubic approximation and compare the graphs. How do the x², x, and constant terms compare with your quadratic approximation?
• Find a quartic approximation and compare the graphs.
• Find quadratic approximation near (p, f(p)). It's easiest to write your quadratic in the form a(x-p)² + b(x-p) + c. Compare the graph with the graph of f.
• Find cubic and quartic approximations to f near (p, f(p)). Compare the graphs.
• Repeat the process for a different function.
• Repeat the process for an unspecified function. (Start a new problem but don't define f.) You won't be able to see the graphs, of course, and the calculator isn't much help. But you can probably discover a formula for the coefficients of the approximating polynomial.
• Use the Taylor polynomial utility on your calculator to check your answers. For the quartic approximation to the function f above near the point (, f()), you'd enter taylor(sin(2x)+3cos(x),x,4,p). The p here is called the center of the Taylor polynomial. If you don't tell the calculator what the center is, it assumes the center is 0, which is often what you want anyway.)
• Compare the graphs of f and its Taylor polynomials of degree 8, 9, and 10 centered at 0.