proj018

Calculus Project #18 Calculating Derivatives

The derivative of a function at a number is the slope of the line tangent to the graph of the function at the point with that number as x-coordinate. The derivative also measures the rate of change of the output of that function (relative to change in the input) near that particular input. This project investigates derivatives graphically, numerically, and algebraically. The techniques we'll use here only work if the function has some sort of formula.

First, define a function f such that f(x) = x² - 3x + 4.

We're interested in finding the derivative of f at 1, which is usually denoted f'(1).
It's the slope of the line tangent to the graph with equation y = f(x) at the point (1, 2).

Graphical: Graphy1(x) = f(x), and zoom in on the point (1, 2) until the graph looks like a straight line. Use the Trace feature to find a second point on this "line," and then find the slope.

Numerical: There are several different numerical approaches to derivatives on the calculator:

If you still have the slope function saved from project #3, type slope(1,2,1+h,f(1+h))|h=.1 and hit ENTER. Edit the command line to make the value of h closer and closer to zero until you can make a reasonable guess for the value of the derivative. (If you don't have a slope function stored in your calculator, define one first.)
There are a couple of built-in functions which will do the same job. One is avgRC, for "average rate of change." Use this syntax: avgRC(f(x),x,.1)|x=1. Then change the .1 to estimate the derivative. (You can find avgRC in the CATALOG under A. If you type it in, you don't need to capitalize R and C.)
Another built-in function is nDeriv; enter nDeriv(f(x),x,.1)|x=1. This computes the quantity (f(x+h)-f(x-h))/2h, in this case for x =1 and h=.1. (Do some algebra to verify that it doesn't matter what value you use for h with this particular formula for f(x).) You can find nDeriv in the CATALOG under N. If you type it in, you don't need to capitalize the D.)

Algebraic: On the command line of the HOME screen, enter (f(1+h) - f(1))/h. Then take the limit of that expression as h goes to zero. You can do that by hand, or by using the calculator's limit command. If you highlight the result of the computation and hit enter, you transfer that expression to the command line.

There is also a built-in differentiation command. Enter d (f(x),x)|x=1.
If you use the letter d instead of the differentiation operator, the calculator won't tell you anything.

Exercises:

1. Repeat the process for different values of x.

2. Repeat the process for f(x) = sin(x) + 2cos(x) at x = 0 and at x = p/2.

3. Repeat the process for f(x) = 3.127sin(1.3x) + 2.485cos(1.8x) at any convenient input.

4. Repeat the process for f(x) = x^x at x = 2 and at x = 2.1.

Return to top of page.
Go to project index.