A tangent line is a line which approximates a curve well near a particular point. This project investigates tangent lines algebraically. Algebra requires some sort of formula.

Define the function f by the formula f(x) = x² - 3x + 1.

We'll find an equation for the line tangent to the graph of f at the point (1, -1).

Every non-vertical line through the point (1, -1) has an equation of the form y = m*(x-1) - 1.
The only thing which distinguishes one line from another is the value of the slope, m.
Here's a process to determine the right value for m.

• On the command line of the HOME screen, type the equation y = f(x) and store it in e, for "equation."
• Type in the formula m*(x-1) - 1 and store it in the variable y. We'll find the best linear approximation to f at (1,-1) by assuming f is linear. (The assumption is false, but we'll fix that later.)
• Enter e and see what it looks like now.
• Solve for m. You can do that manually, by adding or multiplying the equation you see by appropriate expressions. But the quickest way is to use the calculator's Solve command. Enter solve(e,m).
• This gives an expression with an x in it. We're only interested in what happens at (1, -1), so we want to evaluate the expression when x = 1. On the calculator, highlight the last result and push ENTER to move it to the command line. Then add |x=1 at the end and ENTER.
• If you store the resulting value in m and then type y, you'll have the formula you want.

Exercises:

1. Repeat the process at a couple of different points on the curve.

2. Repeat the process for the curve with equation y = sin(x) + 2cos(x) at (0, 2). This time you'll get undef when you substitute 0 for x. Instead of evaluating the expression at x = 0, you can find the limit of the expression as x approaches 0. The syntax is limit(expression, variable, limit).

3. Repeat the process for the curve with equation y = x^x at the point (2, 4).

4. Repeat the process for the curve with equation y = sin(1/x) at the point (0, 0).
    (This formula doesn't make sense when x=0, but the calculator usually doesn't mind.)

Return to top of page.
Go to project index.