2.6 Tangent Lines and Their Slopes

Summary

The study of calculus begins in earnest with a solution of the Tangent Line problem. Its solution leads to the derivative and the rich subject of differential calculus.

By the end of your studying, you should know:

A statement of the Tangent Line problem.
The solution of the Tangent line problem.
How to find the equation of the tangent line to the graph of a function at a point.
The limit definition of the slope of the tangent line at a point on the graph of a function.
Typical examples where the tangent line does not exist at a point on the graph of a function.

On-screen applet instructions: Note that the tangent line is the dotted blue line. Use the slider to control the position of the point Q (hence the secant line and its slope m).

Examples

Can you find a tangent line to f(x) = |x| at x = 0?

A practice ski jump hill follows the shape of a given curve. Come up with a formula for the angle the skier's skis make with the horizontal, and find how far from the top of the jump he is when this angle is the greatest.

A potted cactus is thrown upward with a velocity of 40 feet per second. Its height in feet at time t is given by the formula h(t) = 40t – 16t². Find its velocity 2 seconds after it is released.