## 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.

### Applets

Secant and Tangent Lines

### Videos

See short videos of worked problems for this section.

### Exercises

See Exercises for 2.6 Tangent Lines and Their Slopes (PDF).

Work online to solve the exercises for this section, or for any other section of the textbook.

#### Interesting Application

The Koch Snowflake is a continuous curve that does not have a tangent line at any point.

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Copyright © 2005 Donald L. Kreider, C. Dwight Lahr, Susan J. Diesel