2.6 Tangent Lines and Their Slopes



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:

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).


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.


Secant and Tangent Lines


See short videos of worked problems for this section.


Take a quiz.


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.

2.5 Continuity Table of Contents 2.7 The Derivative

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