Rates of Change and Tangent Lines Devil’s Tower, Wyoming Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 1993

The slope of a line is given by: The slope at (1,1) can be approximated by the slope of the secant through (4,16). We could get a better approximation if we move the point closer to (1,1). ie: (3,9) Even better would be the point (2,4).

The slope of a line is given by: If we got really close to (1,1), say (1.1,1.21), the approximation would get better still How far can we go?

slope slope at The slope of the curve at the point is:

The slope of the curve at the point is: is called the difference quotient of f at a. If you are asked to find the slope using the definition or using the difference quotient, this is the technique you will use.

The slope of a curve at a point is the same as the slope of the tangent line at that point. In the previous example, the tangent line could be found using . If you want the normal line, use the negative reciprocal of the slope. (in this case, ) (The normal line is perpendicular.)

limit ((1/(a + h) – 1/ a) / h, h, 0) Example 4: Let a Find the slope at . limit ((1/(a + h) – 1/ a) / h, h, 0) Note: If it says “Find the limit” on a test, you must show your work!

Example 4: Let b Where is the slope ?

Example 4: Let b Where is the slope ? tangent equation

Review: p velocity = slope These are often mixed up by Calculus students! average slope: slope at a point: average velocity: So are these! instantaneous velocity: If is the position function: velocity = slope p