2.4 Rates of Change and Tangent Lines

Photo by Vickie Kelly, 1993 Greg Kelly, Hanford High School, Richland, Washington 2.4 Rates of Change and Tangent Lines Devil’s Tower, Wyoming

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

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

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 slope of a curve at a point is the same as the slope of the tangent line at that point. (The normal line is perpendicular.)

Let F3 a Find the slope at . On the TI-89: 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: Calc

Let WINDOW b Where is the slope ? Y= On the TI-89: GRAPH Example 4: y = 1 / x

Let WINDOW F5 b Where is the slope ? ENTER ENTER Y= On the TI-89: GRAPH Example 4: We can let the calculator plot the tangent: Math y = 1 / x A: Tangent 2 Repeat for x= -2 tangent equation

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

2.4 Rates of Change and Tangent Lines