2.4 Rates of Change and Tangent Lines Devils Tower, Wyoming Greg Kelly, Hanford High School, Richland, WashingtonPhoto 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:
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.
In the previous example, the tangent line could be found using. The slope of a curve at a point is the same as the slope of the tangent line at that point. If you want the normal line, use the opposite signed reciprocal of the slope. (in this case, ) (The normal line is perpendicular.)
Example 4: a Find the slope at. Let On the TI-89: limit ((1/(a + h) – 1/ a) / h, h, 0) F3Calc Note: If it says Find the limit on a test, you must show your work!
Example 4: b Where is the slope ? Let
Example 4: c What are the tangent line equations when and ?
Example 4: Graph the curve and the tangents on theTI-89: Y= y = 1 / x WINDOW GRAPH
Graph the curve and the tangents on theTI-89: Example 4: Y= y = 1 / x WINDOW GRAPH F5Math A: Tangent ENTER 2 Repeat for x = -2 tangent equation We can let the calculator plot the tangents:
Review: average slope: slope at a point: average velocity: instantaneous velocity: If is the position function: These are often mixed up by Calculus students! So are these! velocity = slope