Presentation on theme: "Warmup 1)describe the interval(s) on which the function is continuous 2) which of the following points of discontinuity of are not removable? a)0c) -2e)"— Presentation transcript:
Warmup 1)describe the interval(s) on which the function is continuous 2) which of the following points of discontinuity of are not removable? a)0c) -2e) -5 b)6d) 4
2.4 Rates of Change and Tangent Lines
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. Sometimes, you will see the problem already in the difference quotient, and have to figure out the limit.
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 (perpendicular line), use the negative reciprocal of the slope. (in this case, )
Example 4: b Where is the slope ? Let
Review: average slope: slope at a point: average velocity: (slope) instantaneous velocity: (slope at 1 point) If is the position function: These are often mixed up by Calculus students! So are these! velocity = slope
(slope of the tangent line to graph at the point
The End p. 87 (1-5 odd, 7, 9-23 odd) Day 1: Day 2: p. 87 (2-6 even, even, 29, 30) Suggested Review Problems for Ch.2 from the book: p. 91 (6, 7, 9, 21-32, 39-43, 52)