1. Section 2.4 Rates of Change and Tangent Lines Calculus

2. Find the average rate of change of this function over the interval [1, 3] Since f(1) = 0 and f(3) = 24 Then:

3. Finding Average Rate of Change is easy to find, but not very accurate. We need to come up with a more accurate way of finding the Rate of change at a specific point. INSTANTANEOUS RATE OF CHANGE So the average rate of change:

4. 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). EXAMPLE 1:

5. 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? SAME EXAMPLE CONTINUED:

6. slope slope at The slope of the curve at the point is: SAME EXAMPLE CONTINUED: This is called the “definition” or “Difference quotient”

7. 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. (The normal line is perpendicular.) Some more key notes:

8. Example 4: a Find the slope at “x”. Let Let’s find an equation that would tell us find the slope at any point on this line. So let’s use the point (x, y)

9. Example 4: b Where is the slope ? Let So now we can use what we just found to help find where certain slopes are:

10. Write an equation for the normal to the curve below at x = 1 So the slope is -2, then the perpendicular slope is 1/2

11. 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 