2.4 Rates of Change and Tangent Lines

1. 2.4 Rates of Change and Tangent Lines

2. Average Rate of Change • Average Rate of Change = Amount of change divided by the time it takes. Or, • where Δy = the amount of change and Δx = the time it takes. • This idea is used to find the tangent of a curve at a certain point.

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

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 of a Tangent

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

6. 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.

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.

8. Let a Find the slope at . Note: If it says “Find the limit” on a test, you must show your work! Example 1:

9. Let b Where is the slope ? Example 1 (cont.):

10. ALL OF THESE ARE THE SAME: • The slope of y=f(x) at x=a • The slope of the tangent to y=f(x) at x=a • The instantaneous rate of change of f(x) with respect to x at x=a • Limit (at h approaches) 0 of [f(a+h) –f(a)]/h

11. Example 2: • For y = x2 at x = -2, find • the slope of the curve, • the equation of the tangent, and • an equation of the normal line. A normal line to a curve is the line perpendicular to the tangent at that point. • Then draw the graph of the curve, tangent line, and normal line on the same graph.

12. Example 2 (cont.) f(x) = x2 at x = -2 So, now we know y = -4x + b, but we do not know the value of b.

13. Example 2 (cont.) To find the value of b, and hence the equation of the tangent, use y = mx + b to find the y-intercept. At the given value of x = -2, y = (-2)2 or 4. So, substituting into y = mx + b, gives 4 = -4(-2) + b, which means b = -4. Therefore, the tangent line has an equation of y = -4x – 4 by substituting the values for m and b into y = mx + b.

14. Example 2 (cont.) Since the normal line is perpendicular to the tangent line, the slope is the opposite reciprocal of -4 or ¼. Also, the normal line goes through the same point as the tangent line or (-2, 4). So, use y = mx + b to find the y-intercept of the normal line. y = ¼x + b 4 = ¼(-2) + b 4 = -½ + b b = 9/2 This give an equation for the normal line of y = ¼x + 9/2.

15. Example 3 Free Fall. A rock breaks loose from the top of a tall cliff and falls 16t2 feet after t seconds. How fast is it falling 3 seconds after it starts to fall?

16. These are often mixed up by Calculus students! If is the position function: velocity = slope Review: average slope: slope at a point: average velocity: And, so are these! instantaneous velocity: