Warm Up

1. Warm Up Page 92 Quick Review Exercises 9, 10

2. 2.4 Rates of Change & Tangent Lines What you’ll learn Average rate of change Tangent to a curve Slope of a curve Normal to a curve (perpendicular to tangent) Speed Revisited

3. Example 1 Finding Average Rate of Change Finding the average rate of change of a function over an interval is simply finding the slope of the line containing the endpoints of the interval. Ave. Rate of change over the interval [a, b] = f(b) – f(a) b – a • Find the average rate of change of f(x) = x3 – x over the interval [1, 3] You try exercise #1.

4. Example 2 Growing Drosophila in a Laboratory When looking at a graph you can compute the “average rate of change” from P to Q by identifying and using their coordinates. This is also known as finding the slope of the secant line PQ. We can always think of an average rate of change as the slope of a secant line. • Given P(23, 150) and Q(45, 340), find the average rate of change from P to Q. • What is the slope of the secant line PQ? • Examine the graph on page 87, is our answer reasonable?

5. Tangent Line to a Curve We define the rate at which the value of the function y = f(x) is changing with respect to x at any particular value x = a to be the slope of the tangent to the curve y = f(x) at x = a. How do we find the slope? • Start with the slope of the secant through P and a point Q nearby on the curve. • Find the limiting value of the secant slope (if it exists) as Q approaches P along the curve. (the derivative at P – use slope formula and pts (a, f(a)) and (a+h, f(a+h)) • We define the slope of the curve at P to be this number and define the tangent to the curve at P to be the line through P with this slope. Method by Pierre Fermat 1629

6. A line is tangent to a curve at the point P if the following 3 conditions are met: • P is both on the curve and on the line • The curve is smooth at the point P. (For example, if the curve is the graph of y = f(x), then f must be differentiable at the x-coordinate of P.) 3. The curve lies on one side of the line within some punctured neighborhood of P (a little circle, excluding P itself).

7. Example 3 Finding Slope and Tangent line Find the slope of the parabola y = x2 at the point P(2,4). Write an equation for the tangent to the parabola at this point. HOW? • Find slope by finding the limit of y = x2 at x = 2. the difference quotient of f at a 2. Use the slope and P(2,4) to write the equation in point slope form. Simplify this to slope intercept form.

8. Example 4 Exploring Slope & Tangent Let f(x) = 1/x • Find the slope of the curve at x = a. • Where does the slope equal -1/4? • What happens to the tangent to the curve at the point (a, 1/a) for different values of a?

9. Example 5 Finding a Normal Line (a line perpendicular to the tangent) Write an equation for the normal to the curve f(x) = 4 - x2 at x= 1. • Find slope, the limit at x=1. • Find the opposite reciprocal slope to use in the normal equation. • Find f(1) to get a point (1, f(1)) • Write an equation in point slope form and simplify it to slope intercept form. Hint: Anytime you are asked to find both a tangent line equation and a normal line equation, graph them both in a square viewing window to verify your results.

10. Example 6 Investigating Free Fall A body’s average speed along a coordinate axis for a given period of time is the average rate of change of its position y = f(t). Its instantaneous speed at any time t is the instantaneous rate of change of position with respect to time t, or ♦A rock breaks loose from the top of a tall cliff with position given from the equation y = 16t 2. Find the speed of the falling rock at t = 1 sec. How? Find the limit at t = 1 – the rate of change of the position of the rock at the instant t=1 second.

11. Homework Page 92 Exercises 1-29 odds, 35-40 all No Calculator!

12. Today’s Agenda Warm Up Quick Review Page 92 Exercises 1-3, 9, 10 Teamwork Free Response Question: 2005 Form B #5 BC 1 form per group, homework quiz grade Complete Page 92 for Wednesday & STUDY FOR QUIZ 2.3 & 2.4