1. RATES OF CHANGE & TANGENT LINES DAY 1 AP Calculus AB

2. LEARNING TARGETS  Define and determine the average rate of change  Find the slope of a secant line  Create the equation of the secant line  Find the slope of a tangent line  Explain the relationship of the slope of a secant line to the slope of a tangent line  Create the equation of the tangent line  Define and determine the normal line  Define and determine the instantaneous rate of change  Solve motion problems using average/instantaneous rates of change

3. AVERAGE RATE OF CHANGE  In a population experiment, there were 150 flies on day 23 and 340 flies on day 45. What was the average rate of change between day 23 and day 45?  In other words, what was the average rate of the flies growth between day 23 and day 45? 8.6 flies/day

4. AVERAGE RATE OF CHANGE: DEFINITION  The average rate of change of a function over an interval is the amount of change divided by the length of the interval.  In this class, we will often reference average rate of change as “AROC”. This is not necessarily a shortcut that is adapted across the board.

5. EXAMPLE

6. One of the classic problems in calculus is the tangent line problem. You are probably very good at finding the slope of a line. Since the slope of a line (and line always implies straight in the world of math) is the same everywhere on the line, you could pick two points on the line and calculate the slope. With a curve, however, the slope is different depending where you are on the curve. You aren't able to just pick any old two points and calculate the slope. Think about the parabola, Whose graph looks like Pretend you are a little bug and this graph is a mountain. The slope of the mountain is different at each point on the mountain. Think of how easy/hard it looks for the bug to climb. Just look at the bug’s angle.

7. Without calculus, we could estimate the slope at a particular point by choosing an additional point close to our point in question and then drawing a line between the points and finding the slope. So if I was interested in finding out the slope at point P, I could estimate the slope by using a point Q, drawing a secant line (which crosses the graph in two places), and then finding the slope of that secant line. P

8. SECANT LINE & SLOPE OF THE SECANT LINE DEFINITION

9. SECANT LINE CONNECTION TO AROC & EXAMPLE  Notice that the slope of the secant line is the average rate of change!  We could say that f(x) is the function which describes the growth of flies. A is at 23 days, b is at 45 days, c is 150, and d is 340.  We can see that the AROC is simple the slope of the secant line between the points (23, 150) and (45, 340)

10. HOW CAN WE FIND THE SLOPE OF TANGENT LINE?

11. HOW CAN WE FIND THE SLOPE OF THE TANGENT LINE?

12. TANGENT LINE & SLOPE OF THE TANGENT LINE DEFINITION

13. SLOPE OF TANGENT EXAMPLE

14. INSTANTANEOUS RATE OF CHANGE DEFINITION  The rate of change at a particular instant/moment  In other words, it is the slope of the tangent line to the curve.  In this class, we will reference this as “IROC”

15. INSTANTANEOUS RATE OF CHANGE EXAMPLE

16. PRACTICE 1

17. PRACTICE 2

18. PRACTICE 3

19. PRACTICE 4