1. 11-2 Key to evens 2a) -5 2b) -3 2c) 0 4a) 0 4b) 1 4c) -2 6) -1/10 8) -5 10) 27 12) - 7/14 14) 1/8 16) 1/16 18) 0 20) 1/4 22) -1/6 24) 4 26) -1/4 28) 1 30) 0 32) 3 34) 0 36)-2/3 38)7.389

2. Review 4) Think through this limit :

3. Review 5) Think through this limit :

4. Definition of a Horizontal Asymptote? The line y = b is a horizontal asymptote of the function y = f(x), if y approaches b as x approaches ±∞.

5. Limits at infinity Find the following limits

6. 11.3 Tangent Lines 2014

7. Precalculus Warm-up Find the equation of the line between the points: (2,5) (3,8) y=3x-1

8. Lesson One Sided Limits

9. Lesson Why? Because the limit from the left of 0 does not agree with the limit from the right of 0.

10. Because both one sided limits agree, Graph to verify

11. Precalculus Warm-up At time t = 0, a diver jumps from a platform diving board that is 32 feet above the water. The position of the diver is given by where s is measured in feet, and t in seconds. a)When does the diver hit the water? b)What is the diver’s average rate of change on the dive? c)What is the diver’s velocity at impact? ?

12. Slope and the Limit Process In the warm-up we found average velocity of a graph by finding the slope of a secant line between two points. How can we take the idea of the slope of a secant line and use it to find the velocity at any point on the graph? With limits!

13. 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).

14. 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?

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

16. is called the difference quotient of f at x.  If you are asked to find the slope using the limit definition or using the difference quotient, this is the technique you will use.  The slope of a curve at a point is the same as the slope of the tangent line at that point.

17. Example: Find a formula for the slope of the graph of using the limit process.

18. Example Find a formula for the slope of the graph of using the difference quotient. Then find the slope at the point ? How about at ?

19. Derivatives

20. The limit we used to define the slope of a tangent line is also used to define one of 2 fundamental operations in calculus-differentiation. The derivative of a function of x is also itself a function of x. This “new” function gives the slope of the tangent line to the graph of the function at a point, provided that the graph has a tangent line at this point. So, the derivative describes the slope of a function at any point along the graph of the function.

21. “The derivative of f with respect to x is …” This equation finds the formula for the derivative of a function at any point.

22. “The derivative of at is …” There are many ways to write the derivative of This equation finds the derivative of a function at a.

23. “f prime x”or “the derivative of f with respect to x” “y prime” “dee why dee ecks” or “the derivative of y with respect to x” “dee eff dee ecks” or “the derivative of f with respect to x” “dee dee ecks uv eff uv ecks”or “the derivative of f of x”

24. dx does not mean d times x ! dy does not mean d times y !

25. does not mean ! (except when it is convenient to think of it as division.) does not mean ! (except when it is convenient to think of it as division.)

26. (except when it is convenient to treat it that way.) does not mean times !

27. Example: Use the limit of the difference quotient to find the derivative of What is the derivative of the function at x = 2? What does this mean?

28. You must be able to do this : Use the limit process to find the derivative of f(x). Find the equation of a line tangent to the graph at (-2,-1)

29. Example: Use the limit process to find the derivative of and write the equation of the line tangent to the graph of f(x) at x=4.

30. You Try: Find the derivative of using the limit process. a) Find any values of x where the tangent line is horizontal. b) Find the equation of the line tangent at x=1.

31. Big Idea: Velocity (slope of a tangent line to a curve at a point) can be found by taking the limit of difference quotient as the change in x (we are calling this h) goes to 0. Back to the warm-up problem. What is the diver’s instantaneous rate of change at impact? Use the limit process to find the diver’s velocity at impact. Recall that impact occurred at t = 2. Finally:

32. Recap of ideas: 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 

33. Homework 11.3 : Day 1: pg. 770 1-17 odd Day 2: pg. 771 25-37 odd, 43, 45

34. Example: Find a formula for the slope of the graph of using the limit process. Use the derivative to determine any points on the graph of f at which the tangent line is horizontal and find the equation of each tangent line.

35. Derivatives 11.3 Day 2

36. Homework 11.3 : Day 1: pg. 770 1-17 odd Day 2: pg. 771 25-37 odd, 43, 45 Quiz tomorrow on 11.3