1. 1 Discuss with your group

2. 2.1 Limit definition of the Derivative and Differentiability 2015 Devil’s Tower, Wyoming Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1993

3. Warm-Up Find the following limit (without a calculator):

4. HWQ Find the following limit (without a calculator):

6. 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).

7. 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?

8. slope slope at The slope of the curve at any point is:

9. The slope of the curve at the point is: is called the difference quotient of f at a. If you are asked to find the slope at a point using the definition or using the difference quotient, this is the technique you will use.

10. Recap : Average rate of change between (a, f(a)) and (b, f(b) Is also called the average velocity: Instantaneous rate of change at (a, f(a)) is also called The actual velocity at that point or the derivative:

11. Alternatively, you can find the instantaneous rate of change at (x, f(x)) and evaluate it at x = a.

12. Tangent lines can be found using the point-slope form of a line:. The slope of a curve at a point is the same as the slope of the tangent line at that point. If you want the normal line, use the opposite reciprocal of the slope. (The normal line is perpendicular.)

13. 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 

14. is the derivative of at. We write: “The derivative of f with respect to x is …” Derivatives

15. Differentiability and Continuity

16. The slope of the curve at any point is: Slope at any point on the graph of a function:

17. Slope at a specific point on the graph of a function: The slope of the curve at the point is:

18. Using the limit of the difference quotient, find the slope of the line tangent to the graph of the given function at x= -1, then use the slope to find the equation of the tangent line: Example:

19. Try this one:

20. Another Example ? : Find the equation of a line tangent to the graph at (-2,-1)

21. Differentiability and Continuity The following statements summarize the relationship between continuity and differentiability. 1. If a function is differentiable at x = c, then it is continuous at x = c. So, differentiability implies continuity. 2. It is possible for a function to be continuous at x = c and not be differentiable at x = c. So, continuity does not imply differentiability.

22. 1.If a function is differentiable at x = c, then it is continuous at x = c. Differentiability implies continuity. 2.It is possible for a function to be continuous at x = c and not differentiable at x = c. So, continuity does not imply differentiability. 3.Continuous functions that have sharp turns, corner points or cusps, or vertical tangents are not differentiable at that point. Very Important, so we’ll say it again:

23. Differentiability and Continuity The following alternative limit form of the derivative is useful in investigating the relationship between differentiability and continuity. The derivative of f at c is provided this limit exists (see Figure 2.10). Figure 2.10

24. Differentiability and Continuity Note that the existence of the limit in this alternative form requires that the one-sided limits exist and are equal. These one-sided limits are called the derivatives from the left and from the right, respectively. It follows that f is differentiable on the closed interval [a, b] if it is differentiable on (a, b) and if the derivative from the right at a and the derivative from the left at b both exist.

25. Figure 2.11 Differentiability and Continuity If a function is not continuous at x = c, it is also not differentiable at x = c. For instance, the greatest integer function is not continuous at x = 0, and so it is not differentiable at x = 0 (see Figure 2.11).

26. The function shown in Figure 2.12 is continuous at x = 2. Example of a function not differentiable at every point – A Graph with a Sharp Turn Figure 2.12

27. However, the one-sided limits and are not equal. So, f is not differentiable at x = 2 and the graph of f does not have a tangent line at the point (2, 0). cont’d Example of a function not differentiable at every point –A Graph with a Sharp Turn

28. For example, the function shown in Figure 2.7 has a vertical tangent line at (c, f(c)). Figure 2.7 Example of a function not differentiable at c.

29. Example: Determine whether the function is continuous at x=0. Is it differentiable there? Use to analyze the derivative at x=0. Not differentiable at x=0 Vertical tangent line

31. Example: Determine whether the function is differentiable at x = 2. 1)f(x) is continuous at x=2 and 2)The left hand and right hand derivatives agree Differentiable at x = 2 because:

32. A function is differentiable if it has a derivative everywhere in its domain. It must be continuous and smooth. Functions on closed intervals must have one-sided derivatives defined at the end points. A function will not have a derivative 1)Where it is discontinuous 2)Where it has a sharp turn 3)Where it has a vertical tangent 

33. Recap: To be differentiable, a function must be continuous and smooth. Derivatives will fail to exist at: cornercusp vertical tangent discontinuity

34. The derivative is the slope of the original function. The derivative is defined at the end points of a function on a closed interval.

36. Homework P. 103 8,17,23,25,30,35-47 odd, 81-87 odd