1. Calculus Warm-Up Find the derivative by the limit process, then find the equation of the line tangent to the graph at x=2

2. Warm-Up Find a and b such that f(x) is continuous on the entire real line.

3. Calculus Warm-Up For a function to be differentiable at x=c, what conditions must be met?

4. Calculus HWQ Is this function continuous as x = 2? Is the function differentiable at x = 2?

5. 2.2 Basic Differentiation Rules 2015 Colorado National Monument Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003

6. Calculus Warm-Up For what value of a is the function continuous at x=4?

7. Recall from last lesson: To be differentiable, a function must be continuous and smooth. Derivatives will fail to exist at: cornercusp vertical tangent discontinuity

8. Continuous & Differentiable Continuous but NOT differentiable everywhere: **Remember: differentiablility implies continuity but continuity does NOT imply differentiability** Continuous AND differentiable everywhere :

9. Discontinuous

10. You try … At point c, decide whether these graphs are: a.Continuous b.Differentiable c.Both d.Neither e)

11. If the derivative of a function is its slope, then for a constant function, the derivative must be zero. example: The derivative of a constant is zero.

12. If we find derivatives with the difference quotient: (Pascal’s Triangle) We observe a pattern: …

13. examples: power rule We observe a pattern:…

14. examples: constant multiple rule: When we used the difference quotient, we observed that since the limit had no effect on a constant coefficient, that the constant could be factored to the outside.

15. (Each term is treated separately) constant multiple rule: sum and difference rules:

16. Example: Find the horizontal tangents of: Horizontal tangents occur when slope = zero. Plugging the x values into the original equation, we get: (The function is even, so we only get two horizontal tangents.)

22. First derivative (slope) is zero at:

23. Often it is helpful to rewrite the function before differentiating:

24. Practice:

26. product rule: Notice that this is not just the product of two derivatives. This is sometimes memorized as:

27. quotient rule: or

28. Consider the function We could make a graph of the slope: slope Now we connect the dots! The resulting curve is a cosine curve.

29. We can do the same thing for slope The resulting curve is a sine curve that has been reflected about the x-axis.

31. Practice:

32. Higher Order Derivatives: is the first derivative of y with respect to x. is the second derivative. (y double prime) is the third derivative.is the fourth derivative. We will learn later what these higher order derivatives are used for. 

33. Example:

34. Homework: Pg.115 Larson Calculus 8 th ed. : 31-63 odd, 75, 89 http://college.cengage.com/mathematics/blackboard/cont ent/larson/calc8e/calc8e_solution_main.html?CH=07&SE CT=a&TYPE=se