1. 4.4 Concavity and Inflection Points Wed Oct 21 Do Now Find the 2nd derivative of each function 1) 2)

2. Applications of the 2nd derivative So far, we’ve only talked about one application of the 2nd derivative, which is the acceleration function The second derivative can also be used to describe the behavior of functions as well.

3. Concavity and Inflections The 1st derivative is used to describe slope. But since it is also a function, it also has its own “slope” or derivative. The 2nd derivative can be used to model the behavior of the slope, as it is ALSO changing with the function –Some slopes can be steep, while others rather flat

4. Concavity The 2nd derivative can be used to describe concavity Concavity is the rate at which the slope increases or decreases There are two types of concavity –Concave up (looks like a smile) –Concave down (looks like a frown)

5. Concavity and f’’(x) Thm- Suppose f(x) is differentiable on an interval I and f ’’(x) exists, –If f ’’(x) > 0, then the graph is concave up –If f ’’(x) < 0, then the graph is concave down Note: Second derivative only

6. Inflection Points An inflection point is a point on the graph where a graph alternates between concave up and concave down We can find inflection points when f ’’(x) = 0

7. Example 1 Determine where the graph is concave up and concave down

8. Example 2 Determine where the graph is concave up and down, and find any inflection points

9. Ex 5.3 Determine the concavity and inflection points of

10. 2nd Derivative Test The 2nd Derivative can also be used to determine if a critical point is a local max or min. Thm- Suppose that f(x) is continuous on an interval (a,b) and f’( c) = 0, then –If f’’( c) < 0, then c is a local max Concave down means a local max! –If f’’( c) > 0, then c is a local min Concave up means a local min!

11. Warning! The 2nd derivative test does not always work. It will not work if f’’(c) = 0 If the 2nd derivative test does not work, you must use the table

12. Ex Analyze the critical points of

13. Ex 2 Use the 2nd derivative test to find the local max and mins for

14. Closure: Hand in: Find intervals of increase, decrease, and concavity, local extrema, and inflection points of HW: p.239 #1 5 13 25 33 37 43 47 53- 56 65 4.1-4.4 Quiz tomorrow