Extrema, MVT, Understanding Graphs

1. Extrema, MVT, Understanding Graphs

3. What can we say about f given the graph of f’(x)

4. If the graph is a velocity function of a particle moving, where would the the particle momentarily stop moving?

5. Critical Value Remember, it has to EXIST in the original function to be a critical value. Find the critical value(s) of the function (if any)

6. Draw an example of both statements and think of a function which agrees with the 2nd one

7. Determine the extrema in each. Justify your answer.

8. Analytical Example: Find critical values and determine extrema. No calculator

9. True: since p ‘(x) = 0 at x = 1, and it’s a simple root, so it passes through the x-axis p has a relative extrema though we don’t know the type

11. Looking at the graph of f’(x) where is f concave up?

12. Looking at f “(x), where is f concave up?

13. (and the point actually exists) Given f “(x) Identify points of inflection, justify your answer

14. Given f ‘(x) Identify points of inflection, justify your answer Given a function is continuous If there were f ‘(x), are there any POI? If there were f “(x), are there any POI?

15. Given h(t) is continuous and differentiable, where does h have POI (if any)? No-calc

16. Analytically Find intervals of concavity and POI Justify your answer

17. Connect A to B, while keeping it continuous and a function Where is the highest and lowest ?

18. Analytical Example. No Calc

19. Mean Value Theorem: “He is mean!” “Nah, he is just AVERAGE”

20. Remember, it has to be [cont.] and (diff) on the interval Given find all values of c that satisfy the Mean Value Theorem.

24. solution

25. done