A07-derivarive-how to finde some values of

presentation

1. CALCULUS PREP PROF. DAVI MISTURINI CLASS 07: INTRODUCTION TO INTEGRALS  

2. MAXIMUM AND MINIMUM VALUES A function has an absolute maximum (or global maximum ) at if for all . A function has a local maximum (or relative maximum ) at if when is near .   Example : Analysing with a graphic calculator the minimum and maximum values of where .  

3. The extreme value theorem : If is continuous on a closed interval , then attains an absolute maximum value and an absolute minimum value , for some numbers .   Fermat’s Theorem : If as a local maximum or minimum at , and if exists, then .  

4. The extreme value theorem : If is continuous on a closed interval , then attains an absolute maximum value and an absolute minimum value , for some numbers .   Fermat’s Theorem : If as a local maximum or minimum at , and if exists, then .   If for some , then is a maximum/minimum?  

5. A critical number of a function is a number in the domain of such that either or does not exist. If has a local maximum or minimum at then is a creitical number.   Example : Find the critical numbers of .  

6. Example : Find the absolute maximum and minimum values of the function where .  

7. Example : Use a graphing device to estimate the absolute minimum and maximum values of the function , Use calculus to find the exact minimum and maximum values.  

8. THE MEAN VALUE THEOREM

9. MOSTRAR GEOGEBRA

10. Theorem: If for all in an interval then is a constant on  

11. HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH Example : Find where the function is increasing and where it is decreasing.  

12. Example : Discuss the curve with respect to concavity, points of inflection, and local maxima and minima. Use this information to sketch the curve.  

13. L’HOSPITAL’S RULE

14. Example : Evaluate A) B ) C)  

15. QUESTIONS?