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Calculus I (Assignment Problems) / Applications of Derivatives / The Shape of a Graph, Part II   [Notes] [Practice Problems] [Assignment Problems]

Calculus I - Assignment Problems
Derivatives Previous Chapter   Next Chapter Integrals
The Shape of a Graph, Part I Previous Section   Next Section The Mean Value Theorem

 The Shape of a Graph, Part II

For problems 1 & 2 the graph of a function is given.  Determine the open intervals on which the function is concave up and concave down.

 

1. 

 

2.

 

 

For problems 3  5 the graph of the 2nd derivative of a function is given.  From this graph determine the open intervals in which the function is concave up and concave down.

 

3.

 

4.

 

5.

 

 

 

 

For problems 6  18 answer each of the following.

(a) Determine the open intervals on which the function is concave up and concave down.

(b) Determine the inflection points of the function.

 

6.  

 

7.  

 

8.  

 

9.  

 

10.  

 

11.  on  

 

12.  on  

 

13.  on  

 

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18.  

 

For problems 19  33 answer each of the following.

(a) Identify the critical points of the function.

(b) Determine the open intervals on which the function increases and decreases.

(c) Classify the critical points as relative maximums, relative minimums or neither.

(d) Determine the open intervals on which the function is concave up and concave down.

(e) Determine the inflection points of the function.

(f) Use the information from steps (a)  (e) to sketch the graph of the function.

 

19.  

 

20.  

 

21.  

 

22.  

 

23.    

 

24.  on   

 

25.  on  

 

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28.   

 

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34. Answer each of the following questions.

(a) What is the minimum degree of a polynomial that has exactly two inflection points.

(b) What is the minimum degree of a polynomial that has exactly three inflection points.

(c) What is the minimum degree of a polynomial that has exactly n inflection points.

 

35. For some function, , it is known that there is an inflection point at .  Answer each of the following questions about this function.

(a) What is the simplest form that the 2nd derivative of this function?  . 

(b) Using your answer from (a) determine the most general form that the function itself can take.

(c) Given that  and  find a function that will have an inflection point at .

 

For problems 36  39  is a polynomial.  Given the 2nd derivative of the function classify, if possible, each of the given critical points as relative minimums or relative maximum.  If it is not possible to classify the critical point(s) clearly explain why they cannot be classified.

 

36. .  The critical points are : ,  and .

 

37. .  The critical points are : ,  and

 

38. .  The critical points are : ,  and .

 

39. .  The critical points are : ,  and .  

 

40. Use  for this problem.

(a) Determine the critical points for the function.

(b) Use the 2nd derivative test to classify the critical points as relative minimums or relative maximums.  If it is not possible to classify the critical point(s) clearly explain why they cannot be classified.

(c) Use the 1st derivative test to classify the critical points as relative minimums, relative maximums or neither.

 

41. Given that  and  are concave down functions.  If we define  show that  is a concave down function.

 

42. Given that  is a concave up function.  Determine a condition on  for which    will be a concave up function.

 

43.  For a function  determine conditions on  for which  will be a concave up function.  Note that there are several sets of conditions that can be used here.  How many of them can you find?

 

The Shape of a Graph, Part I Previous Section   Next Section The Mean Value Theorem
Derivatives Previous Chapter   Next Chapter Integrals

Calculus I (Assignment Problems) / Applications of Derivatives / The Shape of a Graph, Part II    [Notes] [Practice Problems] [Assignment Problems]

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