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

Calculus I - Assignment Problems
Derivatives Previous Chapter   Next Chapter Integrals
Finding Absolute Extrema Previous Section   Next Section The Shape of a Graph, Part II

 The Shape of a Graph, Part I

For problems 1  4 the graph of a function is given.  Determine the open intervals on which the function increases and decreases.

 

1.

 

2.

 

3.

 

4.

 

For problems 5  7 the graph of the derivative of a function is given.  From this graph determine the open intervals in which the function increases and decreases.

 

5.

 

6.

 

7.

 

 

For problems 8  10 The known information about the derivative of a function is given. From this information answer each of the following questions.

(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.

 

8.

 

 

 

9.

 

 

 

10.

 

 

 

For problems 11  28 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.

 

11.  

 

12.  

 

13.  

 

14.  

 

15.  

 

16.  

 

17.  

 

18.  on   

 

19.  on  

 

20.  on  

 

21.  on  

 

22.   

 

23.   

 

24.   

 

25.    

 

26.   

 

27.   

 

28.   

 

29. Answer each of the following questions.

(a) What is the minimum degree of a polynomial that has exactly one relative extrema?

(b) What is the minimum degree of a polynomial that has exactly two relative extrema?

(c) What is the minimum degree of a polynomial that has exactly three relative extrema?

(d) What is the minimum degree of a polynomial that has exactly n relative extrema?

 

30. For some function, , it is known that there is a relative minimum at .  Answer each of the following questions about this function.

(a) What is the simplest form that the derivative of this function?  Note : There really are many possible forms of the derivative so to make the rest of this problem as simple as possible you will want to use the simplest form of the derivative. 

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

(c) Given that  find a function that will have a relative minimum at Note : There are many possible answers here so just give one of them.

 

31. For some function, , it is known that there is a relative maximum at .  Answer each of the following questions about this function.

(a) What is the simplest form that the derivative of this function?  Note : There really are many possible forms of the derivative so to make the rest of this problem as simple as possible you will want to use the simplest form of the derivative. 

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

(c) Given that  find a function that will have a relative maximum at Note : There are many possible answers here so just give one of them.

 

32. For some function, , it is known that there is a critical point at  that is neither a relative minimum or a relative maximum.  Answer each of the following questions about this function.

(a) What is the simplest form that the derivative of this function?  Note : There really are many possible forms of the derivative so to make the rest of this problem as simple as possible you will want to use the simplest form of the derivative. 

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

(c) Given that  find a function that will have a critical point at  that is neither a relative minimum or a relative maximum.  Note : There are many possible answers here so just give one of them.

 

33. For some function, , it is known that there is a relative maximum at  and a relative minimum at .  Answer each of the following questions about this function.

(a) What is the simplest form that the derivative of this function?  Note : There really are many possible forms of the derivative so to make the rest of this problem as simple as possible you will want to use the simplest form of the derivative. 

(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 a relative maximum at  and a relative minimum at Note : There are many possible answers here so just give one of them.

 

34.  Given that  and  are increasing functions will  always be an increasing function?  If so, prove that  will be an increasing function.  If not, find increasing functions,  and , so that  will be a decreasing function and find a different set of increasing functions so that  will be an increasing function.

 

35. Given that  is an increasing function.  There are several possible conditions that we can impose on  so that  will be an increasing function.  Determine as many of these possible conditions as you can.

 

 

36. For a function   determine a set of conditions on , different from those given in #15 in the practice problems, for which  will be an increasing function.

 

37. For a function  determine a single condition on  for which  will be an increasing function.

 

38.  Given that  and  are positive functions. Determine a set of conditions on them for which  will be an increasing function.   Note that there are several possible sets of conditions here, but try to determine the “simplest” set of conditions.

 

39.  Repeat #38 for .

 

40.  Given that  and  are increasing functions prove that  will also be an increasing function. 

 

Finding Absolute Extrema Previous Section   Next Section The Shape of a Graph, Part II
Derivatives Previous Chapter   Next Chapter Integrals

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

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