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

Calculus I - Practice Problems
Derivatives Previous Chapter   Next Chapter Integrals
Finding Absolute Extrema Previous Section   Next Section 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 increases and decreases.

 

1.

[Solution]

 

2.

[Solution]

 

3.  Below is the graph of the derivative of a function.  From this graph determine the open intervals in which the function increases and decreases.

[Solution]

 

4. This problem is about some function.  All we know about the function is that it exists everywhere and we also know the information given below about the derivative of the function.  Answer each of the following questions about this function.

(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.

 

 

[Solution]

 

For problems 5  12 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.

 

5.  [Solution]

 

6.  [Solution]

 

7.  [Solution]

 

8.  on   [Solution]

 

9.  on  [Solution]

 

10.  [Solution]

 

11.  [Solution]

 

12.  [Solution]

 

13. 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 for 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 that you can come up with. 

(b) Using your answer from (a) determine the most general form of the function.

(c) Given that  find a function that will have a relative maximum at .  Note : You should be able to use your answer from (b) to determine an answer to this part. 

[Solution]

 

14. Given that  and  are increasing functions.  If we define  show that  is an increasing function.  [Solution]

 

15. Given that  is an increasing function and define .  Will  be an increasing function?  If yes, prove that  is an increasing function.  If not, can you determine any other conditions needed on the function  that will guarantee that  will also increase? [Solution]

 

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

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

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