Calculus/Differentiation/Applications of Derivatives/Exercises

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Differentiation/Applications of Derivatives/Exercises

Contents

Relative ExtremaEdit

Find the relative maximum(s) and minimum(s), if any, of the following functions.

1.  

none

2.  

Minimum at the point  

3.  

Relative minimum at  

4.  

Relative minimum at  
Relative maximum at  

5.  

Relative minimum at  

6.  

Relative minimum at  
Relative maximum at  

Solutions

Range of FunctionEdit

7. Show that the expression   cannot take on any value strictly between 2 and -2.

 
 
 
 
 
Since   is negative,   corresponds to a relative maximum.
 
 
For   ,   is positive, which means that the function is increasing. Coming from very negative  -values,   increases from a very negative value to reach a relative maximum of   at   .
For   ,   is negative, which means that the function is decreasing.
 
 
 
Since   is positive,   corresponds to a relative minimum.
 
Between   the function decreases from   to   , then jumps to   and decreases until it reaches a relative minimum of   at   .
For   ,   is positive, so the function increases from a minimum of   .
The above analysis shows that there is a gap in the function's range between   and   .

Absolute ExtremaEdit

Determine the absolute maximum and minimum of the following functions on the given domain

8.   on  

Maximum at   ; minimum at  

9.   on  

Maximum at   ; minimum at  

Solutions

Determine Intervals of ChangeEdit

Find the intervals where the following functions are increasing or decreasing

10.  

Increasing on   ; decreasing on  

11.  

Decreasing on   ; increasing elsewhere

12.  

Increasing on   ; decreasing elsewhere

13.  

Increasing on   ; decreasing elsewhere

14.  

Decreasing on   ; increasing elsewhere

15.  

Decreasing on   ; increasing elsewhere

Solutions

Determine Intervals of ConcavityEdit

Find the intervals where the following functions are concave up or concave down

16.  

Concave down everywhere

17.  

Concave down on   ; concave up on  

18.  

Concave up on   ; concave down on  

19.  

Concave up on   ; concave down on  

20.  

Concave down on   ; concave up on  

21.  

Concave down on   ; concave up on  

Solutions

Word ProblemsEdit

22. You peer around a corner. A velociraptor 64 meters away spots you. You run away at a speed of 6 meters per second. The raptor chases, running towards the corner you just left at a speed of   meters per second (time   measured in seconds after spotting). After you have run 4 seconds the raptor is 32 meters from the corner. At this time, how fast is death approaching your soon to be mangled flesh? That is, what is the rate of change in the distance between you and the raptor?

 

23. Two bicycles leave an intersection at the same time. One heads north going   and the other heads east going   . How fast are the bikes getting away from each other after one hour?

 

24. You're making a can of volume   with a gold side and silver top/bottom. Say gold costs 10 dollars per m  and silver costs 1 dollar per   . What's the minimum cost of such a can?

$878.76

Solutions

Graphing FunctionsEdit

For each of the following, graph a function that abides by the provided characteristics

25.    
There are many functions that satisfy all the conditions. Here is one example:
26.  
There are many functions that satisfy all the conditions. Here is one example:

Solutions

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