Project #17870 - Calculus I

1. Let  .   Find the -  and -intercepts,  the  critical points of  ,  determine where  the is increasing and where   is decreasing  and then find and identify local extreme values (give both   and  ).  Then determine where    is concave up and where it is concave down and identify any inflection points.  Finally use this information to sketch the graph of  .

 

2.  Let  .   State the domain of  ,  then find the -  and -intercepts and the vertical and horizontal asymptotes of  .   Next find the critical points of   and  determine where 𝑔  is increasing and where    is decreasing  and then find and identify local extreme values (give both   and  ).  Then determine where    is concave up and where it is concave down.  Finally use this information to sketch the graph of  .

 

3.  Let   .  State the domain of    and find the -  and -intercepts,  the  critical points of  ,  determine where  the is increasing and where it is decreasing  and find and identify local extreme values (give both   and  ).  Then determine the concavity of  . Finally use this information to sketch the graph of  .

 

4.  A rectangular pen with  4 sides and an interior divider in the middle is to be built so that total area of the pen is  600  (constraint).  The cost of fencing for the four sides is  $2.00 per foot and the cost of fencing for the divider is $1.00 per foot.   Determine the dimensions of the pen that will minimize the total cost of the fencing.  What is this minimum cost of fencing?  Be sure to include a diagram ,  an explanation of the variables used,  the domain of the cost function and a concluding statement.  Also be sure to show that your answer is the minimum value of the cost function.  Round the lengths to one decimal place and the cost to two decimal places.

 

5.  a.  A boat on the ocean is  5 miles from the nearest point on a straight shoreline;  that point is  12 miles from a restaurant on the shore.  A woman plans to row the boat straight to a point on the shore and then walk along the shore to the restaurant.  If she walks at  4 miles per hour and rows at  2 miles per hour,  at which point on the shore should she land to minimize the total travel time?  What is the minimum travel time?

Note that time equals distance divided by speed  ( ).  Be sure to include a diagram ,  an explanation of the variables used,  the domain of the time function and a concluding statement.  Also be sure to show that your answer is the minimum value of the time function.  Round your answer to one decimal place.   (This is similar to Problem  #15, p. 262).

                                                                                     

6.  a. Write the equation of the line that represents the linear approximation of the function   at  .  Then use this to approximate    Use the actual value of    (rounded to three  decimal places)  to find the percent error in your approximation.

b.  Do the same for   .  Which approximation gives a smaller percent error?

7.  The volume of a sphere of radius    is  .  Find the differential of  .  (Note that    is the independent variable.)   Then use the differential to approximate the change in the volume of the sphere if the radius increases from    feet to   feet.  (Round to one decimal place.)

b.  Approximate what happens to the volume of the sphere if the radius decreases from    feet to   feet.

 

8.  Let    on  .  Determine the value of     between  1  and  3  that is guaranteed to exist by the Mean Value Theorem.

 

9.  a.  Show that the point    guaranteed to exist by the Mean Value Theorem for   on   is the arithmetic mean of    and  ,  that is   .

b.  Show that the point    guaranteed to exist by the Mean Value Theorem for     on  ,  where  ,   is the geometric mean of    and  ,  that is   .

 

 

 

 

 

 

 

 

 

 

Subject Mathematics
Due By (Pacific Time) 11/23/2013 07:00 pm
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