business calculus in 6 hours !
milanesta96
04writtenhwm130.pdf
Written Homework 04 (Applying Derivatives and Max-Min problems)
Show all work for full credit!
1. Draw a graph to match the description of the function given (Four functions f(x), g(x), h(x), u(x)). Each
function is continuous over , . Label points where points are specified for the given function.
a. f(x) is decreasing over ( , 2] and increasing over [2, ]
b. g(x) has a negative derivative over ( , 1) , a negative derivative over ( 1,3) and a positive derivative over
(3, ) and. ( 1)g does not exist. (3) 0g .
c. h(x) is decreasing over [ 1,3] but is otherwise increasing. h(3)=1
d. u(x) is increasing and concave up on ,1 and decreasing and concave up on 1, .
2. Find where [at which value(s) of x] each function has a relative extremum (relative max or relative min). In
each case what is the value of the relative maximum or minimum. Then sketch the graph of the function and
label the extreme points. Also, state the intervals where the function is increasing and where it is decreasing.
2( ) 2 6 3f x x x 3 2( ) 6 8g x x x 13( ) ( 1)h x x
3. Find where (at which value of x) each function has a relative extremum (relative max or relative min). In
each case what is the value of the relative maximum or minimum. Then sketch the graph of the function and
label the extreme points. Also, state the intervals where the function is increasing and where it is decreasing.
Finally, state the intervals where the function is concave up and concave down.
3 2( ) 2 3 36 28f x x x x 4 3( ) 4 10g x x x
4. Show All Work. A person coughs when a foreign object is in the windpipe. The velocity of the cough
depends on the size of the object. Supppose a person has a windpipe with a 20-mm radius. If a foreign object
has a radius r, in millimeters, then the velocity V, in millimeters/second, needed to remove the object by a
cough is given by:
2 3( ) (20 ), 0 20,V r k r r r
Where k is some positive constant. For what size object is the maximum velocity needed to remove the object?
5. a. Determine the vertical asymptote(s) of each function. If none exists, state that fact.
2 3 ( )
3 5
x f x
x
2
6 ( )
7 6
x g x
x x
2
7 ( )
49 h x
x
b. Determine the horizontal asymptote of each function.
2
2
3 ( )
6 1
x f x
x
3 ( ) 4g x
x
6. Sketch the graph of 2
2
2 ( )
16
x f x
x
. Indicate the intervals over which f(x) is increasing or decreasing. Also
indicate where any relative extrema occur, where asymptotes occur, where graph is concave up or concave
down and where any points of inflection occur. Also find x and y intercepts for f(x).
7. Find the absolute maximum and absolute minimum values of f(x) on the interval [-3,4] where
2 3( ) 17 9 3f x x x x
8. Using derivatives, Solve and SHOW ALL WORK! When a theater owner charges $5 for admission, there is
an average attendance of 180 people. For every $0.10 increase in admission, there is a loss of 1 customer from
the average number (of attendees). What admission should be charged in order to maximize revenue? And what
is the maximum revenue
9. Using derivatives, Solve and SHOW ALL WORK!
The local SPCA is constructing a kennel with 12 cages built as a chain link rectangular enclosure with chain
link dividers as shown below. The cost of the chain link fence for the outside rectangular lot borders is
$10/linear foot. The cost of the chain link fence for the inside dividers is $5/linear foot. Assuming the SPCA
has allocated $8000 for the outside fencing and inside dividers, find the length and width of the rectangular
enclosure of maximum area. What is that maximum area? Note, you must identify a variable to be maximized
or minimized and then use calculus methods to solve for length and width. Be sure to label on the diagram
below what is length and what is width, Also, be sure to verify whether you have a maximum or minimum
value!
10. Bon Temps Surf and SCUBA Shop sells 360 surfboards per year. It costs $8 to store one surfboard for a
year. Each reorder costs $10, plus an additional $5 for each surfboard ordered. How many times per year
should the store order surfboards, and in what lot size, in order to minimize inventory costs?
