Math
160

Test
3 Name: __ Key __

You must Show All Work to receive credit.

Numerical answers should be accurate to at least 4
significant digits (unless otherwise specified)

Include units in your answers where appropriate

All
question on this sheet refer to the function

1) a) Find all critical number(s) of f(x)

2) Use
either the 1^{st} derivative test or the 2^{nd} derivative test
(your choice) to determine whether each critical number is a local maximum,
local minimum, or neither.

1^{st}
derivative test: based on the factorization of f’(x) found above, we see

*so f has a local maximum at
x=-1 and a local minimum at x=2*

2^{nd}
derivative test: , *confirming the above results.*

3) Find with justification the locations of the
absolute maximum and minimum values of f(x) on the domain [-3 3]

*We just need to check the
endpoints and critical points:*

*f(-3) = -44 **ß** absolute minimum*

*f(-1) = 8 **ß** absolute maximum*

*f(2) = -19*

*f(3) = -9*

4) Find the point(s) of inflection of f(x) and
its intervals of concavity.

*f”(x) = 12x-6 . This equals 0 when x= ½ . If x< ½ , f”(x) < 0 so the graph is
concave down. *

*If x> ½ f”(x) >0 so it
is concave up. The point of inflection
is at x= ½ *

5) A 13 foot ladder is leaning against a
wall. The bottom of the ladder is being
pulled away from the wall at a rate of ½ ft/min. At what rate is the top of the ladder sliding
down when the base is 5 feet from the wall?

6) Evaluate the following limits for the
function shown in the graph:

2

4

-∞

∞

7) For the function from the previous graph,
where (approximately) are the points of inflection? (hint – you should find 2 of them) *The
two points of inflection are where the concavity changes – approximately at
x=-4 and x=1*

8) The Math Club is selling “I ♥ Math”
shirts. The demand function for these
shirts is where x is the number of shirts sold and p is
the price, in dollars. The cost of
making x shirts is dollars.

a. Find R(x), the revenue function.

b. Find
P(x), the profit function.

c. Determine the number of shirts that the club
should make to maximize its profit, the price it should charge, and the maximum
possible profit.

*So the club should make 80
shirts and sell them for $12 apiece.
This will produce a profit of *

9) What is the elasticity of demand for the
t-shirts from the previous problem when the price is $8?

*When p = $8, the quantity
sold will be 120, because 8 = -1/10 (120)+20*

No top

10) You want to build a box with a square base
and rectangular sides but

__no top__.
The volume of the box must be 500 cm^{3}. Determine with justification the dimensions
that would minimize the amount of material needed to construct this box. *Hint: start by identifying the objective function
and constraint equation(s)*

*Note: picture meant to be helpful, but not
drawn to scale. Feel free to draw a
better picture!*

So
the square base is 10 cm on a side, and the height is then 5 cm.

This
is the only critical point, and it is a local and absolute minimum (as x gets
smaller, 2000/x blows up, and as x gets larger x^{2} gets very large),
so these are the optimal dimensions.