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)

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

1st 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

2nd 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 cm3.  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 x2 gets very large), so these are the optimal dimensions.