Download Sec 14

Third Midterm Exam Math 1330 Section 14, 6 Nov 2006
Relax, relax, RELAX, relax, relax…
Please show all of your work so you can receive partial credit.
Note that x^2 means x-squared, i.e., x2, and that exp(x) means the natural exponential
function, i.e., ex. Please ask if any other notation is unfamiliar.
1. For the graph below, estimate
(a.) the intervals on which the function is increasing.
(b.) the intervals on which the function is decreasing.
(c.) the approximate local maxima and minima (both x and y coordinates).
(d.) the intervals on which the function is concave up.
(e.) the intervals on which the function is concave down.
(f.) the approximate inflection points, if any.
2. Sketch the slope graph of this graph.
3. You have determined that you can sell 800 mugs if the price is $3.00, 300 mugs if the
price is $5.00, and 100 mugs if the price is $7.00. The supplier will sell you mugs for
$2.50 each. You also must pay for fixed cost of $50 for the initial design setup. A
reasonable demand model for selling mugs using the power function trendline is
p(q) = 46.899 q^(-0.4054)
where the p is the price in dollars to sell q mugs.
(a.) Determine the revenue function and the cost function and the profit function in
terms of the variable q.
(b.) How many mugs should you order and sell to maximize profit. Use any method
to find this maximum.
(c.) More generally, we can handle an infinite number of such scenarios at once. Let
p(q) = A q^(-b)
where p is the price to sell q units of the product, and with A and b positive constants.
Let C be the fixed costs, and let U be the unit cost of each item (assume C and U are
positive constants). Explain why the profit function is given by
π( q) = A q^(1-b) – U q – C.
(d.) Find the quantity q that will maximize profit, and find the corresponding price p
(use the symbolic method, including verifying via the second derivative that the point is a
maximum.)
4. Use symbolic calculus methods to find the optimal points for these functions on these
domains. In particular, find the derivative, set it equal to zero, find the critical points,
calculate the functions at the critical points and endpoints, find the second derivative, use
the second derivative to verify if a point is a local max or min.
(a.) x * exp(-x^2 / 2) for -5 ≤ x ≤ 5
5. Lay out the mathematical argument that the revenue is maximized when the elasticity
is one (more precisely, when η = -1). Explain with a few words each step.