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This is a math 112 exam from spring 2007, consisting of 12 questions covering derivatives, partial derivatives, tangent lines, local maxima/minima, total cost, average cost, and linear programming. Calculator and one page of notes are allowed.
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(a) (4 pts) G(w, b) = b^3 m + 4 ln(m) −
b^2
− m^3 b^6 + 7m + 14
∂b
∂m
(b) (4 pts) y = ln
( e^3 x^5 +
3 x − 2
)
dy dx
(c) (4 pts) F (x) =
(x^2 + 1)^12 1 + ln(2x + 1)
F ′(x) =
T C(q) =
q^3 3
− 5 q^2 + 21q +
The Average Cost is given by AC(q) =
T C(q) q
(a) (6 pts) Find the two values of x at which the Total Cost graph has a horizontal tangent. Use the Second Derivative Test to determine whether T C(q) reaches a local maximum or a local minimum at each value.
ANSWER: q = gives a local of T C(q);
q = gives a local of T C(q). (b) (4 pts) Give the global maximum and global minimum values of Total Cost on the inter- val from q = 0 to q = 5.
ANSWER: MAX = dollars; MIN = dollars. (c) (4 pts) Is the Average Cost graph concave up, concave down, or neither at q = 3? Justify your answer. Guessing the answer with no supporting work receives zero pts.
ANSWER (circle one): CONCAVE UP CONCAVE DOWN NEITHER
4 x + 6y ≤ 1800 , x ≤ 300 , and y ≤ 150.
(a) (4 pts) Sketch the feasible region.
(b) (4 pts) Find the exact coordinates of the vertices of the feasible region. Label all of them on your graph.
(c) (4 pts) Subject to the given constraints, find the maximum value of the objective function: f(x, y) = 14x + 20y.
ANSWER: maximum =