Math 1P97. Assignment 3 (4%). Student's name: ID: Box #: Due date: Monday, June 5, 6:45pm Math 1P97. Assignment 3 (4%). Due date: Monday, June 5, 6:45pm Student's name: Total mark: NOTE: Students are expected to complete ALL questions on the assignment. However, only a subset of questions will be considered for marking. Marks will be deducted for incomplete assignments. Show all your work. 1. (Marginal cost, revenue, and profit) The quantity demanded each day of the Walter Serkin recording of Beethovens Moonlight Sonata, manufactured by Phonola Record Industries, is related to the price/compact disc. The equation p = 0.38x + 60.4 (0 x 100) where p denotes the unit price in dollars and x is the number of discs demanded, relates the demand to the price. The total monthly cost (in dollars) for pressing and packaging x copies of this classical recording is given by C(x) = 20 + 0.4x 0.005x2 (0 x 100) a) Find the revenue function R = R(x) and the profit function P = P (x). b) Find the marginal profit function. c) How many copies should Phonola produce each month to maximize the profit? 2. Refer to the following graph of a function y = f (x): a) find the points where f (x) = 0 b) find the intervals where f 0 (x) > 0 c) find the intervals where f 0 (x) < 0 d) find the points/intervals where f 0 (x) = 0 e) find the points where f 0 (x) doesn't exist. f) find the points/intervals where f 00 (x) = 0. j) find the intervals where f 00 (x) > 0. h) find the intervals where f 00 (x) < 0. 3. Find the interval(s) where the function f (x) is increasing and the interval(s) where it is decreasing. a) f (x) = x3 + 3x2 9x + 10 b) f (x) = x(x2 x(x 1) + 1) c) f (x) = (4 x2 )2 4. Find the relative maxima and relative minima, if any, of each function. a) f (x) = x3 3x2 9x + 10 b) f (x) = x4 8x2 c) f (x) = (x2 + 4x + 4)2 5. Find the absolute maximum value and the absolute minimum value of each function. a) f (x) = x2 x + 6 on [1, 3] b) f (x) = x 16 x2 on [4, 0] 6. (Price-Demand) Suppose the quantity demanded weekly of the Super Titan radial tires is related to its unit price by the equation p + x2 = 144 where p is measured in dollars and x is measured in units of a thousand. How fast is the quantity demanded changing when p = 63, and the price/tire is increasing at the rate of $2/week? 7. (Maximizing yield) An apple orchard has an average yield of 36 bushels of apples/tree if tree density is 22 trees/acre. For each unit increase in tree density, the yield decreases by 2 bushels/tree. How many trees should be planted in order to maximize the yield? 8. (Minimizing packaging costs) A rectangular box is to have a square base and a volume of 40 f t3 . If the material for the base costs 0.3 $/square foot, the material for the sides costs 0.1 $/square foot, and the material for the top costs 0.2 $/square foot, determine the dimensions of the box that can be constructed at minimum cost. 9. Find the horizontal and vertical asymptotes of the graph of the function. a) f (x) = b) f (x) = c) f (x) = 1+x 1x x2 x 1 x3 3x (x 1)(x2 + 2)