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Homework 4 1 Calculus Homework 4 Due Date: November 7 (Wednesday) 1. Reading assignments (a) 2.3 Rates of Changes: Velocity and Marginals (b) 2.7 Implicit Differentiation (c) 2.8 Related Rates (d) 3.1 Increasing and Decreasing Functions (e) 3.2 Extrema and the First-Derivative Test 2. The profit P (in dollars) from selling x units of a product is given by √ 1 P = 36, 000 + 2048 x − 2 , 8x 150 ≤ x ≤ 275. Find the marginal profit for each of the following sales. (a) x = 150 (b) x = 200 (c) x = 300. (It does not make any sense when x = 300.) Solution: 3. The profit P (in dollars) from selling x laptop computers is given by P = −0.04x2 + 25x − 1500. 2 Calculus H415611 Fall 2012 (a) Find the additional profit when the sales increase form 150 to 151 units. (b) Find the marginal profit when x = 150. (c) Compare the results from parts (a) and (b). Solution: 4. Finding the derivative of the function. (a) f (x) = (x3 − 3x)(2x2 + 3x + 5) (b) f (x) = x3 +3x+2 x2 −1 (c) f (x) = (3x3 + 4x)(x − 5)(x + 1) (d) f (x) = (x+1)(2x−7) 2x+1 (e) f (x) = (2x3 − 1)2 (f) f (x) = x2 −2x+5 √ x (g) f (x) = 6−5x 2 2 x −1 (h) f (x) = q 2x x+1 5. The ordering and transportation cost C per unit (in thousands of dollars) of the components used in manufacturing a product is given by C = 100 200 x + 2 x x + 30 where x is the order size (in hundreds). Find the rate of change of C with respect to x for each order size. What do these rates of change imply about increasing the size of an order? Of the given order sizes, which would you choose? Explain (a) x = 10 (b) x = 20 (c) x = 30 Solution: Homework 4 3 6. The value V of a machine t years after it is purchased is inversely proportional to the square root of t + 1. The initial value of the machine is $10,000. (a) Write V as a function of t. (b) Find the rate of depreciation when t = 1. Solution: