Show all work necessary for your answers. 1. Compute the derivative of each of the following functions. (a) f(x) = 4√
mathematics
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Show all work necessary for your answers. 1. Compute the derivative of each of the following functions. (a) f(x) = 4√ x − 5 x 3 (b) y = 5x − 4x 2 2x 2 − 4x + 7 (c) f(x) = (5x 3 − 3x + 2) · e x (d) y = 7x 2 + 2x x + ln(x) 2. Suppose that the function f is given by f(x) = 12, 000 + 20x − 0.005x 2 . (a) If x changes from x = 100 to x = 103, find the following: ∆x, ∆y, and dy. (b) Estimate the change in y as x changes from x = 100 to x = 103. (c) To compute ∆y in part (a), you computed that f(100) = 13, 950. Given your answer to part (a), estimate f(103). 3. The daily cost function for a firm which produces blenders is given by C(x) = 12, 000 + 20x − 0.005x 2 , where x is the number of blenders produced daily. The derivative of C is given by C 0 (x) = 20 − 0.01x. (a) Find the average cost of increasing production from x = 100 to x = 110 blenders per day. (b) Compute C(100) and C 0 (100). (c) Explain the meaning of C(100) and the meaning of C 0 (100). 4. Suppose that the daily price-demand equation for a raincoat is given by p = 30 − 0.05x. Find the daily revenue function for the raincoats. 5. Suppose that the daily revenue equation for the production and sale of x personal speakers is given by the function R(x) = 25x − 0.07x 2 and the daily cost function is given by C(x) = 9000 + 7x + 0.03x 2 . Find the daily profit function, P(x). 6. Suppose that the daily revenue equation for the production and sale of x skillets is given by R(x) = 28x − 0.04x 2 0 ≤ x ≤ 700
