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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