Reference no: EM131038302

Calculus-

1. The average cast a(q) of producing q items is defined as the total cost C(q) divided by the number of Items, q, so a(q) = C(q)/q. Explain why the average tea Ls minimized when the marginal can Ls equal to the average cost In:

(a) mathematical terms

(b) economic terms.

2. The average cost per item to produce q items is given by a(q) = 0.01q² - 0.6q + 13. for q> 0.

(a) What Is the total cost, C(q), of producing q goods?

(b) What is the minimum marginal cost? What is the practical interpretation of this result?

(c) At what production level is the average cost a minimum? What is the lowest avenge cost?

(d) Compute the marginal met at q = 30. How don this relate to your answer to pan (c)? Explain this relationship both analytically and in words.

3. Consider the function f(x) = x³ - 3x².

(a) Find the equation of the derivative f'(x).

(b) Find the equation of the tangent to the graph of f at x = 2.

(c) Find the equation of the normal at x = 2. (The normal is the line perpendicular to the tangent).

(d) Sketch f(x),f'(x)', the tangent at x = 2. and the normal at x = 2. Label any critical points of f.

4. Find the derivatives of the following (Unction using the various rules for computing derivatives

(a) f(x) = x³ - 3x² + 7x-6

(b) g(x) = x + √(x² + 1/x²)

(c) r(t)=(t² - t + 1)(5 + 2t - t³)

(d) q(s) = (2s - 7)⁵(3s + 1)⁶ (Factor the answer as much as possible.)

(e) w(z) = z² + 8z + 1/z² - 1

5. Find the global maxima and minima of the following functions over the stated interval.

(a) f(x) = 2x² +6x +2 on [-1, 3].

(b) g(x) =  x³ - x² +12x - 7 on [-4, 4].