Assignment 2: Differentiation
Due date: Tue, Feb. 9

1. A firm's cost function is C = Q3 - 5Q2 + 14Q + 100, where Q is output.

  1. Obtain the variable cost function.
  2. Obtain the marginal cost function. Provide a sketch.
  3. Obtain C(10), C'(10) and C''(10).
  4. What is the intrepretation of C'(10)?

2. Consider the function, y = (9x2 - 2)(3x + 1). Obtain dy/dx using the product rule.

3. Consider the function, y = (x + 7) / x. Obtain dy/dx using the quotient rule.

4. Given the function f(x) = ax + b, obtain the derivatives of:

  1. f(x)
  2. xf(x)
  3. 1/f(x)
  4. f(x)/x

5. Consider the functions, y = f(u) = 5u2 and u = 3 ln(x) + 2. Obtain dy/dx.

6. A consumer derives utility from the consumption of two goods X and Y. The consumer's utility function is: u(x,y) = x1/3 y2/3, where u represents utility, and x and y represent the amounts consumed of the two goods.

  1. Obtain u1. What does it represent?
  2. If x = 8 and y = 27, what is the marginal utility of good Y?
  3. Obtain u11 and u22. Show that they are negative.
  4. Show that u12 = u21.

7. A firm's production function is given by Y = F(K,L) = RKa Lb, where y is output; K and L are inputs of capital and labor resp.; R, a and b are parameters.

  1. Obtain the partial derivatives of Y w.r.t K and L.
  2. What do F1 and F2 represent?

8. Consider a special case of the production function in Question 7. Let a = 0.7, b = 0.3 (Note: a + b = 1).

  1. Find F1 and F2.
  2. Obtain the partial derivatives of F1 and F2 w.r.t K and L.
  3. What are the signs of F11, F12 and F22?
  4. Is F12 = F21? Show this.

9. Consider the implicit function, x2 + 3y2 = 50.

  1. Totally differentiate to obtain dy/dx (except for y = 0).
  2. Why is the function not differentiable at y = 0?
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