Calculus Evaluation Self Test

Mineral and Energy Economics Program

Golden, CO, USA

A semester of calculus is a prerequisite for graduate program in Mineral and Energy Economics. The following self test is to help you evaluate how well you can apply calculus. You will be expected to know all the rules in the following questions. Since these tools will be applied to economic problems in your core courses and many other courses in the program, it is important that you have mastery of them prior to starting the program. If you need to brush up on these rules and others in an economic context see:

Reference: Dowling, Edward T. Schaum's Outline Series (Sh) Introduction to Mathematical Economics available at http://www.amazon.com/.

Derivatives

In Economics we use a variety of functions to represent utility, cost, production, demand and supply, etc. Often we need to know how an independent variable changes a dependent variable. Derivatives give us such information. If we have a function y = f (x), mathematically a derivative is

also written as y', df(x)/dx or dy/dx. The derivative of a function, f'(x) measures the slope of a function f(x) at a given point, which is the instantaneous rate of change of f(x) at that point.

Practice Derivative Rules

Constant-Function Rule

1. True False Given f(Q) = 1,500; df/dQ = 0

2. True False Given I(t) = e0.03, dI/dt = 0.03e0.03

Power-Function Rule

3. True False Given y = 0.5x4, dy/dx = 2x3

4. True False Given y = x -3, dy/dx = -3 x-2

Sum-Difference Rule

5. True False d/dx(3x5 + 2x) = 15x4

6. True False d/dx(4x3 - 2x2 -5) = 12x2 - 4x

7. True False d/dx(ax2 - bx + c) = ax - b

Product Rule

8. True False d/dx[(3x + 2)(2x3)] = 24x3 + 12x2

9. True False d/dx[(x-3 - 3x2)(6x + 1)] =-18x-1 - 3x-2 - 42x

10. True False d/dx[cx2(ax + b)] = 3acx2 + 2bcx

Quotient Rule

11. True False d/dx[(2x-3)/(x-1)] = (x-1)-2

12. True False d/dx[ax2/(bx+c)] = ax (bx - c)/(bx + c)2

13. True False d/dx[(5x2-3x)/(x2 + 1)] = (x2 + 1)-2(10x3-3x2)

Chain Rule

14. True False Given z = y3, where y = 2x + 3, so z = (2x+3)3, then dz/dx = 3(2x + 3)2

15. True False If z = 3y - 2, where y = x2 - 4, then z = 3(x2 - 4) -2 and dz/dx = 6x

16. True False If TC = f(w), where w = g(L), then dTC/dL = d(f(g(L)) = f'(w)*g'(L)

Inverse Function Rule

17. True False If y = 3x + 20, then dx/dy = 1/3.

18. True False If y = -3x(1/3) + 5, then dx/dy = x2/3.

19. True False If y = ax3 + b, where a, b > 0, then dx/dy = 3ax2

Economic Interpretations of Derivatives

20. True False The total cost function is given by TC = 2Q2 + 5Q +10, where Q is units of output produced. The derivative of the total cost function with respect to quantity dTC/dQ = 2Q + 5. (Note the economic expression for dTC/dQ is the marginal cost.)

21. True False Suppose the marginal cost function is given by MC = Q2 +3Q. The value of dTC/dQ = MC at Q=2, or the value of marginal cost for the second unit produced is 10.

22. True False The elasticity of demand is defined as e = [dQ/dP]*P/Q. It tells us how responsive quantity is to price. If the demand function is given by Q = 20 - 4P, then the elasticity of demand at P = 3 is equal to -1.5.

23. True False Suppose the average revenue function is given as AR = f(Q). Then the total revenue will be given by TR = f(Q)*Q. Therefore, the marginal revenue function is given by MR = Q*f'(Q).

24. True False If the total function is defined as TC = TC(Q), then the average cost function will be the quotient = AC = TC(Q)/Q. Therefore, the rate of change of AC with respect to Q will be given by dAC/dQ = [Q*TC'(Q) - TC(Q)]/Q2.

25. True False Suppose the demand function is given by Q = 3 - 0.5P. Then the slope of the inverse demand function P = P(Q) is equal to -0.5.

Optimization

In general, to optimize a function y=f(x), set the first derivative equal to zero, solve f'(x)=0 for x0 to find the optimal level of the independent variable, and check the sign of the second derivative at the optimal point:

f(x0) is a relative maximum if f"(x0)<0.

f(x0) is a relative maximum if f"(x0)>0.

f(x0) is a either a relative minimum, or a relative maximum, or an inflection point if f"(x0)=0.

26. True False x=6 is a maximum of f(x) = x2 - 12x + 13.

27. True False. Given y = -2x2 + 4x + 9, x0 = 1 is a global maximum.

Economic Applications of Optimization

28. True False. The optimal size plant is the output where the average total cost is at a minimum. Suppose the total cost function is given by TC = Q4/3 - 6Q3 +11Q2 +30Q. Average cost is AVC = TC/Q. The optimal level of output is found by setting dAVC(Q)/dQ=0 and is found to be equal to 7. At this optimal point AVC(Q) is at its minimum.

29. True False. Suppose your profit function is p = -Q2 + 12Q - 25. Your optimal or profit maximizing output is 6.

Partial Differentiation

Consider a function y = f(x1, x2,. . . ., xn), where the variables xi (i=1,2, . . .,n) are all independent of each other. If the variable x1 is changed by Dx1 while x2, . . ., xn remain fixed, there will be a corresponding change y, Dy. In this case, the partial derivative of y with respect to x1 is defined as

The partial ∂y/∂xi is also written as fi. Taking partial derivatives is very straight forward. It is just like taking total derivatives except you treat all other variables that are not changing as if they were constant. All the rules of total derivatives are applicable to partial derivatives.

30. True False If y = f(x1, x2) = 2x1+ x1x2 +3x22, then f1 = 2 + x2 and f2 = x1+ 6x2

31. True False If y = f(u,v) = (u + 3)(2u +v2), then fu = 3u + v2 + 6 and fv = 2v(u+3)

32. True False If y = f(u,v) = (3u - 2v)/(u+3), then fu = (9 - 2v)/(u + 3)2 and fv = 2u/(u+3)2

Economic Application of Partial Derivatives

33. True False Suppose utility is derived from the consumption of two goods, x and y. Given the utility function U = -x2 + 200xy - y2, partial derivatives of U(x,y), known as marginal utilities of goods x and y are Ux = -2x +200y, Uy = 200x - 2y.

34. True False Suppose you have a production function Q = K1/2L1/4E1/3. Where Q is output, L is labor, and E is energy. The partial derivative of the production function with respect to K capital is the marginal product of capital. ∂Q/∂K = 1/2K-1/2L1/4E1/3 + 1/4K1/2L-3/4E1/3 + 1/3 K1/2L1/4E-2/3.

Derivatives of Exponential and Logarithmic Functions

35. True False Let y = lnt. dy/dt = d(lnt)/dt = 1/t = t-1

36. True False d/dt(et) = tet-1

37. True False d/dt(e-0.025 t) = -0.025e-0.025t

38. True False d(exp(x2))/dx = 2x*(exp(x2))

39. True False d(lnt4)/dt = 4t3lnt4

40. True False d/dt(t3lnt2) = (1 + 3lnt)2t2

Economic Examples of Derivatives of Logs and Exponents

41. True False Suppose the value of wine grows according to Vt = Kexp(t0.5), i.e. at t=0, Vt = K. Then the optimal time of selling the wine is t* = (4r2)-1.

42. True False. Suppose that the value of a certain asset grows according to St = S0ert. Then the rate of growth of St is r.

The Indefinite Integral.

Frequently in Economics, we know the rate of change of a function F'(x)and want to find the original function (F(x)). For example if F'(x) was marginal utility then F would be total utility. Reversing the process of differentiation and finding the original function from the derivative is called integration. The original function F(x) is called the integral of F'(x). Thus when we integrate a function we look for a function such that when we take the derivative we get the function in the integral. (i.e. ∫F'(x) = F(x))

Integral of constant

43. True False ∫3dx = 0

Integral of a power

44. True False ∫3x2dx = x3 + c

Integral of an inverse

45. True False ∫x-1dx = ln(x) + c

# Integral of Logs and Exponents

46. True False ∫32xdx = 32x/ln(3) + c

47. True False ∫e9xdx = e9x/9+c

Integration by Parts

48. True False ∫10x(x+1)3dx = 2.5x(x+1)4 - 0.5(x+1)5 + c

49. True False ∫3xexdx =3xex - ex + c

Integration by Substitution

50. True False ∫15x2(x3+1)dx = 3(x3+1)2 + c

The Definite Integral

The area under a graph of a continuous function f(x) from a to b (a<b) can be expressed as a definite integral of f(x) over the interval a to b:

51. True False

Economic Examples of Integrals

52. True False. If marginal cost MC = dTC/dQ = 16 + 9Q - 6Q2 and FC = 20, then TC =

53. True False. Consumer surplus is a measure of consumer welfare and is the area under the demand curve and above price. Given the inverse demand function P= 34 - 2Q and the equilibrium price P* = 8, consumer surplus at equilibrium price CS = ∫P(Q)dQ - Q*P* = 169.

This test was last updated: August 9, 2011.