Calculus Evaluation Test

A semester of calculus is a prerequisite for graduate program in Mineral 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 and will be examined on them in EBGN 409 in the first few weeks of classes. 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 f'(x) = lim_Dx®0 [f (x + Dx) - f(x)]/Dx, also written as y', d f(x)/dx , or dy/dx. The derivative of a function, f'(x) measures the slope of a function f(x) at a given point, i.e. 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) = e^0.03, dI/dt = 0.03e^0.03

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 = x^2/3.

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…, x_n), where the variables x_i (i=1,2,…,n) are all independent of each other. If the variable x₁ is changed by Dx₁ while x₂, …, x_n 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 ¶y/¶x₁ = lim_Dxi®0Dy/Dx= lim_Dxi®0[f(x₁+Dx₁, x₂…, x_n)/Dx₁]. The partial ¶y/¶x_i is also written as f_i. 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(x_1, 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 f_u = 3u + v² + 6 and f_v = 2v(u+3)

32. True False If y = f(u,v) = (3u - 2v)/(u+3), then f_u = (9 - 2v)/(u + 3)² and f_v = 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 U_x = -2x +200y, U_y = 200x –2y.

34. True False Suppose you have a production function Q = K^1/2L^1/4E^1/3. 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^-1/2L^1/4E^1/3 + 1/4K^1/2L^-3/4E^1/3 + 1/3 K^1/2L^1/4E^-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(e^t) = te^t-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 V_t = Kexp(t^0.5), i.e. at t=0, V_t = 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 S_t = S₀e^rt. Then the rate of growth of S_t 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 3^2xdx = 3^2x/ln(3) + c

47. True False e^9xdx = e^9x/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 3xe^xdx =3xe^x - e^x + 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:

= = F(b) - F(a) where dF/dx = f(x).

51. True False = 40

Economic Examples of Integrals

52. True False. If marginal cost MC = dTC/dQ = 16 + 9Q - 6Q² and FC = 20, then TC = 16Q+4.5 Q² - 2Q³ + 20

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.