**Calculus Evaluation Self Test**

**Mineral and Energy Economics Program**

**Colorado School of Mines**

**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) = e^{0.03},
dI/dt = 0.03e^{0.03}

__Power-Function
Rule__

3. True False Given y = 0.5x^{4},
dy/dx = 2x^{3}

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

__Sum-Difference
Rule__

5. True False d/dx(3x^{5}
+ 2x) = 15x^{4}

6. True False d/dx(4x^{3}
- 2x^{2} -5) = 12x^{2} - 4x

7. True False d/dx(ax^{2}
- bx + c) = ax - b

__Product Rule__

8. True False d/dx[(3x + 2)(2x^{3})] = 24x^{3}
+ 12x^{2}

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

10. True False d/dx[cx^{2}(ax + b)] = 3acx^{2}
+ 2bcx

__Quotient Rule__

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

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

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

__Chain Rule__

14. True False Given z = y^{3}, 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 = x^{2 }- 4, then
z = 3(x^{2 }- 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 = ax^{3}
+ b, where a, b > 0, then dx/dy = 3ax^{2}

**Economic
Interpretations of Derivatives**

20. True False The total cost function is given by TC = 2Q^{2}
+ 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 = Q^{2} +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)]/Q^{2}.

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 x_{0} to find the
optimal level of the independent variable, and check the sign of the second
derivative at the optimal point:

f(x_{0}) is a relative maximum if f"(x_{0})<0.

f(x_{0}) is a relative maximum if f"(x_{0})>0.

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

26.
True False x=6 is a
maximum of f(x) = x^{2} - 12x + 13.

27.
True False. Given y =
-2x^{2 }+ 4x + 9, x_{0} = 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 = Q^{4}/3 - 6Q^{3 }+11Q^{2} +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 = -Q^{2} + 12Q - 25. Your optimal or
profit maximizing output is 6.

**Partial Differentiation**

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

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, }x_{2}) = 2x_{1}+
x_{1}x_{2 }+3x_{2}^{2}, then f_{1 }= 2
+ x_{2} and f_{2 }= x_{1}+ 6x_{2}

31. True False If y = f(u,v) = (u + 3)(2u +v^{2}), then
f_{u} = 3u + v^{2} + 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 = -x^{2} + 200xy -
y^{2}, 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^{1/2}L^{1/4}E^{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^{-1/2}L^{1/4}E^{1/3}
+ 1/4K^{1/2}L^{-3/4}E^{1/3} + 1/3 K^{1/2}L^{1/4}E^{-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(x^{2}))/dx = 2x*(exp(x^{2}))

39. True False d(lnt^{4})/dt
= 4t^{3}lnt^{4}

40. True False d/dt(t^{3}lnt^{2})
= (1 + 3lnt)2t^{2}

**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* = (4r^{2})^{-1}.

42.
True False. Suppose
that the value of a certain asset grows according to S_{t} = S_{0}e^{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__

__Integral of a power__

44. True False ∫3x^{2}dx = x^{3 }+ c

__Integral of an inverse__

45. True False ∫x^{-1}dx = ln(x) + c

46. True False ∫3^{2x}dx = 3^{2x}/ln(3) +
c

47.
True False ∫e^{9x}dx
= e^{9x}/9+c

__Integration
by Parts__

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

49.
True False ∫3xe^{x}dx
=3xe^{x} - e^{x }+ c

__Integration
by Substitution__

50.
True False ∫15x^{2}(x^{3}+1)dx
= 3(x^{3}+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*:

_{}

__Economic Examples of Integrals__

52.
True False. If marginal
cost MC = dTC/dQ = 16 + 9Q - 6Q^{2 }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.