AGEC 317

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AGEC 317

• Review of Key Algebraic Concepts and Applications
  to Economic Issues

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Readings

• Review of fundamental algebra concepts (Consult
  any math textbook)
• Chapter 2, pp. 23-44, Managerial Economics

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Topics

• Mathematical operations with algebraic
  expressions
• Addition, subtraction, multiplication, division
• Solving equations
• Linear functions
• Graphical analysis
• Slope
• Intercept
• Nonlinear functions
• Graphical analysis
• Rate of change (marginal effects)
• Optimal points (minima, maxima)

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Addition and Subtraction

• Express in simplistic terms
• 1. (2ac) (-6ac) (9ac) 2. 3x (-7x)
• 3. (-8a) (-3a) (2a) 4. (5x) (6x) (7x)
• Add the two expressions in each problem
• 5. x 2y 8 6. 11m 7n 13
• 3x 4y 9 3m 8n 21
• Subtract the two expressions in each problem
• 7. x 2y 8 8. 11m 7n 13
• 3x 4y 9 3m 8n 21
• 9. 10a 17b 24c 13a 14b 16c

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Addition, Subtraction, Multiplication

• Add the three expressions in each of the
  problems
• 10. 7a 3b 11c -14a 10b 10c 8a 8b
  13c
• Combine like terms
• 11. 3x 7y 3z 6xc 8y 7z 5 -1
• 12. 9xy 3x 4a 5ax 10a 7x 3yx 6xa
• Remove the symbols of grouping and simplify by
  combining terms
• 13. 3a (b c) (a b c) 14. -x (3
  x) (4 3x)
• 15. -5a b 3b (c - 2b a) 4a c
• Carry out these multiplicative operations
• 16. (5)(-4)(-2) 17. (3ab)(2a)
• 18. (6a2b)(3ab2) 19. (3xy2)(5x2y)(xy)
• 20. 3x2y(2xy2 y) 21. -4mn(3-5m 6mn 3n)
• 22. (a 3b)(3a2 6ab 4b2) 23. (3a 2)(a
  2)(2a 1)

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Multiplication and Division

• Simplify each expression
• 24. x2/x2 25. -y3/y3 26. a8/a5
• 27. acz/ac 28. 12a4/6a 29. -93d4/3c5d3
• 30. 34a3b2/17a2b 31. (9x2 6x3 3x4)/(-3x2)
• Perform the indicated operations, giving the
  results in simplistic terms
• 32. 33. 34.
• 35. 36. 37.

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Division, Solving Equation(s)

• Simplify the complex fractions and other
  expressions
• 40. 41. 42. 43.
• Perform the indicated operations, giving the
  results in simplistic terms
• 44. 8x-153x 45. x 45 2x
• 46. 47. (3x-1)(x1)3x2
• Solve the following systems
• 48. -4x 5y 14 0 49. 4xy 2y 11 0
• -2x 2y 7 0 2xy 3y 4
  0

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Solving Equation(s), Complex fractions

• 50. 3x y 6 0 51. 4x 5y 9
• 4x 3y 21 0 3x 4y 8
• Simplify the complex fractions
• 52. 53.
• 54. 55.

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Derivatives

• Yf(x)
• The value of the ratio of for extremely
  small change in X.

• Derivative of Y with respect to X at point A is
  the slope of a line that is tangent to the curve
  at the point A.

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Illustration of Derivatives
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Rules of Derivatives
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Linear Function YaXb

• Slope dY/dX a
• Interpretation a one unit increase in X leads
  to an increase in Y of a units.
• Intercepts
• On x-axis the value of X if y 0
• aX b 0, the x intercept is -b/a put another
  way (-b/a, 0)
• On y-axis the value of Y if x 0
• The y intercept is b put another way (0,b)
• Graph
• Y -2X 2, x intercept is 1 or (1,0) y
  intercept is 2 or (0,2)
• Y 2X 4, x intercept is -2 or (-2,0) y
  intercept if 4 or (0,4)

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Application of Linear Function Revenue Output
Total Revenue Output 1.50 1 3.00 2 4.50 3 6
.00 4 7.50 5 9.00 6

• Questions
• Slope
• Intercepts

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Nonlinear function Total, Marginal, and Average
Profits
Curvilinear expression of profit and output
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Locating Maximum and Minimum Values of a Function
Step 1 Find the derivative of the function with
respect to the independent variable. For
example, suppose that profit ( p ) a bQ
cQ2 Then the derivative (marginal profit) -b
2cQ The independent variable is Q and the
dependent variable is p Step 2 Set the
derivative expression from step 1 to 0
(first-order condition) so, -b 2cQ 0 Step
3 Find the value of the independent variable
that solves the derivative expression -b 2cQ
0 2cQ b Q b/2c
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Locating Maximum and Minimum Values of a Function
(Cont)

• Step 4 How to discern whether the value(s) from
  step 3 correspond to minimum values of the
  function or maximum values of the function
• Calculate the second derivative with respect to
  the independent variable
• first derivative -b 2cQ
• second derivative 2c
• If the second derivative at the value of the
  independent variable that solves the first
  derivative expression (step 3) is positive, then
  that value of the independent variable
  corresponds to a minimum.
• If the second derivative is negative at this
  point, then that value of the independent
  variable corresponds to a maximum.

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Locating Maximum and Minimum Values of a Function
(cont)
Step 5 Finding the Maximum of Minimum Value of
the Function Simply replace the optimum value
of the independent variable into the
function a bQ b/2c from step 3, Q
b/2c from step 4, if c gt 0, then Q b/2c
corresponds to a minimum value if c lt 0, then Q
b/2c corresponds to a maximum value. The
minimum (maximum) value of then is
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Profit
Locating Maximum and Minimum Values of a Function
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Summary of Algebraic Review

• Mathematical operations with algebraic
  expressions
• Solving equations
• Linear functions
• Nonlinear functions
• Applications to revenue and profit functions