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AGEC 317

AGEC 317. Introductory Calculus: Marginal Analysis. Readings. Review of fundamental algebra concepts (Consult any math textbook) Chapter 2, pp. 23-44, Managerial Economics.

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AGEC 317

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  1. AGEC 317 Introductory Calculus: Marginal Analysis

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

  3. Marginal profit is the derivative of the profit function (the same is true for cost and revenue). We use this marginal profit function to estimate the amount of profit from the “next” item. Topics Derivatives Linear functions Graphical analysis Slope Intercept Nonlinear functions Graphical analysis Rate of change (marginal effects) Optimal points (minima, maxima) Applications to revenue and profit functions

  4. Derivatives are all about change ... • They show how fast something is changing (called the rate of change) at any point. • Y=f(x) • First Derivative • The value of the ratio of for extremely small • Marginal profit is the first derivative of the profit function (the same is true for cost, utility and revenue, etc.). We use this marginal profit function to estimate the amount of profit from the “next” item. Derivatives

  5. Y=f(x) • First Derivative Derivatives • 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.

  6. Second Derivative of a function ƒ is the derivative of the derivative of. ƒ.It measures how the rate of change of a quantity is itself changing Derivatives • On the graph of a function, the second derivative corresponds to the curvature or concavity of the graph. • The graph of a function with positive second derivative curves upwards, while the graph of a function with negative second derivative curves downwards.

  7. Rules of Derivatives

  8. Linear Function: Y=aX+b 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)

  9. Total RevenueOutput$1.50 1 3.00 2 4.50 3 6.00 4 7.50 5 9.00 6 Application of Linear Function: Revenue & Output • Questions: • Slope • Intercepts

  10. Nonlinear function

  11. Step 1: Find the derivative of the function with respect to the “independent” variable. For example, suppose that profit ( π) = a – bQ + cQ2 The “independent” variable is Q and the “dependent” variable is π Then the derivative (marginal profit) = -b + 2cQ 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 Locating Maximum and Minimum Values of a Function

  12. 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. Locating Maximum and Minimum Values of a Function (Con’t)

  13. 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 > 0, then Q = b/2c corresponds to a minimum value if c < 0, then Q = b/2c corresponds to a maximum value. The minimum (maximum) value of then is Locating Maximum and Minimum Values of a Function (con’t)

  14. Profit Locating Maximum and Minimum Values of a Function

  15. Mathematical operations with algebraic expressions • Solving equations • Linear functions • Nonlinear functions • Applications to revenue and profit functions Summary of Algebraic Review

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