Padmanabhan Seshaiyer

1
Introductory Calculus with Business Applications
Math 108 (Section 003)
Padmanabhan Seshaiyer Mathematical
Sciences George Mason University Email
pseshaiy_at_gmu.edu Lecture 9 (November 02, 2009)
2
Higher Order Derivatives
Padmanabhan Seshaiyer George Mason University

• Consider the following function
• P(x) (x 1)(4x 3)
• Q(x) 1/x
• What is P(x)?
• What is Q(x)?
• What is the fifth derivative of Q(x)?

3
Chain Rule Application
Padmanabhan Seshaiyer George Mason University

• The cost of producing x units of a particular
  commodity is
• and the production level t hours into a
  particular production run is
• At what rate is cost changing with respect to
  time after 4 hours?

4
2.4 The Chain rule
Padmanabhan Seshaiyer George Mason University
5
Apply Chain rule
Padmanabhan Seshaiyer George Mason University

• Find the derivative of the following functions
• Differentiate the function
• and simplify your answer. Also find all values
  of xc for which the tangent line to the graph of
  f (x) at (c, f (c)) is horizontal.

6
Apply Chain rule
Padmanabhan Seshaiyer George Mason University

• The manager of an appliance manufacturing firm
  determines that when blenders are priced at p
  dollars apiece, the number sold each month can be
  modeled by
• The manager estimates that t months from now,
  the unit price of the blenders will be
• At what rate will the monthly demand for
  blenders D(p) be changing 25 months from now?
  Will it be increasing or decreasing at this time?

7
2.5 Marginal Analysis
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• A manufacturer estimates that when x units of a
  particular commodity are produced, the total cost
  will be
• Furthermore, all x units will be sold when the
  price is
• dollars per unit.
• Find the marginal cost and the marginal revenue.
• Use marginal cost to estimate the cost of
  producing the ninth unit.
• What is the actual cost of producing the ninth
  unit?
• Use marginal revenue to estimate the revenue
  derived from the sale of the ninth unit.
• What is the actual revenue derived from the sale
  of the ninth unit?

8
2.6 Implicit Differentiation
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• Find if
• Find the slope of the tangent line to the circle
• at the point (3, 4). What is the slope at the
  point (3, 4).

9
Application to Economics
Padmanabhan Seshaiyer George Mason University

• Suppose the output at a certain factory is
• units, were x is the number of hours of skilled
  labor used and y is the number of hours of
  unskilled labor. The current labor force consists
  of 30 hours of skilled labor and 20 hours of
  unskilled labor. Use calculus to estimate the
  change in unskilled labor y that should be made
  to offset a 1-hour increase in skilled labor x so
  that output will be maintained at its current
  level.

10
Announcements
Padmanabhan Seshaiyer George Mason University

• Suggested Problems
• 2.3 1 39 (odd problems), 42 51 (odd
  problems), 57 61 (odd problems)
• 2.4 1 66 (odd problems)
• 2.5 Example 2.5.1 (page 157)
• 2.6 1 30 (Odd problems)
• Quiz VI (Rules of differentiation, Chain rule,
  Implicit Differentiation)