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 11 (November 16, 2009)
2
4.1 Exponential Functions
Padmanabhan Seshaiyer George Mason University

• Definition of an exponents
• Integer powers
• Fractional powers
• Negative powers
• Zero power
• Exponential Functions
• Graph of the function
• Properties of the graph

3
Exponential Rules
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• Equality rule
• Product rule
• Quotient rule
• Power rule
• Multiplication rule
• Division rule

4
Word Problems
Padmanabhan Seshaiyer George Mason University

• If, find all values of x such
  that f (x) 64.
• Biologists have determined that the number
• of bacteria in a culture is given by
• where t is the number of minutes after
  observation begins. What is the average rate of
  change of the bacterial population during the
  second hour?
• Suppose 1,000 is invested at an annual interest
  rate of 10. Compute the balance after 10 years
  if the interest is compounded annually.

5
Compound interest
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• P Amount of money invested
• r Annual Interest Rate
• B(t) Accumulated (Future) value in the account
  after t years
• k Number of times per year interest is
  compounded

6
Continuous Compounding
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• P Amount of money invested
• r Annual Interest Rate
• B(t) Accumulated (Future) value in the account
  after t years

7
Computing the Balance
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• Suppose 1,000 is invested at an annual interest
  rate of 6. Compute the balance after 10 years if
  the interest is compounded
• Annually
• Quarterly
• Monthly
• Daily
• Continuously

8
Present Value
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• Compounded k times a year
• Compounded continuously
• Sue is about to enter college. When she graduates
  4 years from now, she wants to take a trip to
  Europe that she estimates will cost 5,000. How
  much should she invest now at 7 to have enough
  for the trip if interest is compounded
• Quarterly
• Continuously

9
4.2 Logarithmic Functions
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• Suppose you invest 1,000 at 8 compounded
  continuously and wish to know how much time must
  pass for your investment to double in value to
  2,000.
• An investor has 1500 and wishes it to grow to
  2000 in 5 years. At what annual interest rate r
  compounded continuously must he invest to achieve
  this goal?

10
Logarithmic Functions
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• Definition
• Logarithmic Functions
• Graph of the function
• Properties of the graph
• Inverse Relationship with exponential functions

11
Logarithmic Functions
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• Logarithmic Rules
• Equality Rule
• Product Rule
• Power Rule
• Quotient Rule
• Inversion Rule

12
Word Problems
Padmanabhan Seshaiyer George Mason University

• Suppose you invest 1,000 at 8 compounded
  continuously and wish to know how much time must
  pass for your investment to double in value to
  2,000.
• An investor has 1500 and wishes it to grow to
  2000 in 5 years. At what annual interest rate r
  compounded continuously must he invest to achieve
  this goal?

13
Radioactive Decay (Half-life)
Padmanabhan Seshaiyer George Mason University

• It has been experimentally determined that a
  radioactive sample of initial size Q0 grams will
  decay to
• in t years. Here r is the rate of decay
  corresponding to the amount of time th for half
  a given sample to decay. Find the half-life of
  the sample.

14
Announcements
Padmanabhan Seshaiyer George Mason University

• Suggested Problems
• 4.1 3-28 (Odd Problems) 35-43 (Odd Problems)
• 4.2 1-53 (Odd Problems)
• Exam III (Chapter 2)