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Differential Equations and Mathematical Modeling

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AP CALCULUS AB CHAPTER 6: DIFFERENTIAL EQUATIONS AND MATHEMATICAL MODELING SECTION 6.4: EXPONENTIAL GROWTH AND DECAY What you ll learn about Separable Differential ... – PowerPoint PPT presentation

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Title: Differential Equations and Mathematical Modeling


1
AP CALCULUS AB
  • Chapter 6
  • Differential Equations and Mathematical Modeling
  • Section 6.4
  • Exponential Growth and Decay

2
What youll learn about
  • Separable Differential Equations
  • Law of Exponential Change
  • Continuously Compounded Interest
  • Modeling Growth with Other Bases
  • Newtons Law of Cooling
  • and why
  • Understanding the differential equation
  • gives us new insight into exponential growth and
    decay.

3
Separable Differential Equation

4
Example Solving by Separation of Variables

5
Section 6.4 Exponential Growth and Decay
  • Law of Exponential Change
  • If y changes at a rate proportional to the
    amount present
  • and y y0 when t 0,
  • then
  • where kgt0 represents growth and klt0 represents
    decay.
  • The number k is the rate constant of the
    equation.

6
Section 6.4 Exponential Growth and Decay
  • From Larson Exponential Growth and Decay Model
  • If y is a differentiable function of t such that
    ygt0 and ykt, for some constant k, then
  • where C initial value of y, and
  • k constant of proportionality
  • (see proof next slide)

7
Section 6.4 Exponential Growth and Decay
  • Derivation of this formula

8
Section 6.4 Exponential Growth and Decay
  • This corresponds with the formula for
    Continuously Compounded Interest
  • This also corresponds to the formula for
    radioactive decay

9
Continuously Compounded Interest

10
Example Compounding Interest Continuously

11
Example Finding Half-Life

Hint When will the quantity be half as much?
12
Section 6.4 Exponential Growth and Decay
  • The formula for Derivation
  • half-life of a
  • radioactive
  • substance is

13
Newtons Law of Cooling

14
Section 6.4 Exponential Growth and Decay
  • Another version of Newtons Law of Cooling
  • (where Htemp of object
  • Ttemp of outside medium)

15
Example Using Newtons Law of Cooling


16
Example Using Newtons Law of Cooling
Use time for L1 and T-Ts for L2 to fit an
exponential regression equation to the data.
This formula is T-Ts.

17
Section 6.4 Exponential Growth and Decay
  • Resistance Proportional to Velocity
  • It is reasonable to assume that, other forces
    being absent, the resistance encountered by a
    moving object, such as a car coasting to a stop,
    is proportional to the objects velocity.
  • The resisting force opposing the motion is
  • We can express that the resisting force is
    proportional to velocity by writing
  • This is a differential equation of exponential
    change,
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