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

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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.

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Separable Differential Equation

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Example Solving by Separation of Variables

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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.

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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)

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Section 6.4 Exponential Growth and Decay

• Derivation of this formula

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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

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Continuously Compounded Interest

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Example Compounding Interest Continuously

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Example Finding Half-Life

Hint When will the quantity be half as much?
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Section 6.4 Exponential Growth and Decay

• The formula for Derivation
• half-life of a
• radioactive
• substance is

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Newtons Law of Cooling

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Section 6.4 Exponential Growth and Decay

• Another version of Newtons Law of Cooling
• (where Htemp of object
• Ttemp of outside medium)

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Example Using Newtons Law of Cooling

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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.

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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,