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AP CALCULUS AB

AP CALCULUS AB. Chapter 6: Differential Equations and Mathematical Modeling Section 6.4: Exponential Growth and Decay. What you’ll learn about. Separable Differential Equations Law of Exponential Change Continuously Compounded Interest Modeling Growth with Other Bases

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AP CALCULUS AB

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

  2. What you’ll learn about • Separable Differential Equations • Law of Exponential Change • Continuously Compounded Interest • Modeling Growth with Other Bases • Newton’s 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 k>0 represents growth and k<0 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 y>0 and y’=kt, 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. Newton’s Law of Cooling

  14. Section 6.4 – Exponential Growth and Decay • Another version of Newton’s Law of Cooling (where H=temp of object & T=temp of outside medium)

  15. A temperature probe is removed from a cup of coffee and placed in water that • has a temperature of T = 4.5 C. • Temperature readings T, as recorded in the table below, are taken • after 2 sec, 5 sec, and every 5 sec thereafter. • Estimate • the coffee's temperature at the time • the temperature probe was removed. • the time when the temperature • probe reading will be 8 C. o S o Example Using Newton’s Law of Cooling

  16. Example Using Newton’s 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 object’s 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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