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 Newton’s Law of Cooling … and why Understanding the differential equation gives us new insight into exponential growth and decay.

Separable Differential Equation

Example Solving by Separation of Variables

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.

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)

Section 6.4 – Exponential Growth and Decay Derivation of this formula:

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

Continuously Compounded Interest

Example Compounding Interest Continuously

Example Finding Half-Life Hint: When will the quantity be half as much?

Section 6.4 – Exponential Growth and Decay The formula for Derivation: half-life of a radioactive substance is

Newton’s Law of Cooling

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)

Example Using Newton’s Law of Cooling 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

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.

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,