Overview

Topic 7.8 extends student learning from Topic 7.7 by focusing on exponential modeling.  When “the rate of change of a quantity is proportional to the quantity present,” we are in the presence of dy/dt = ky.  After separation of variables, the antiderivative expression will contain a logarithm quantity.  Solving for y then produces a model for exponential growth or decay. An initial condition will determine the constant of integration and allow students to select the one true solution equation.

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Teaching Tips

When dy/dt = ky, help students reason why the function y must be exponential:  eu is the only function that appears in its own derivative. When an initial condition is present (y(0) = y0), the solution will be 
y = y0ekt.  After students experience several examples in this pattern, they should be able to “eyeball” the solution without too much analytic work. 

Remember: Rounding off at intermediate steps is not encouraged! Students who round their value for C may lose points later for not presenting sufficient correct decimal digits. 

If you would like to challenge students with a bit of BC material, this is the best opportunity to introduce logistic growth and integration using partial fractions. Many real-world populations follow the logistic growth model which students find very interesting.

Continue to repeat the mantra of separation of variables: Separate (the variables), Integrate (both sides using +C), Isolate (solve for y), but add two new commands: Separate, Integrate, Solve (for C), Isolate (solve for y), and Select (the correct version of y depending on the initial condition). Now we have SISIS!

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Exam Insights

Exponential growth applications appear on both the MC and FRQ sections. Often, differential equations are paired with slope fields on FRQs (for example, 2006 AB 5 is an interesting question because the solution appears to contain exponential growth but resolves to a linear expression; 2009 BC 4c requires factoring before separating variables.)

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Student Misconceptions

Watch for integrals with dy or dx in the denominator after separation of variables. The differential term must never sit in the denominator. For best preparation, allow students to work with many forms representing exponential growth or decay: dy/dx = k(y – h) or dP/dt + 5 = P.