AP Calculus AB · Summary

Unit 7 summary

1. "Rate proportional to Q" means $\frac{dy}{dt} = kQ$. To check a solution, substitute the function and its derivative and show the two sides agree.
2. A slope-field segment at $(x, y)$ has slope $\frac{dy}{dx}$ there. Flat segments sit where $\frac{dy}{dx} = 0$; an equation in x alone matches down columns, in y alone along rows.
3. For concavity, differentiate $\frac{dy}{dx}$ with respect to x, use the chain rule on y, and substitute $\frac{dy}{dx}$ back in.
4. Separate, integrate both sides with one constant, use the initial condition to find the constant, then solve for y and keep the branch through the given point.
5. $\frac{dy}{dt} = ky$ gives $y = y_0e^{kt}$. Doubling time and half-life are $\frac{\ln 2}{|k|}$, from any starting amount.
6. k is not the growth factor: each unit of time multiplies y by $e^k$. Exponential decay approaches 0 but never reaches it.