Code Scholars Tutoring · AP Calculus BC
9 topics across 6 lessons, with explanations, worked examples, and self-checks that develop complete mathematical reasoning.
Topics 7.1, 7.2
A differential equation relates a quantity to one or more derivatives. Translate proportional growth as a derivative proportional to the quantity; a rate depending only on time is a different model. To check a proposed solution, compute its derivative and substitute both function and derivative into the equation throughout the claimed domain. An initial condition picks a member of a solution family when uniqueness holds.
Topics 7.3, 7.4
At each point a slope field draws a short segment with slope given by the differential equation. If the rate depends only on , slopes agree along horizontal lines. Zero slopes reveal equilibrium solution candidates, which must satisfy the equation. Trace solution curves tangent to the field; do not connect segment endpoints as if they were data values.
Topics 7.5 · BC only
Euler's method approximates a solution of a differential equation by following short tangent-line segments. Each step uses the slope evaluated at the current approximate point. It produces a sequence of approximate values, not an exact formula for the solution curve.
Topics 7.6, 7.7
For a separable equation move the factors involving to the side and the factors involving to the side, then integrate. Keep an integration constant until applying the initial condition. Dividing by a function of can discard equilibrium solutions, so inspect those separately. State the interval containing the initial point on which the expression solves the equation.
Topics 7.8
For a constant proportional rate, the solution is exponential. Determine the constant from data using logarithms and interpret its sign and units. Half-life and doubling time describe multiplicative changes, not fixed additive changes. A model in which a temperature difference decays uses that difference as the dependent quantity, producing a shifted exponential.
Topics 7.9 · BC only
A logistic model describes growth whose rate depends on both the current quantity and the remaining capacity. For a positive intrinsic growth parameter, a population between zero and its carrying capacity increases, while a population above capacity decreases toward it. The equilibria and the sign of the derivative explain the long-term behavior.
