AP Calculus BC · Lesson 3 of 4
Applications and differential equations
Connect derivatives and integrals to motion, accumulation, and models whose rate depends on the quantity itself.
Accumulation in context
If a rate is given, the definite integral of that rate is a net change. Position is not the same as distance traveled if the velocity changes sign. Distance uses the integral of speed, which is the absolute value of velocity.
Keep units. An acceleration in meters per second squared, integrated over seconds, becomes a velocity in meters per second. An initial condition supplies the missing constant when you go backward from a rate to the quantity.
Differential equations
A slope field shows the direction of solutions without giving the explicit formula. A solution curve follows those segments and must pass through the given point if an initial condition is present.
Logistic and other proportional models are interpreted, not only solved. The phrase “rate proportional to the amount present” becomes dy/dt = ky. Euler’s method approximates a solution with short tangent steps; the step size affects the accuracy, and the result is an approximation.
Worked example
Velocity is v(t) = t − 2 on 0 ≤ t ≤ 4, and the position at t = 0 is 1. What is the position at t = 4?
- An antiderivative of t − 2 is t²/2 − 2t.
- The net change in position is (8 − 8) − 0 = 0.
- The position at t = 4 is the initial position plus net change, which is 1. The distance traveled is larger because the velocity is negative and then positive.
Why this works. Net change can be zero while the object still moves. Distance and displacement answer different questions.
Check your understanding
- Add an initial value after integrating a rate.
- Use absolute value when the question asks for total distance.
- Check a differential-equation solution against both the equation and the initial condition.