Calculus I Foundations (MATH 151) · Module 6 of 11
Motion, Related Rates & Optimization
Calculus in action: describing movement, relating changing quantities, and finding the best possible outcome.
Position, Velocity & Acceleration
- If s(t) is position, then v(t) = s'(t) is velocity and a(t) = v'(t) = s''(t) is acceleration; an object is speeding up when velocity and acceleration share a sign, slowing down when they have opposite signs.
An object's velocity is positive and its acceleration is negative. Is the object speeding up or slowing down?
Solution. Slowing down — since velocity and acceleration have opposite signs, the acceleration is working against the direction of motion, decreasing the speed.
Optimization Problems
- Optimization problems use derivatives to find the maximum or minimum value of a quantity, typically by writing an equation for it, finding critical points via the first derivative, then verifying it's a max or min.
What are the general steps to solve an optimization problem using calculus?
Solution. Write an equation for the quantity to be optimized, often in terms of one variable using any given constraints, take its derivative, set the derivative to zero to find critical points, and use the first or second derivative test to confirm the desired maximum or minimum.
Linear Approximation
- Linear approximation uses the tangent line at a point to estimate function values nearby: f(x) ≈ f(a) + f'(a)(x−a) for x close to a.
How does linear approximation use the derivative to estimate a function's value near a point?
Solution. It uses the tangent line at a known point a — whose slope is f'(a) — as a stand-in for the function itself nearby, since the tangent line closely follows the curve very close to the point of tangency.
Check yourself
2 of this module's 5 quiz questions. Pick your answers, then check them in the app, with an explanation for every question.
1. If s(t) is position, what does s''(t) represent?
2. An object's velocity and acceleration have the same sign. Is it speeding up or slowing down?