Topic 3 of 6
Applications of Derivatives
Use derivatives to model motion, optimize quantities, approximate values, and read the shape of a graph.
01Method guideRelated rates: translate the geometry before differentiatingConnect changing quantities with one equation, differentiate with respect to time, and substitute only after the rates appear.02Method guideA calculus optimization workflow that does not skip the modelTurn a word problem into one objective function, then use calculus and endpoint checks to justify the best feasible value.03Concept explainerLinear approximation: use the tangent line as a local calculatorNear a known input, a smooth function behaves like its tangent line—and the derivative measures the approximation’s sensitivity.04Concept explainerWhat the Mean Value Theorem actually guaranteesSome instantaneous rate must match the average rate—provided continuity and differentiability hold where required.05Method guideCurve sketching from first and second derivativesBuild a graph from domain, intercepts, asymptotes, monotonicity, extrema, concavity, and end behavior—in that order.