Topic 3 of 6
Applications of Derivatives
Use derivatives to model motion, optimize quantities, approximate values, and read the shape of a graph.
01Calculus · Method guideRelated rates: translate the geometry before differentiatingConnect changing quantities with one equation, differentiate with respect to time, and substitute only after the rates appear.02Calculus · Method 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.03Calculus · Concept 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.04Calculus · Concept explainerWhat the Mean Value Theorem actually guaranteesSome instantaneous rate must match the average rate—provided continuity and differentiability hold where required.05Calculus · Method guideCurve sketching from first and second derivativesBuild a graph from domain, intercepts, asymptotes, monotonicity, extrema, concavity, and end behavior—in that order.