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Related rates and optimisation

Turning a word problem into a derivative: what is changing, what is fixed, what is extreme.

Related rates differentiate a relation with respect to time; optimisation sets a derivative to zero. Both begin by naming the variables and writing the one equation that ties them. Picture it: the graph of the quantity being optimised; the peak is where the tangent is flat. Think it: Lagrange multipliers extend the same idea to constrained problems in many variables.

Toimiva esimerkki · maximum of -x^2 + 4x

Calculus · Extrema

Critical points of -x^2 + 4x

- x^{2} + 4 x

Askel kerrallaan

  1. f(x) = - x^{2} + 4 x

    Critical points are where f′(x) = 0 or is undefined.

  2. f'(x) = 4 - 2 x

    Differentiate.

  3. x = 2

    Solve f′(x) = 0.

  4. f''(x) = -2

    Second-derivative test: f″ < 0 → maximum, f″ > 0 → minimum.

  5. f''(2) = -2 \Rightarrow (2, 4) \text{ is a local maximum}

Paljasta vastaus
(2, 4)\ \text{local maximum}

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Miten: Related rates and optimisation

  1. Critical points are where f′(x) = 0 or is undefined.
  2. Differentiate.
  3. Solve f′(x) = 0.
  4. Second-derivative test: f″ < 0 → maximum, f″ > 0 → minimum.

Kysymyksiä

What is a derivative in one sentence?

The slope of the graph at a point, the rate at which the output is changing there. Speed is the derivative of position.

What is an integral in one sentence?

The accumulated total of a rate: the area under the curve. Distance travelled is the integral of speed.

Why are derivatives and integrals opposites?

That is the fundamental theorem of calculus: accumulating a rate and then measuring how fast the accumulation grows gets you back the rate. Integration undoes differentiation up to a constant.

When do I use substitution and when integration by parts?

Substitution when part of the integrand is the derivative of another part (u and du both present). Parts when the integrand is a product of two unrelated kinds of function, a polynomial times an exponential, log or trig function.

Lisää Calculus

LimitsDerivativesIntegralsDefinite integralsTaylor seriesSeries and sumsMaxima and minimaThe chain ruleImplicit differentiationIntegration techniques: substitution, parts, partial fractionsApplications of integration: area, volume, arc lengthInfinite series and convergence tests