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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.

Kugwira ntchito chitsanzo · maximum of -x^2 + 4x

Calculus · Extrema

Critical points of -x^2 + 4x

- x^{2} + 4 x

Gawo ndi Gawo

  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}

Kusonyeza yankho
(2, 4)\ \text{local maximum}

Tsopano inu Sankhani vuto, kapena kulemba kapena kujambula wanu. Pakalipano, yankho limeneli lidzapezeka patsamba lino.

Kusunga ntchito yanu yokha

M’malo mwake, maakaunti aulere amawonjezera malemba pazophunzira zonse, mnda wa zomwe mwamaliza, mavuto anu omaliza m’malo limodzi, ndi m’bale amene mungafunse za nkhaniyo. Maphunziro a matekinoloje ndi otsegulira kwa aliyense, olembetsa kapena osalembetsa.

Kulembetsa Kulowa

Zithunzi

Pitani pa dzina lililonse lachifaniziro kuti mudziwe tanthauzo lake, chithunzi chake, ndi zimene limatanthauza.

Momwe mungagwiritsire ntchito: 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.

Mafunso omwe anthu amafunsa

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.

Zambiri pa 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