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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.

වැඩ කළ උදාහරණය · maximum of -x^2 + 4x

Calculus · Extrema

Critical points of -x^2 + 4x

- x^{2} + 4 x

පියවරෙන් පියවර

  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}

උත්තරය හෙළි කරන්න
(2, 4)\ \text{local maximum}

දැන් ඔයා ප් රශ්නයක් තෝරන්න, හෝ වර්ගය හෝ ඔබේම අඳින්න. සෑම පියවරක්, පින්තූරයක්, ඔබ ඉල්ලා තෙක් පිළිතුර සැඟවුණු.

ඔයාගෙ වැඩේ කරගෙන යන්න

නිදහස් ගිණුමක් සෑම පාඩමක් මත සටහන් එකතු, ඔබ අවසන් කර ඇති දේ වාර්තාවක්, එක් ස්ථානයක ඔබේ විසඳා ගැටළු, හා ඔබ මෙම පිටුව ගැන විමසීමට හැකි ගුරුවරයෙකු. ගණිතය ම සියලු දෙනාට විවෘත වේ, ඇතුලත් හෝ නැත.

ලියාපදිංචි වන්න පිවිසුම්

මෙහිදී භාවිත කරන සංකේත

සම්පූර්ණ අර්ථ දැක්වීම සඳහා ඕනෑම සංකේතයක් ටැප්, පින්තූරයක්, සහ එය සෑම අකුරු අදහස් කරන්නේ කුමක්ද.

කොහොමද: 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.

ජනතාව අහනවා ප්රශ්න

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.

තවත් 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