maths.free › Calculus › Related rates and optimisation

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.

उदाहरण · maximum of -x^2 + 4x

Calculus · Extrema

Critical points of -x^2 + 4x

- x^{2} + 4 x

चरणद्वारा चरण

  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}

जवाफ प्रकट गर्नुहोस्
(2, 4)\ \text{local maximum}

अब तपाईँ समस्या रोज्नुहोस्, वा टाइप गर्नुहोस् वा तपाईँको आफ्नै रेखाचित्र गर्नुहोस् । प्रत्येक चरण, एउटा तस्वीर, तपाईँले सोध्नु अघि उत्तर लुकेको छ ।

तपाईँको आफ्नै काम गर्ने राख्नुहोस्

एक निःशुल्क खाता प्रत्येक पाठ मा नोट थप्छ, तपाईं के समाप्त गरेको छ को एक रेकर्ड, एक ठाउँमा आफ्नो समाधान समस्या, र एक शिक्षक तपाईं यो पृष्ठ बारेमा सोध्न सक्नुहुन्छ. गणित आफैलाई सबैलाई खुला छ, मा साइन इन वा छैन.

साइन अप गर्नुहोस् लगइन

यहाँ प्रयोग गरिएको प्रतीक

पूर्ण परिभाषा, एउटा तस्वीर, र यो प्रत्येक अक्षर अर्थ के लागि कुनै पनि प्रतीक ट्याप गर्नुहोस्।

कसरी: 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.

प्रश्नहरू मानिसहरूले सोध्छन्

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.

यसमा थप 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