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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.
مثال کار شده · maximum of -x^2 + 4x
قدم به قدم
- f(x) = - x^{2} + 4 x
Critical points are where f′(x) = 0 or is undefined.
- f'(x) = 4 - 2 x
Differentiate.
- x = 2
Solve f′(x) = 0.
- f''(x) = -2
Second-derivative test: f″ < 0 → maximum, f″ > 0 → minimum.
- f''(2) = -2 \Rightarrow (2, 4) \text{ is a local maximum}
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چطور: Related rates and optimisation
- Critical points are where f′(x) = 0 or is undefined.
- Differentiate.
- Solve f′(x) = 0.
- Second-derivative test: f″ < 0 → maximum, f″ > 0 → minimum.
سوالاتي که مردم ميپرسن
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.
بیشتر در 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