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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.

工作范例 · maximum of -x^2 + 4x

Calculus · Extrema

Critical points of -x^2 + 4x

- x^{2} + 4 x

一步

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}

发送答案

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

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如何: 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.

人们对人们提出的问题

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.