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Maxima and minima

Critical points and the second-derivative test.

A smooth function peaks or bottoms out only where its derivative is zero. Find those critical points, then use the second derivative to tell a maximum (f″ < 0) from a minimum (f″ > 0). This is the whole of optimisation in one variable.

Àwọn ààtò àwọn ìṣàmúlò-ètò · critical points of x^3 - 3x

Calculus · Extrema

Critical points of x^3 - 3x

x^{3} - 3 x

Àwọn ìṣàmúlò-ètò

f(x) = x^{3} - 3 x
Critical points are where f′(x) = 0 or is undefined.
f'(x) = 3 x^{2} - 3
Differentiate.
x = -1, x = 1
Solve f′(x) = 0.
f''(x) = 6 x
Second-derivative test: f″ < 0 → maximum, f″ > 0 → minimum.
f''(-1) = -6 \Rightarrow (-1, 2) \text{ is a local maximum}
f''(1) = 6 \Rightarrow (1, -2) \text{ is a local minimum}

Fi àwọn àgbèwọlé hàn

(-1, 2)\ \text{local maximum},\; (1, -2)\ \text{local minimum}

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Bii o ṣe le: Maxima and minima

Critical points are where f′(x) = 0 or is undefined.
Differentiate.
Solve f′(x) = 0.
Second-derivative test: f″ < 0 → maximum, f″ > 0 → minimum.

Àwọn Àtòjọ-ẹ̀yàn

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.

Diẹ̀ nínú Calculus

Limits Derivatives Integrals Definite integrals Taylor series Series and sums The chain rule Implicit differentiation Related rates and optimisation Integration techniques: substitution, parts, partial fractions Applications of integration: area, volume, arc length Infinite series and convergence tests