[B.8-4 Taylor and Maclaurin Series](#page-7-0)

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B.8-4 Taylor and Maclaurin Series

$$f(x) = f(a) + \frac{(x-a)}{1!}\dot{f}(a) + \frac{(x-a)^2}{2!}\ddot{f}(a) + \dots = \sum_{k=0}^{\infty} \frac{(x-a)^k}{k!}f^{(k)}(a)$$ $$f(x) = f(0) + \frac{x}{1!}\dot{f}(0) + \frac{x^2}{2!}\ddot{f}(0) + \dots = \sum_{k=0}^{\infty} \frac{x^k}{k!}f^{(k)}(0)$$

B.8-5 Power Series

$$e^{x} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \dots + \frac{x^{n}}{n!} + \dots$$

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$$\sin x = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} + \dots$$

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$$\cos x = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} + \frac{x^{8}}{8!} - \dots$$

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$$\tan x = x + \frac{x^{3}}{3} + \frac{2x^{5}}{15} + \frac{17x^{7}}{315} + \dots \qquad x^{2} < \pi^{2}/4$$

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$$\tanh x = x - \frac{x^{3}}{3} + \frac{2x^{5}}{15} - \frac{17x^{7}}{315} + \dots \qquad x^{2} < \pi^{2}/4$$

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$$(1 + x)^{n} = 1 + nx + \frac{n(n-1)}{2!}x^{2} + \frac{n(n-1)(n-2)}{3!}x^{3} + \dots + \binom{n}{k}x^{k} + \dots + x^{n}$$

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$$(1 + x)^{n} \approx 1 + nx \qquad |x| \ll 1$$

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$$\frac{1}{1 - x} = 1 + x + x^{2} + x^{3} + \dots \qquad |x| < 1$$

B.8-6 Trigonometric Identities

$$e^{\pm jx} = \cos x \pm j \sin x$$

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$$\cos x = \frac{1}{2} [e^{jx} + e^{-jx}]$$

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$$\sin x = \frac{1}{2j} [e^{jx} - e^{-jx}]$$

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$$\cos (x \pm \frac{\pi}{2}) = \pm \sin x$$

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$$\sin (x \pm \frac{\pi}{2}) = \pm \cos x$$

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$$2 \sin x \cos x = \sin 2x$$

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$$\sin^2 x + \cos^2 x = 1$$

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$$\cos^2 x - \sin^2 x = \cos 2x$$

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$$\cos^2 x = \frac{1}{2} (1 + \cos 2x)$$

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$$\sin^2 x = \frac{1}{2} (1 - \cos 2x)$$

$$\cos^3 x = \frac{1}{4} (3 \cos x + \cos 3x)$$

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$$\sin^3 x = \frac{1}{4} (3 \sin x - \sin 3x)$$

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$$\sin (x \pm y) = \sin x \cos y \pm \cos x \sin y$$

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$$\cos (x \pm y) = \cos x \cos y \mp \sin x \sin y$$

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$$\tan (x \pm y) = \frac{\tan x \pm \tan y}{1 \mp \tan x \tan y}$$

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$$\sin x \sin y = \frac{1}{2} [\cos (x - y) - \cos (x + y)]$$

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$$\cos x \cos y = \frac{1}{2} [\cos (x - y) + \cos (x + y)]$$

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$$\sin x \cos y = \frac{1}{2} [\sin (x - y) + \sin (x + y)]$$

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$$\sin x \cos x + b \sin x = C \cos (x + \theta) \qquad C = \sqrt{a^2 + b^2}, \theta = \tan^{-1} (\frac{-b}{a})$$

B.8-7 Common Derivative Formulas

$$\frac{d}{dx}f(u) = \frac{d}{du}f(u)\frac{du}{dx}$$

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$$\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$$

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$$\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$$

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$$\frac{dx^n}{dx} = nx^{n-1}$$

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$$\frac{d}{dx}\ln(ax) = \frac{1}{x}$$

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$$\frac{d}{dx}\log(ax) = \frac{\log e}{x}$$

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$$\frac{d}{dx}e^{bx} = be^{bx}$$

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$$\frac{d}{dx}a^{bx} = b(\ln a)a^{bx}$$

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$$\frac{d}{dx}\sin ax = a\cos ax$$

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$$\frac{d}{dx}\cos ax = -a\sin ax$$

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$$\frac{d}{dx}\tan ax = \frac{a}{\cos^2 ax}$$

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$$\frac{d}{dx}(\sin^{-1}ax) = \frac{a}{\sqrt{1 - a^2x^2}}$$

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$$\frac{d}{dx}(\cos^{-1}ax) = \frac{-a}{\sqrt{1 - a^2x^2}}$$

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$$\frac{d}{dx}(\tan^{-1}ax) = \frac{a}{1 + a^2x^2}$$

B.8-8 Indefinite Integrals

$$\int u dv = uv - \int v du$$

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$$\int f(x)\dot{g}(x) dx = f(x)g(x) - \int f(x)g(x) dx$$

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$$\int \sin ax dx = -\frac{1}{a} \cos ax \qquad \int \cos ax dx = \frac{1}{a} \sin ax$$

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$$\int \sin^2 ax dx = \frac{x}{2} - \frac{\sin 2ax}{4a} \qquad \int \cos^2 ax dx = \frac{x}{2} + \frac{\sin 2ax}{4a}$$

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$$\int x \sin ax dx = \frac{1}{a^2} (\sin ax - ax \cos ax)$$

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$$\int x \cos ax dx = \frac{1}{a^2} (\cos ax + ax \sin ax)$$

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$$\int x^2 \sin ax dx = \frac{1}{a^3} (2ax \sin ax + 2 \cos ax - a^2x^2 \cos ax)$$

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$$\int x^2 \cos ax dx = \frac{1}{a^3} (2ax \cos ax - 2 \sin ax + a^2x^2 \sin ax)$$

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$$\int \sin ax \sin bx dx = \frac{\sin (a - b)x}{2(a - b)} - \frac{\sin (a + b)x}{2(a + b)} \qquad a^2 \neq b^2$$

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$$\int \sin ax \cos bx dx = -\left[ \frac{\cos (a - b)x}{2(a - b)} + \frac{\cos (a + b)x}{2(a + b)} \right] \qquad a^2 \neq b^2$$

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$$\int \cos ax \cos bx dx = \frac{\sin (a - b)x}{2(a - b)} + \frac{\sin (a + b)x}{2(a + b)} \qquad a^2 \neq b^2$$

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$$\int e^{ax} dx = \frac{e^{ax}}{a^2} (ax - 1)$$

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$$\int x^2 e^{ax} dx = \frac{e^{ax}}{a^2} (a^2x^2 - 2ax + 2)$$

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$$\int e^{ax} \sin bx dx = \frac{e^{ax}}{a^2 + b^2} (a \sin bx - b \cos bx)$$

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$$\int e^{ax} \cos bx dx = \frac{e^{ax}}{a^2 + b^2} (a \cos bx + b \sin bx)$$

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$$\int \frac{1}{x^2 + a^2} dx = \frac{1}{a} \ln(x$$

B.8-9 L’Hôpital’s Rule

If lim f(x)/g(x) results in the indeterministic form 0/0 or ∞/∞, then

$$\lim \frac{f(x)}{g(x)} = \lim \frac{\dot{f}(x)}{\dot{g}(x)}$$

B.8-10 Solution of Quadratic and Cubic Equations

Any quadratic equation can be reduced to the form

$$ax^2 + bx + c = 0$$

The solution of this equation is provided by

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

A general cubic equation

$$y^3 + py^2 + qy + r = 0$$

may be reduced to the depressed cubic form

$$x^3 + ax + b = 0$$

by substituting

$$y = x - \frac{p}{3}$$

This yields

$$a = \frac{1}{3}(3q - p^2) \qquad b = \frac{1}{27}(2p^3 - 9pq + 27r)$$

Now let

$$A = \sqrt[3]{-\frac{b}{2} + \sqrt{\frac{b^2}{4} + \frac{a^3}{27}}} \qquad B = \sqrt[3]{-\frac{b}{2} - \sqrt{\frac{b^2}{4} + \frac{a^3}{27}}}$$

The solution of the depressed cubic is

$$x = A + B$$

, $x = -\frac{A+B}{2} + \frac{A-B}{2}\sqrt{-3}$ , $x = -\frac{A+B}{2} - \frac{A-B}{2}\sqrt{-3}$

and

$$y = x - \frac{p}{3}$$