Ch2-Preliminary calculus - Keii Chin's notes

Mathematical Methods for Physics and Engineering

1. Basic derivatives of some functions [Page 44]

$$\frac{d}{dx}sec(ax)=asec(ax)tan(ax)\\ \frac{d}{dx}tan(ax)=asec^2(ax)\\ \frac{d}{dx}cosec(ax)=-acosec(ax)cot(ax)\\ \frac{d}{dx}cot(ax)=-acosec^2(ax)\\ \frac{d}{dx}sin^{-1}\frac{x}{a}=\frac{1}{\sqrt{a^2-x^2}}\\ \frac{d}{dx}cos^{-1}\frac{x}{a}=\frac{-1}{\sqrt{a^2-x^2}}\\ \frac{d}{dx}tan^{-1}\frac{x}{a}=\frac{a}{a^2+x^2}\\ Leibnitz’ theorem:(uv)^{(n)}=\sum\limits_{r=0}^{n}{}^{n}C_ru^{(r)}v^{(n-r)}$$

2. Some properties of curves [Page 44,49,51,53-56]

$\frac{df}{dx}=0,\frac{d^2f}{dx^2}>0\Rightarrow Minimum;\frac{df}{dx}=0,\frac{d^2f}{dx^2}<0\Rightarrow Maximum$ .

$$\frac{df}{dx}=0,\frac{d^2f}{dx^2}=0\Rightarrow a\ stationary\ point\ of\ inflection\\(\frac{d^2f}{dx^2} changes sign through the point, normally requires \frac{d^3f}{dx^3}\neq0).$$

For a curve s(x), $\frac{ds}{dx}=[1+(\frac{df}{dx})^2]^{1/2}$.

The radius of curvature:

$\rho=\lim\limits_{\Delta\theta\rightarrow0}\frac{\Delta s}{\Delta\theta}=\frac{ds}{d\theta}=\frac{ds}{dx}\frac{dx}{d\theta}=\frac{[1+(f{‘})^2]^{3/2}}{f{”}}$ and $\rho=\frac{(a^4y^2+b^4x^2)^{3/2}}{a^4b^4}$ for $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.

Rolle’s theorem:

if a function f(x) is continuous and differentiable in the range (a,c), and satisfies f(a)=f(c), then for at least one point x = b, where $b\in(a,c)$,f'(b) = 0.

Mean value theorem:

if a function f(x) is continuous and differentiable in the range (a,c) then $f'(b)=\frac{f(c)-f(a)}{c-a}$ for at least one value b where $b\in(a,c)$.

3. Basic integrals of some functions [Page 63]

$$\int a\ tan(bx)dx=-\frac{a}{b}ln[cos(bx)]+c\\ \int \frac{a}{a^2+x^2}dx,\int \frac{-1}{\sqrt{a^2-x^2}}dx,\int \frac{1}{\sqrt{a^2-x^2}}dx,\ see\ section\ 1.$$

4. Methods and properties of integrations [Page 63-70,72-73]

Sinusoidal functions:

sin^2x = 1−cos^2x for odd degree of sinusoidal functions; cos^2x=\frac{1 + cos2x}{2} for even degree of sinusoidal functions. [p63]

Sometimes you can find:

$\int \frac{f'(x)}{f(x)}dx=lnf(x)+c$. [p64]

Using partial fractions (see ch1-preliminary calculus sections 6,7 and 8).

Integration by substitution:

x = sinu or $\tan\frac{x}{2}=t$ (see ch1-preliminary calculus section 4), etc.

Integration by parts:

$\int u\frac{dv}{dx}dx=uv-\int v\frac{du}{dx}dx$. [p67-69]

Reduction formula:

find relations between I_n and I_{n-1}, and then find I_0 to finally find I_n. [p69]

Deal with infinite and improper integrals: exclusion. For example, if the integrand f(x) is infiniteat x = c,c $\in [a,b]$ then $\int_{a}^{b}f(x)dx=\lim\limits_{\delta\rightarrow0}\int_{a}^{c-\delta}f(x)dx+\lim\limits_{\epsilon \rightarrow0}\int_{c+\epsilon}^{b}f(x)dx$.

Evaluation of an integral: If $\phi_1(x)<f(x)<\phi_2(x),x\in[a,b]$, then $\int_a^b\phi_1(x)dx<\int_a^bf(x)dx<\int_a^b\phi_2(x)dx$. [p72]

Mean value of a function: $m=\frac{1}{b-a}\int_a^bf(x)dx$. [p72-73]

5. Some integrals of curves, area and volume [Page 63-70]

Area in plane polar coordinates [p71]:

Area: $A=\int_{\phi_1}^{\phi_2}\frac{1}{2}\rho^2(\phi)d\phi$

Length of a curve in x-y coordinates and plane polar coordinates [p73]:

$$ds=\sqrt{(dx)^2+(dy)^2}=\sqrt{(dr)^2+(rd\phi)^2}\\ \rightarrow s=\int_a^b\sqrt{1+(\frac{dy}{dx})^2}dx=\int_{r_1}^{r_2}\sqrt{1+r^2(\frac{d\phi}{dr})^2}dr$$

Surfaces of revolution [p74-75]:

Rotate about x-axis:

$S=\int_a^b2\pi yds=\int_a^b2\pi y\sqrt{1+(\frac{dy}{dx})^2}dx$

Rotate about y-axis:

$S=\int_a^b2\pi xds=\int_a^b2\pi x\sqrt{1+(\frac{dx}{dy})^2}dy$

Volumes of revolution [p75-76]: