Formula Sheet
Every key formula of the course on one page, grouped by chapter, with a link back to the section that develops it.
Chapter 5 · Techniques of Integration
Powers of sine and cosine§ 5.1 →
$\int\sin^mx\cos^nx\,dx$: $n$ odd → save $\cos x$, $u=\sin x$; $m$ odd → save $\sin x$, $u=\cos x$; both even → $\sin^2x=\tfrac{1-\cos2x}{2}$, $\cos^2x=\tfrac{1+\cos2x}{2}$
Squares of sine and cosine§ 5.1 →
$\int\sin^2x\,dx=\tfrac{x}{2}-\tfrac{\sin 2x}{4}+C,\quad\int\cos^2x\,dx=\tfrac{x}{2}+\tfrac{\sin 2x}{4}+C$
Integrals of tan, sec, cot, csc§ 5.1 →
$\int\tan x\,dx=\ln|\sec x|+C,\ \int\sec x\,dx=\ln|\sec x+\tan x|+C$ (where $\cos x\ne0$); $\int\cot x\,dx=\ln|\sin x|+C,\ \int\csc x\,dx=-\ln|\csc x+\cot x|+C$ (where $\sin x\ne0$)
Integral of sec³x§ 5.1 →
$\int\sec^3x\,dx=\tfrac12\sec x\tan x+\tfrac12\ln|\sec x+\tan x|+C$
Reduction formulas ($n\ge2$)§ 5.1 →
$\int\sin^nx\,dx=-\tfrac1n\sin^{n-1}x\cos x+\tfrac{n-1}{n}\int\sin^{n-2}x\,dx$; $\int\cos^nx\,dx=\tfrac1n\cos^{n-1}x\sin x+\tfrac{n-1}{n}\int\cos^{n-2}x\,dx$; $\int\sec^nx\,dx=\tfrac{\sec^{n-2}x\tan x}{n-1}+\tfrac{n-2}{n-1}\int\sec^{n-2}x\,dx$
Product-to-sum identities§ 5.1 →
$\sin A\cos B=\tfrac12[\sin(A-B)+\sin(A+B)]$, $\sin A\sin B=\tfrac12[\cos(A-B)-\cos(A+B)]$, $\cos A\cos B=\tfrac12[\cos(A-B)+\cos(A+B)]$
Trigonometric substitution ($a\gt0$)§ 5.2 →
$\sqrt{a^2-x^2}$: $x=a\sin\theta$, $-\tfrac\pi2\le\theta\le\tfrac\pi2$, root $=a\cos\theta$; $\sqrt{a^2+x^2}$: $x=a\tan\theta$, $-\tfrac\pi2\lt\theta\lt\tfrac\pi2$, root $=a\sec\theta$; $\sqrt{x^2-a^2}$ ($x\ge a$): $x=a\sec\theta$, $0\le\theta\lt\tfrac\pi2$, root $=a\tan\theta$
Inverse-trig integrals ($a\gt0$)§ 5.2 →
$\int\frac{dx}{\sqrt{a^2-x^2}}=\sin^{-1}\frac xa+C\ (|x|\lt a)$, $\ \int\frac{dx}{a^2+x^2}=\frac1a\tan^{-1}\frac xa+C$
Logarithmic forms ($a\gt0$)§ 5.2 →
$\int\frac{dx}{\sqrt{x^2+a^2}}=\ln\left(x+\sqrt{x^2+a^2}\right)+C$, $\ \int\frac{dx}{\sqrt{x^2-a^2}}=\ln\left|x+\sqrt{x^2-a^2}\right|+C\ (|x|\gt a)$
Integral of $\sqrt{a^2-x^2}$§ 5.2 →
$\int\sqrt{a^2-x^2}\,dx=\frac{a^2}{2}\sin^{-1}\frac xa+\frac x2\sqrt{a^2-x^2}+C\quad(|x|\le a)$
Ellipse and circular segment§ 5.2 →
Ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$: $A=\pi ab$. Part of a disc of radius $r$ below height $h$: $A=h\sqrt{r^2-h^2}+r^2\sin^{-1}\frac hr+\frac{\pi r^2}{2}$
Partial fraction forms ($\deg P\lt\deg Q$)§ 5.3 →
$(ax+b)^m\ \to\ \dfrac{A_1}{ax+b}+\cdots+\dfrac{A_m}{(ax+b)^m}$; $\quad(ax^2+bx+c)^m,\ b^2-4ac\lt0\ \to\ \dfrac{B_1x+C_1}{ax^2+bx+c}+\cdots+\dfrac{B_mx+C_m}{(ax^2+bx+c)^m}$
Cover-up (non-repeated factor $x-r$)§ 5.3 →
$A=\displaystyle\lim_{x\to r}(x-r)\frac{P(x)}{Q(x)}$
Linear building blocks§ 5.3 →
$\displaystyle\int\frac{dx}{x-a}=\ln|x-a|+C,\quad\int\frac{dx}{(x-a)^n}=\frac{-1}{(n-1)(x-a)^{n-1}}+C\ (n\ge2)$
Quadratic building blocks ($c\gt0$)§ 5.3 →
$\displaystyle\int\frac{dx}{x^2+c^2}=\frac1c\tan^{-1}\frac xc+C,\quad\int\frac{x\,dx}{x^2+c^2}=\frac12\ln(x^2+c^2)+C$
$\int dx/(x^2-a^2)$, $a\gt0$§ 5.3 →
$\displaystyle\int\frac{dx}{x^2-a^2}=\frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right|+C$
Midpoint rule ($\Delta x=\frac{b-a}{n}$, $m_i$ = midpoints)§ 5.4 →
$M_n=\Delta x\left[f(m_1)+f(m_2)+\cdots+f(m_n)\right]$
Trapezoidal rule§ 5.4 →
$T_n=\dfrac{\Delta x}{2}\left[f(x_0)+2f(x_1)+\cdots+2f(x_{n-1})+f(x_n)\right]$
Simpson's rule ($n$ even)§ 5.4 →
$S_n=\dfrac{\Delta x}{3}\left[y_0+4y_1+2y_2+4y_3+\cdots+2y_{n-2}+4y_{n-1}+y_n\right]=\dfrac{2M_{n/2}+T_{n/2}}{3}$
Error bounds for $M_n$, $T_n$ ($f''$ continuous, $|f''|\le K$ on $[a,b]$)§ 5.4 →
$|E_M|\le\dfrac{K(b-a)^3}{24n^2},\qquad|E_T|\le\dfrac{K(b-a)^3}{12n^2}$
Simpson error bound ($f^{(4)}$ continuous, $|f^{(4)}|\le K$)§ 5.4 →
$|E_S|\le\dfrac{K(b-a)^5}{180n^4}$
Absolute and relative error§ 5.4 →
$|I-A|\qquad\text{and}\qquad\dfrac{|I-A|}{|I|}$
Improper integrals: infinite limits ($f$ continuous)§ 5.5 →
$\displaystyle\int_a^\infty f\,dx=\lim_{b\to\infty}\int_a^bf\,dx,\qquad\int_{-\infty}^{\infty}f\,dx=\int_{-\infty}^{c}f\,dx+\int_c^\infty f\,dx$ (both must converge)
Improper integrals: $f$ unbounded at $b$ (continuous on $[a,b)$)§ 5.5 →
$\displaystyle\int_a^bf\,dx=\lim_{t\to b^-}\int_a^tf\,dx$; at an interior point $c$: split $\displaystyle\int_a^c+\int_c^b$, both must converge
$p$-integral on $[1,\infty)$§ 5.5 →
$\displaystyle\int_1^\infty\frac{dx}{x^p}=\frac{1}{p-1}$ if $p\gt1$; diverges if $p\le1$
$p$-integral on $(0,1]$§ 5.5 →
$\displaystyle\int_0^1\frac{dx}{x^p}=\frac{1}{1-p}$ if $p\lt1$; diverges if $p\ge1$
Comparison test ($f,g$ continuous, $0\le f\le g$ on $[a,\infty)$)§ 5.5 →
$\int_a^\infty g$ converges $\Rightarrow\int_a^\infty f$ converges; $\quad\int_a^\infty f$ diverges $\Rightarrow\int_a^\infty g$ diverges
Exponential tail ($k\gt0$)§ 5.5 →
$\displaystyle\int_0^\infty Ce^{-kt}\,dt=\frac{C}{k}$
Chapter 6 · Applications of Integration
Area between curves (vertical strips)§ 6.1 →
$f,g$ continuous, $f(x)\ge g(x)$ on $[a,b]$: $\ A=\displaystyle\int_a^b\left[f(x)-g(x)\right]dx$
Area between curves (horizontal strips)§ 6.1 →
$u,v$ continuous, $u(y)\ge v(y)$ on $[c,d]$: $\ A=\displaystyle\int_c^d\left[u(y)-v(y)\right]dy$
Curves that cross§ 6.1 →
$A=\displaystyle\int_a^b\left|f(x)-g(x)\right|dx$ — split $[a,b]$ at every crossing point and integrate top − bottom on each piece
Consumer and producer surplus§ 6.1 →
At equilibrium $D(q_e)=S(q_e)=p_e$: $\ CS=\displaystyle\int_0^{q_e}\left[D(q)-p_e\right]dq,\quad PS=\int_0^{q_e}\left[p_e-S(q)\right]dq$
Volume by slicing§ 6.2 →
Cross-sectional area $A(x)$ continuous on $[a,b]$: $\ V=\displaystyle\int_a^bA(x)\,dx$
Disk method§ 6.2 →
About the $x$-axis: $V=\displaystyle\int_a^b\pi\left[f(x)\right]^2dx$; about the $y$-axis: $V=\displaystyle\int_c^d\pi\left[g(y)\right]^2dy$ ($f$, $g$ continuous)
Washer method§ 6.2 →
$f(x)\ge g(x)\ge0$ continuous, about the $x$-axis: $\ V=\displaystyle\int_a^b\pi\left(\left[f(x)\right]^2-\left[g(x)\right]^2\right)dx$
Revolving about another line§ 6.2 →
Axis $y=k$ (not through the region): $V=\displaystyle\int_a^b\pi\left(R(x)^2-r(x)^2\right)dx$, with $R$, $r$ the distances from $y=k$ to the far and near boundaries; for $x=h$ use horizontal slices and $dy$
Shell method about the $y$-axis§ 6.3 →
$f$ continuous, $f\ge0$ on $[a,b]$, $0\le a\lt b$: $\ V=\displaystyle\int_a^b2\pi x\,f(x)\,dx$
Shells about a vertical line $x=c$§ 6.3 →
$g\le f$ continuous on $[a,b]$, $c\le a$ or $c\ge b$: $\ V=\displaystyle\int_a^b2\pi\,|x-c|\,\big(f(x)-g(x)\big)\,dx$
Shells about a horizontal line $y=k$§ 6.3 →
Region $v(y)\le x\le u(y)$, $p\le y\le q$ ($u,v$ continuous), $k\le p$ or $k\ge q$: $\ V=\displaystyle\int_p^q2\pi\,|y-k|\,\big(u(y)-v(y)\big)\,dy$
Volume of one shell§ 6.3 →
Average radius $r$, height $h$, thickness $\Delta r$: $\ V=2\pi r\,h\,\Delta r$ (exact)
Arc length§ 6.4 →
$f'$ continuous on $[a,b]$: $\ L=\displaystyle\int_a^b\sqrt{1+\big(f'(x)\big)^2}\,dx$; for $x=g(y)$, $g'$ continuous on $[c,d]$: $\ L=\displaystyle\int_c^d\sqrt{1+\big(g'(y)\big)^2}\,dy$
Arc-length function§ 6.4 →
$s(x)=\displaystyle\int_a^x\sqrt{1+\big(f'(t)\big)^2}\,dt$, $\ (ds)^2=(dx)^2+(dy)^2$
Hyperbolic functions§ 6.4 →
$\cosh x=\tfrac{e^x+e^{-x}}{2}$, $\sinh x=\tfrac{e^x-e^{-x}}{2}$, $\cosh^2x-\sinh^2x=1$, $(\cosh x)'=\sinh x$, $(\sinh x)'=\cosh x$
Surface area of revolution§ 6.4 →
$f\ge0$, $f'$ continuous: about the $x$-axis $S=\displaystyle\int_a^b2\pi f(x)\sqrt{1+\big(f'(x)\big)^2}\,dx$; about the $y$-axis ($0\le a$) $S=\displaystyle\int_a^b2\pi x\sqrt{1+\big(f'(x)\big)^2}\,dx$
Frustum of a cone§ 6.4 →
End radii $r_1,r_2$, slant height $\ell$: lateral area $\pi(r_1+r_2)\,\ell$; sphere: $4\pi R^2$
Work done by a variable force§ 6.5 →
$F$ continuous on $[a,b]$: $\ W=\displaystyle\int_a^bF(x)\,dx$ (J $=$ N·m)
Hooke's law§ 6.5 →
$F=kx$ ($x$ from natural length): $\ W=\displaystyle\int_a^bkx\,dx=\tfrac12k\left(b^2-a^2\right)$
Pumping liquid from a tank§ 6.5 →
Layer at height $y$, area $A(y)$, lifted a distance $D(y)$; liquid from $y=c$ to $y=d$: $\ W=\displaystyle\int_c^d\rho\,g\,A(y)\,D(y)\,dy$ ($\rho=1000$ kg/m³ for water, $g=9.8$ m/s²)
Centre of mass of point masses§ 6.5 →
$M=\sum m_i$, $M_y=\sum m_ix_i$, $M_x=\sum m_iy_i$; $\ (\bar x,\bar y)=\left(\tfrac{M_y}{M},\tfrac{M_x}{M}\right)$
Centroid of a region§ 6.5 →
$g\le f$ continuous on $[a,b]$, $A=\displaystyle\int_a^b(f-g)\,dx$: $\ \bar x=\dfrac1A\displaystyle\int_a^bx(f-g)\,dx$, $\ \bar y=\dfrac1A\displaystyle\int_a^b\tfrac12\left(f^2-g^2\right)dx$
Theorem of Pappus§ 6.5 →
Region of area $A$ on one side of an axis, centroid at distance $d$: $\ V=2\pi d\,A$ (curves: $S=2\pi d\,L$)
Chapter 7 · Introduction to Differential Equations
Order and solutions§ 7.1 →
Order = highest derivative present. $y=f(x)$ is a solution on $I$ if substituting it makes the equation true for every $x\in I$.
IVP by direct integration§ 7.1 →
$y^\prime=f(x),\ y(x_0)=y_0\ \Rightarrow\ y=y_0+\displaystyle\int_{x_0}^{x}f(t)\,dt$ ($f$ continuous)
Motion from acceleration§ 7.1 →
Constant $a$: $v=at+v_0$, $\ s=\tfrac12at^2+v_0t+s_0$
Exponential growth / decay§ 7.1 →
$y^\prime=ky,\ y(0)=y_0\iff y=y_0e^{kt}$; half-life (if $k\lt0$) $=\dfrac{\ln2}{|k|}$
Direction field§ 7.2 →
For $y^\prime=f(x,y)$ the segment at $(x,y)$ has slope $f(x,y)$; isoclines: $f(x,y)=c$.
Existence and uniqueness§ 7.2 →
If $f$ and $\partial f/\partial y$ are continuous on an open rectangle containing $(x_0,y_0)$, then $y^\prime=f(x,y)$, $y(x_0)=y_0$ has exactly one solution near $x_0$.
Equilibria of $y^\prime=f(y)$§ 7.2 →
$y=c$ is an equilibrium iff $f(c)=0$. With $f^\prime$ continuous: $f^\prime(c)\lt0\Rightarrow$ stable; $f^\prime(c)\gt0\Rightarrow$ unstable; $f^\prime(c)=0$: use the phase line.
Euler's method§ 7.2 →
$x_{k+1}=x_k+h,\quad y_{k+1}=y_k+h\,f(x_k,y_k)$
Euler's error§ 7.2 →
$|y(b)-y_n|\le Kh$ ($f$ and its first partial derivatives continuous): halving $h$ roughly halves the error.
Separation of variables§ 7.3 →
$\dfrac{dy}{dx}=g(x)h(y)\ \Rightarrow\ \displaystyle\int\frac{dy}{h(y)}=\int g(x)\,dx+C$ where $h(y)\ne0$; every root $c$ of $h$ gives the constant solution $y\equiv c$.
Exponential growth / decay§ 7.3 →
$y^\prime=ky,\ y(0)=y_0\ \Rightarrow\ y=y_0e^{kt}$
Half-life and doubling time§ 7.3 →
$t_{1/2}=\dfrac{\ln2}{|k|}\ (k\lt0),\qquad T_d=\dfrac{\ln2}{k}\ (k\gt0)$
Newton's law of cooling§ 7.3 →
$\dfrac{dT}{dt}=-k(T-T_a),\ k\gt0,\ T(0)=T_0\ \Rightarrow\ T(t)=T_a+(T_0-T_a)e^{-kt}$
Input–output (mixing, infusion)§ 7.3 →
$\dfrac{dy}{dt}=a-by,\ b\gt0,\ y(0)=y_0\ \Rightarrow\ y=\dfrac ab+\left(y_0-\dfrac ab\right)e^{-bt}\to\dfrac ab$; mixing: rate in $=c_{\text{in}}r_{\text{in}}$, rate out $=\dfrac{S}{V}r_{\text{out}}$
Logistic equation and solution§ 7.4 →
$\dfrac{dP}{dt}=rP\left(1-\dfrac PK\right)$, $r,K,P_0\gt0$: $\;P(t)=\dfrac{K}{1+Ae^{-rt}}$, $\;A=\dfrac{K-P_0}{P_0}$, $\;P\to K$
Logistic inflection point§ 7.4 →
If $0\lt P_0\lt\frac K2$: $P=\dfrac K2$ at $t^{*}=\dfrac{\ln A}{r}$; max rate $\dfrac{rK}{4}$
Stability of equilibria§ 7.4 →
$y^\prime=f(y)$, $f^\prime$ continuous, $f(c)=0$: $f^\prime(c)\lt0$ stable; $f^\prime(c)\gt0$ unstable
Logistic fit from $P$ at $t=0,T,2T$§ 7.4 →
$u_i=\dfrac1{P_i}$: $\;q=\dfrac{u_2-u_1}{u_1-u_0}$, $\;\dfrac1K=\dfrac{u_0u_2-u_1^2}{u_0+u_2-2u_1}$, $\;r=-\dfrac{\ln q}{T}$ (needs $0\lt q\lt1$)
Harvesting§ 7.4 →
$\dfrac{dP}{dt}=rP\left(1-\dfrac PK\right)-H$: equilibria exist iff $H\le\dfrac{rK}{4}$ (maximum sustainable yield)
Chapter 8 · Sequences and Series
Limit of a sequence§ 8.1 →
$a_n\to L$ means: for every $\varepsilon\gt0$ there is $N$ with $|a_n-L|\lt\varepsilon$ for all $n\gt N$.
Sequences from functions§ 8.1 →
If $a_n=f(n)$ and $\displaystyle\lim_{x\to\infty}f(x)=L$, then $a_n\to L$ (so L'Hôpital may be used on $f$).
Standard limits§ 8.1 →
$r^n\to0$ if $|r|\lt1$ · $n^{1/n}\to1$ · $x^{1/n}\to1\ (x\gt0)$ · $\left(1+\tfrac xn\right)^n\to e^x$ · $\tfrac{x^n}{n!}\to0$ · $\tfrac{n!}{n^n}\to0$
Squeeze theorem (sequences)§ 8.1 →
If $a_n\le b_n\le c_n$ for all $n\ge n_0$ and $a_n\to L$, $c_n\to L$, then $b_n\to L$.
Monotone Convergence Theorem§ 8.1 →
Monotone and bounded $\Rightarrow$ convergent. (Convergent $\Rightarrow$ bounded.)
Sum of a series§ 8.2 →
$\displaystyle\sum_{n=1}^\infty a_n=\lim_{N\to\infty}S_N$, where $S_N=a_1+\cdots+a_N$ (if the limit exists).
Geometric series§ 8.2 →
$\displaystyle\sum_{n=1}^\infty ar^{n-1}=\frac{a}{1-r}$ if $|r|\lt1$ (diverges if $|r|\ge1$, $a\ne0$); $\ S_N=\dfrac{a(1-r^N)}{1-r}$
Telescoping series§ 8.2 →
$\displaystyle\sum_{n=1}^\infty\left(b_n-b_{n+1}\right)=b_1-\lim_{n\to\infty}b_n$ (when the limit exists); e.g. $\displaystyle\sum\frac1{n(n+1)}=1$
Harmonic series§ 8.2 →
$\displaystyle\sum_{n=1}^\infty\frac1n=\infty$ even though $\tfrac1n\to0$.
Combining convergent series§ 8.2 →
If $\sum a_n=A$, $\sum b_n=B$ converge: $\sum(a_n\pm b_n)=A\pm B$, $\sum ca_n=cA$. Convergent + divergent diverges.
Divergence test§ 8.3 →
If $\lim\limits_{n\to\infty}a_n\ne0$ (or does not exist), then $\sum a_n$ diverges. If $a_n\to0$: no conclusion.
Integral test§ 8.3 →
$f$ positive, continuous, decreasing on $[k,\infty)$, $a_n=f(n)$: $\ \sum_{n=k}^\infty a_n$ and $\int_k^\infty f(x)\,dx$ both converge or both diverge.
$p$-series§ 8.3 →
$\displaystyle\sum_{n=1}^\infty\frac{1}{n^p}$ converges $\iff p\gt 1$ (harmonic series $p=1$ diverges).
Remainder estimate (integral test)§ 8.3 →
$f$ positive, continuous, decreasing on $[N,\infty)$, $\sum a_n$ convergent: $\ \displaystyle\int_{N+1}^\infty f\,dx\le R_N\le\int_N^\infty f\,dx$, so $S_N+\int_{N+1}^\infty f\le S\le S_N+\int_N^\infty f$.
Direct comparison test§ 8.4 →
If $0\le a_n\le b_n$ for all $n\ge N_0$: $\ \sum b_n$ converges $\Rightarrow\sum a_n$ converges; $\ \sum a_n$ diverges $\Rightarrow\sum b_n$ diverges.
Limit comparison test§ 8.4 →
$a_n,b_n\gt 0$, $L=\lim\dfrac{a_n}{b_n}$. $\ 0\lt L\lt\infty$: same behaviour. $\ L=0$ and $\sum b_n$ conv. $\Rightarrow\sum a_n$ conv. $\ L=\infty$ and $\sum b_n$ div. $\Rightarrow\sum a_n$ div.
Benchmark series§ 8.4 →
$\sum r^n$ converges $\iff|r|\lt 1$; $\quad\sum\dfrac1{n^p}$ converges $\iff p\gt 1$.
Growth ladder (dominant terms)§ 8.4 →
$\ln n\ll n^p\ (p\gt 0)\ll r^n\ (r\gt 1)\ll n!$ as $n\to\infty$.
Alternating series test§ 8.5 →
If $b_n\gt0$, $b_{n+1}\le b_n$ (eventually) and $b_n\to0$, then $\displaystyle\sum(-1)^{n+1}b_n$ converges.
Alternating series remainder§ 8.5 →
Under the AST hypotheses: $|S-S_N|\le b_{N+1}$, and $S$ lies between $S_N$ and $S_{N+1}$.
Absolute ⇒ convergent§ 8.5 →
$\sum|a_n|$ converges $\Rightarrow$ $\sum a_n$ converges and $\left|\sum a_n\right|\le\sum|a_n|$.
Alternating harmonic series§ 8.5 →
$1-\tfrac12+\tfrac13-\tfrac14+\cdots=\ln2$
Ratio test§ 8.6 →
$a_n\ne0$, $\displaystyle L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|$: $L\lt1$ ⇒ absolutely convergent; $L\gt1$ or $\infty$ ⇒ divergent; $L=1$ ⇒ inconclusive.
Root test§ 8.6 →
$\displaystyle L=\lim_{n\to\infty}\sqrt[n]{|a_n|}$: $L\lt1$ ⇒ absolutely convergent; $L\gt1$ or $\infty$ ⇒ divergent; $L=1$ ⇒ inconclusive.
Limits for ratio/root tests§ 8.6 →
$\left(1+\tfrac1n\right)^n\to e$, $\ \left(1-\tfrac1n\right)^n\to\tfrac1e$, $\ \sqrt[n]{n}\to1$, $\ \dfrac{(n+1)!}{n!}=n+1$
Strategy for testing $\sum a_n$§ 8.6 →
$a_n\not\to0$ ⇒ diverges · geometric ($|r|\lt1$) / $p$-series ($p\gt1$) · factorials, $c^n$ ⇒ ratio · $(b_n)^n$ ⇒ root · alternating ⇒ AST · algebraic ⇒ (limit) comparison · $f(n)$ positive decreasing ⇒ integral test
Chapter 9 · Power Series
Convergence of a power series§ 9.1 →
$\sum c_n(x-a)^n$ converges only at $x=a$ ($R=0$), or for all $x$ ($R=\infty$), or absolutely for $|x-a|\lt R$ and diverges for $|x-a|\gt R$; endpoints $x=a\pm R$ must be checked separately
Radius from the ratio test§ 9.1 →
If all $c_n\ne0$ and $\displaystyle\lim_{n\to\infty}\left|\frac{c_{n+1}}{c_n}\right|=\ell$, then $R=\dfrac1\ell$ ($R=\infty$ if $\ell=0$, $R=0$ if $\ell=\infty$)
Geometric power series§ 9.1 →
$\dfrac{1}{1-x}=\displaystyle\sum_{n=0}^{\infty}x^n=1+x+x^2+\cdots,\quad|x|\lt 1$
Geometric series error after $x^N$§ 9.1 →
$\dfrac{1}{1-x}-\left(1+x+\cdots+x^N\right)=\dfrac{x^{N+1}}{1-x},\quad x\ne1$
Substituting into the geometric series§ 9.1 →
$\dfrac{1}{1+x^2}=\displaystyle\sum_{n=0}^{\infty}(-1)^nx^{2n}\ (|x|\lt 1),\qquad \dfrac{1}{c-x}=\sum_{n=0}^{\infty}\frac{x^n}{c^{n+1}}\ (|x|\lt |c|,\ c\ne0)$
Term-by-term calculus (radius $R\gt 0$, for $|x-a|\lt R$)§ 9.2 →
$f(x)=\sum c_n(x-a)^n\ \Rightarrow\ f^\prime(x)=\sum_{n\ge1} nc_n(x-a)^{n-1},\quad \int f\,dx=C+\sum c_n\dfrac{(x-a)^{n+1}}{n+1}$ — same $R$; recheck the endpoints
Series for $\ln(1+x)$§ 9.2 →
$\ln(1+x)=\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n+1}x^n}{n}=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots,\quad -1\lt x\le1$
Series for $\arctan x$ and $\pi$§ 9.2 →
$\arctan x=\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n+1}}{2n+1},\ -1\le x\le1;\quad \frac{\pi}{4}=1-\frac13+\frac15-\cdots$
Differentiated geometric series§ 9.2 →
$\dfrac{1}{(1-x)^2}=\displaystyle\sum_{n=0}^{\infty}(n+1)x^n,\quad \sum_{n=1}^{\infty}nx^n=\frac{x}{(1-x)^2},\quad|x|\lt 1$
Uniqueness and products§ 9.2 →
If $f(x)=\sum c_n(x-a)^n$ near $a$, then $c_n=\dfrac{f^{(n)}(a)}{n!}$; $\ \left(\sum c_nx^n\right)\left(\sum d_nx^n\right)=\sum_n\Big(\sum_{k=0}^{n}c_kd_{n-k}\Big)x^n$
Taylor polynomial of degree $n$ about $a$§ 9.3 →
$p_n(x)=\displaystyle\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k$ (Maclaurin: $a=0$)
Taylor's theorem — Lagrange remainder§ 9.3 →
If $f^{(n+1)}$ exists on an open interval $I\ni a$, then for $x\in I$: $f(x)=p_n(x)+R_n(x)$, $R_n(x)=\dfrac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}$, $c$ between $a$ and $x$. If $|f^{(n+1)}(t)|\le M$ between $a$ and $x$: $|R_n(x)|\le\dfrac{M|x-a|^{n+1}}{(n+1)!}$.
Taylor series and when it equals $f$§ 9.3 →
$\displaystyle\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n=f(x)\iff\lim_{n\to\infty}R_n(x)=0$ (for $f$ with derivatives of all orders on an open interval containing $a$ and $x$)
Standard Maclaurin series§ 9.3 →
$e^x=\displaystyle\sum_{n=0}^\infty\frac{x^n}{n!}$, $\sin x=\displaystyle\sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{(2n+1)!}$, $\cos x=\displaystyle\sum_{n=0}^\infty\frac{(-1)^nx^{2n}}{(2n)!}$ (all $x$); $\dfrac1{1-x}=\displaystyle\sum_{n=0}^\infty x^n$ ($|x|\lt1$)
Binomial series (any real $k$)§ 9.4 →
$(1+x)^k=\displaystyle\sum_{n=0}^\infty\binom kn x^n=1+kx+\frac{k(k-1)}{2!}x^2+\cdots$ for $|x|\lt1$, $\dbinom kn=\dfrac{k(k-1)\cdots(k-n+1)}{n!}$
Series obtained by integration§ 9.4 →
$\ln(1+x)=\displaystyle\sum_{n=1}^\infty\frac{(-1)^{n+1}x^n}{n}$, $\tan^{-1}x=\displaystyle\sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{2n+1}$ ($|x|\lt1$; also at $x=1$)
Series of $e^{-x^2}$ and its integral§ 9.4 →
$e^{-x^2}=\displaystyle\sum_{n=0}^\infty\frac{(-1)^nx^{2n}}{n!}$, $\displaystyle\int_0^b e^{-x^2}dx=\sum_{n=0}^\infty\frac{(-1)^nb^{2n+1}}{n!\,(2n+1)}$ (all $x$, $b$)
Physics approximations§ 9.4 →
$\sin\theta\approx\theta$, error $\le\dfrac{|\theta|^3}{6}$; $mc^2\left(\dfrac1{\sqrt{1-v^2/c^2}}-1\right)=\dfrac12mv^2+\dfrac38\dfrac{mv^4}{c^2}+\cdots$ ($v\lt c$)
Chapter 10 · Parametric Equations and Polar Coordinates
Line segment from $(x_0,y_0)$ to $(x_1,y_1)$§ 10.1 →
$x=x_0+(x_1-x_0)t,\ \ y=y_0+(y_1-y_0)t,\ \ 0\le t\le1$
Circle and ellipse (counter-clockwise once)§ 10.1 →
$x=h+a\cos t,\ y=k+b\sin t,\ 0\le t\le2\pi$ traces $\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1$ (a circle when $a=b=r$)
Cycloid (wheel of radius $r$)§ 10.1 →
$x=r(\theta-\sin\theta),\ \ y=r(1-\cos\theta)$; one arch: $0\le\theta\le2\pi$, width $2\pi r$, height $2r$
Projectile (gravity only, no air resistance)§ 10.1 →
$x=(v_0\cos\alpha)t,\ \ y=h+(v_0\sin\alpha)t-\tfrac12gt^2$; from ground level the range is $R=\dfrac{v_0^2\sin2\alpha}{g}$
Slope and concavity of $x=x(t)$, $y=y(t)$ (where $x^\prime(t)\neq0$)§ 10.2 →
$\dfrac{dy}{dx}=\dfrac{y^\prime(t)}{x^\prime(t)},\qquad\dfrac{d^2y}{dx^2}=\dfrac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{x^\prime(t)}$; horizontal tangent: $y^\prime=0,\ x^\prime\neq0$; vertical tangent: $x^\prime=0,\ y^\prime\neq0$
Area under a parametric curve§ 10.2 →
$A=\displaystyle\int_a^b y(t)\,x^\prime(t)\,dt$ if $y\ge0$ and the curve is traced once left to right ($x^\prime\ge0$); use $-\displaystyle\int_a^b y\,x^\prime\,dt$ if traced right to left
Arc length and speed§ 10.2 →
$L=\displaystyle\int_a^b\sqrt{x^\prime(t)^2+y^\prime(t)^2}\,dt$ ($x^\prime,y^\prime$ continuous, curve traversed exactly once); speed $\dfrac{ds}{dt}=\sqrt{x^{\prime2}+y^{\prime2}}$
Surface area of revolution (parametric)§ 10.2 →
About the $x$-axis ($y\ge0$): $S=2\pi\displaystyle\int_a^b y\sqrt{x^{\prime2}+y^{\prime2}}\,dt$; about the $y$-axis ($x\ge0$): $S=2\pi\displaystyle\int_a^b x\sqrt{x^{\prime2}+y^{\prime2}}\,dt$
One arch of a cycloid (radius $r$)§ 10.2 →
length $=8r$, area under the arch $=3\pi r^2$
Polar ↔ rectangular§ 10.3 →
$x=r\cos\theta,\ \ y=r\sin\theta;\qquad r^2=x^2+y^2,\ \ \tan\theta=\dfrac yx\ (x\ne0)$, with $\theta$ chosen in the quadrant of $(x,y)$
Other names for a point§ 10.3 →
$(r,\theta)=(r,\theta+2k\pi)=(-r,\theta+\pi+2k\pi)$, $k\in\mathbb Z$; the pole is $(0,\theta)$ for all $\theta$
Symmetry tests (sufficient, not necessary)§ 10.3 →
Polar axis: $(r,-\theta)$ or $(-r,\pi-\theta)$. $\ $ Line $\theta=\tfrac{\pi}{2}$: $(r,\pi-\theta)$ or $(-r,-\theta)$. $\ $ Pole: $(-r,\theta)$ or $(r,\theta+\pi)$ — each must give an equivalent equation.
Slope of $r=f(\theta)$§ 10.3 →
$\dfrac{dy}{dx}=\dfrac{f'(\theta)\sin\theta+f(\theta)\cos\theta}{f'(\theta)\cos\theta-f(\theta)\sin\theta}$, $\ f$ differentiable, denominator $\ne0$. Tangent at the pole: $\theta=\alpha$ if $f(\alpha)=0,\ f'(\alpha)\ne0$.
Roses and limaçons§ 10.3 →
$r=a\cos n\theta$: $n$ petals ($n$ odd), $2n$ petals ($n$ even). $\ r=a+b\cos\theta$ ($a,b\gt0$): inner loop $a\lt b$, cardioid $a=b$, dimpled $b\lt a\lt2b$, convex $a\ge2b$.
Area in polar coordinates§ 10.4 →
$A=\dfrac12\displaystyle\int_\alpha^\beta r^2\,d\theta$, $\ f$ continuous, region swept exactly once, $0\lt\beta-\alpha\le2\pi$
Area between polar curves§ 10.4 →
$A=\dfrac12\displaystyle\int_\alpha^\beta\left(f(\theta)^2-g(\theta)^2\right)d\theta$, $\ 0\le g\le f$ on $[\alpha,\beta]$ — check the pole for extra intersections
Standard polar areas§ 10.4 →
Cardioid $r=a(1\pm\cos\theta)$: $\tfrac32\pi a^2$. $\ $ Rose $r=a\cos n\theta$: $\tfrac{\pi a^2}{2}$ ($n$ even), $\tfrac{\pi a^2}{4}$ ($n$ odd).
Arc length of $r=f(\theta)$§ 10.4 →
$L=\displaystyle\int_\alpha^\beta\sqrt{r^2+\left(\dfrac{dr}{d\theta}\right)^2}\,d\theta$, $\ f'$ continuous, curve traced exactly once. Cardioid $r=a(1+\cos\theta)$: $L=8a$.