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calculus

Limits, differentiation, integration, kinematics, sequences & series, Taylor series.

limits · differentiation · graphs & kinematics · integration · definite integrals · sequences & series · power series · taylor series

Limits & continuity

Limits/continuity. \(\lim_{x\to a}f(x)=L\) iff both one-sided limits \(=L\). Continuous at \(a\): \(f(a)\) defined, limit exists, \(\lim_{x\to a}f(x)=f(a)\). Differentiable \(\Rightarrow\) continuous. Standard: \(\lim_{x\to0}\dfrac{\sin x}{x}=1\), \(\lim_{x\to0}\dfrac{1-\cos x}{x}=0\), \(\lim_{x\to0}\dfrac{e^x-1}{x}=1\), \(\lim_{x\to0}\dfrac{\ln(1+x)}{x}=1\). L'Hôpital: for \(0/0\) or \(\infty/\infty\), \(\lim f/g=\lim f'/g'\) if the latter exists; repeat if needed. Convert \(0\cdot\infty\), \(\infty-\infty\), \(0^0\), \(1^\infty\), \(\infty^0\) to quotient/log form.

Differentiation

Derivative definition/rules. \(f'(x)=\lim_{h\to0}\dfrac{f(x+h)-f(x)}{h}\). Linearity: \(\dfrac{d}{dx}[cf+g]=cf'+g'\). Product: \((uv)'=u'v+uv'\). Quotient: \(\left(\dfrac uv\right)'=\dfrac{u'v-uv'}{v^2}\). Chain: \(\dfrac{d}{dx}f(g(x))=f'(g(x))\,g'(x)\). Inverse: \((f^{-1})'(x)=1/f'(f^{-1}(x))\). Implicit: differentiate both sides, treat \(y=y(x)\), collect \(dy/dx\). Related rates: \(\dfrac{dA}{dt}=\dfrac{dA}{dx}\dfrac{dx}{dt}\).

Standard derivatives. \(\dfrac{d}{dx}x^n=nx^{n-1}\). \((e^x)'=e^x\), \((a^x)'=a^x\ln a\). \((\ln x)'=1/x\), \((\log_ax)'=1/(x\ln a)\). \((\sin x)'=\cos x\), \((\cos x)'=-\sin x\), \((\tan x)'=\sec^2x\), \((\sec x)'=\sec x\tan x\), \((\cosec x)'=-\cosec x\cot x\), \((\cot x)'=-\cosec^2x\). \((\arcsin x)'=\dfrac1{\sqrt{1-x^2}}\), \((\arccos x)'=-\dfrac1{\sqrt{1-x^2}}\), \((\arctan x)'=\dfrac1{1+x^2}\).

Graphs, optimization & kinematics

Graph/optimization. Tangent at \(x=a\): \(y-f(a)=f'(a)(x-a)\); normal slope \(-1/f'(a)\). Increasing \(f'\gt0\), decreasing \(f'\lt0\). Stationary \(f'=0\). Local max/min: sign change of \(f'\); or \(f''(a)\lt0\Rightarrow\) max, \(f''(a)\gt0\Rightarrow\) min. Concave up \(f''\gt0\), concave down \(f''\lt0\). Inflection: \(f''\) changes sign. Optimization: solve critical points plus endpoints/domain restrictions; verify feasible global value.

Kinematics. \(v=\dfrac{ds}{dt}\), \(a=\dfrac{dv}{dt}=\dfrac{d^2s}{dt^2}\). Displacement \(=\int_{t_1}^{t_2}v(t)\dd t\); distance \(=\int_{t_1}^{t_2}|v(t)|\dd t\); speed \(=|v|\). Vector motion \(\vv r(t)\): \(\vv v=\vv r'(t)\), \(\vv a=\vv r''(t)\), speed \(|\vv v|\).

Integration

Standard integrals. \(\int x^n\dd x=\dfrac{x^{n+1}}{n+1}+C\;(n\ne-1)\). \(\int\dfrac1x\dd x=\ln|x|+C\). \(\int e^x\dd x=e^x+C\), \(\int a^x\dd x=\dfrac{a^x}{\ln a}+C\). \(\int\sin x\dd x=-\cos x+C\), \(\int\cos x\dd x=\sin x+C\), \(\int\sec^2x\dd x=\tan x+C\), \(\int\sec x\tan x\dd x=\sec x+C\), \(\int\cosec^2x\dd x=-\cot x+C\), \(\int\cosec x\cot x\dd x=-\cosec x+C\). \(\int\dfrac1{1+x^2}\dd x=\arctan x+C\), \(\int\dfrac1{\sqrt{1-x^2}}\dd x=\arcsin x+C\). \(\int\tan x\dd x=-\ln|\cos x|+C\), \(\int\cot x\dd x=\ln|\sin x|+C\), \(\int\sec x\dd x=\ln|\sec x+\tan x|+C\), \(\int\cosec x\dd x=\ln|\cosec x-\cot x|+C\).

Linear composites. \(\int f(ax+b)\dd x=\dfrac1aF(ax+b)+C\) where \(F'=f\). \(\int(ax+b)^n\dd x=\dfrac{(ax+b)^{n+1}}{a(n+1)}+C\). \(\int e^{ax+b}\dd x=\dfrac1a e^{ax+b}+C\). \(\int\sin(ax+b)\dd x=-\dfrac1a\cos(ax+b)+C\), \(\int\cos(ax+b)\dd x=\dfrac1a\sin(ax+b)+C\). \(\int\dfrac1{ax+b}\dd x=\dfrac1a\ln|ax+b|+C\).

Integration methods. Substitution: let \(u=g(x)\), \(\dd u=g'(x)\dd x\); \(\int kg'(x)f(g(x))\dd x=kF(g(x))+C\). Parts: \(\int u\dd v=uv-\int v\dd u\); definite \(\int_a^b uv'\dd x=[uv]_a^b-\int_a^b u'v\dd x\). Partial fractions then integrate: \(\int\dfrac A{x-a}\dd x=A\ln|x-a|+C\). Completing the square: \(x^2+2px+q=(x+p)^2+(q-p^2)\) for arctan/arcsin forms.

Definite integrals, area & volume

Definite integrals/area/volume. \(\int_a^b F'(x)\dd x=F(b)-F(a)\). \(\int_a^b f\dd x=-\int_b^a f\dd x\); \(\int_a^b f+\int_b^c f=\int_a^c f\). Area to \(x\)-axis: \(A=\int_a^b|f(x)|\dd x\). Between curves: \(A=\int_a^b(\text{upper}-\text{lower})\dd x\) or \(\int_c^d(\text{right}-\text{left})\dd y\). Volume about \(x\)-axis: \(V=\pi\int_a^b y^2\dd x\); about \(y\)-axis: \(V=\pi\int_c^d x^2\dd y\). Washer: \(V=\pi\int(R^2-r^2)\dd x\) or \(\dd y\).

Sequences & series

Sequences & limits. A sequence \(a_0,a_1,a_2,\ldots\) converges to \(L\), \(\lim_{n\to\infty}a_n=L\), if its terms are eventually arbitrarily close to \(L\); otherwise it diverges — to \(\pm\infty\), or by oscillation (\((-1)^n\)). If \(f\) is continuous and \(x_n\to L\), then \(f(x_n)\to f(L)\). Squeeze: \(a_n\le c_n\le b_n\) with \(a_n,b_n\to L\) forces \(c_n\to L\). Standard limits: \(\dfrac1n\to0\); \(r^n\to0\) for \(|r|\lt1\); \(\dfrac{r^n}{n!}\to0\) for every \(r\). If \(\left|\dfrac{a_{n+1}}{a_n}\right|\to R\), then \(R\lt1\Rightarrow a_n\to0\) and \(R\gt1\Rightarrow|a_n|\to\infty\).

Summation notation. Reindex by substitution (\(n=k+1\) shifts the limits accordingly); sums over the same index range add term by term; a product of two sums is a double sum over both indices — never merge two sums under one index.

Series. \(\sum_n a_n=\lim_{N\to\infty}S_N\), the limit of the partial sums \(S_N=\sum_{n\le N}a_n\); finite limit: converges, otherwise diverges. \(n\)th-term test: \(a_n\not\to0\Rightarrow\) the series diverges — it can never prove convergence (\(1/n\to0\) yet \(\sum1/n=\infty\)).

Geometric series. \(\sum_{n=0}^{\infty}ar^n=\dfrac a{1-r}\) for \(|r|\lt1\) (partial sums \(a\,\dfrac{1-r^{N+1}}{1-r}\)); diverges for \(|r|\ge1\). Repeating decimals are geometric series: \(0.999\ldots=1\).

Convergence tests.

Absolute vs conditional. Absolutely convergent: \(\sum|a_n|\lt\infty\); conditionally convergent: converges but not absolutely (alternating harmonic). Positive-term partial sums increase, so bounded above \(\Rightarrow\) convergent — the engine behind the comparison test.

Reference series. Harmonic \(\sum1/n=\infty\) (compare with \(1+\frac12+\frac12+\cdots\), or integral test); \(p\)-series \(\sum1/n^p\) converges iff \(p\gt1\); \(\sum_{n\ge1}1/n^2=\pi^2/6\); \(\sum_{n\ge0}1/n!=e\); alternating harmonic \(\sum(-1)^{n+1}/n=\ln2\); Leibniz \(1-\frac13+\frac15-\frac17+\cdots=\pi/4\).

Remainder estimates. Positive decreasing \(f\): \(\int_{N+1}^{\infty}f\dd x\le\sum_{n=N+1}^{\infty}f(n)\le\int_{N}^{\infty}f\dd x\) brackets the tail. Alternating series: the sum lies between any two consecutive partial sums, so \(|S-S_N|\le a_{N+1}\) (first omitted term).

Power series

Definition. \(\sum_{n=0}^{\infty}c_nx^n\), or centered at \(a\): \(\sum c_n(x-a)^n\) — a function of \(x\) wherever it converges.

Radius of convergence. Every power series has an \(R\) (possibly \(0\) or \(\infty\)) with absolute convergence for \(|x-a|\lt R\) and divergence for \(|x-a|\gt R\); find \(R\) by the ratio test; at the endpoints \(|x-a|=R\) the ratio test is inconclusive — test separately (comparison, integral, alternating). Examples: \(\sum x^n\): \(R=1\); \(\sum x^n/n!\): \(R=\infty\); \(\sum n!\,x^n\): \(R=0\). Substituting complex \(z\) gives convergence on the disk \(|z-a|\lt R\) — hence "radius"; analytic functions extend to complex arguments this way (\(e^{i\theta}=\cos\theta+i\sin\theta\): see complex numbers).

Term-by-term operations. Within the interval of convergence: add series termwise; multiply like long polynomials, collecting powers (coefficient of \(x^n\) in \(fg\) is \(\sum_{k=0}^{n}a_kb_{n-k}\)); differentiate and integrate term by term. From the geometric series \(\frac1{1-x}=\sum x^n\), \(|x|\lt1\): substitute (\(\frac1{1+x}=\sum(-1)^nx^n\), \(\frac1{1+x^2}=\sum(-1)^nx^{2n}\)), differentiate (\(\frac1{(1-x)^2}=\sum(n+1)x^n\)), integrate (\(\ln(1+x)\) and \(\arctan x\) series).

Taylor & Maclaurin series

Taylor's formula. If \(f\) equals a series \(\sum c_n(x-a)^n\), repeated differentiation at \(x=a\) forces \(c_n=f^{(n)}(a)/n!\):

Maclaurin series: the case \(a=0\). Centered elsewhere, e.g. \(\ln x\) at \(a=1\): \(\ln x=(x-1)-\dfrac{(x-1)^2}2+\dfrac{(x-1)^3}3-\cdots\), valid for \(0\lt x\le2\).

Maclaurin series. \(e^x=1+x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\cdots\). \(\sin x=x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\cdots\). \(\cos x=1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}-\cdots\). \(\ln(1+x)=x-\dfrac{x^2}{2}+\dfrac{x^3}{3}-\dfrac{x^4}{4}+\cdots\), \(|x|\lt1\). \(\arctan x=x-\dfrac{x^3}{3}+\dfrac{x^5}{5}-\dfrac{x^7}{7}+\cdots\), \(|x|\lt1\). \((1+x)^p=1+px+\dfrac{p(p-1)}{2!}x^2+\dfrac{p(p-1)(p-2)}{3!}x^3+\cdots\), \(|x|\lt1\). Get variants by the term-by-term operations above; keep terms to required order; use \(O(x^n)\) for omitted higher terms.

Taylor polynomials & error. \(T_N(x)=\sum_{n=0}^{N}f^{(n)}(a)\dfrac{(x-a)^n}{n!}\): \(N=1\) is the tangent line \(f(a)+f'(a)(x-a)\) (linear approximation), \(N=2\) adds the curvature term \(\dfrac{f''(a)}{2}(x-a)^2\). Write \(f(x)=T_N(x)+E_N(x)\); Lagrange's error formula \(E_N(x)=\int_a^x f^{(N+1)}(t)\dfrac{(x-t)^N}{N!}\dd t\) gives the bound \(|E_N(x)|\le M_{N+1}\dfrac{|x-a|^{N+1}}{(N+1)!}\), where \(M_{N+1}=\max|f^{(N+1)}|\) between \(a\) and \(x\). Accuracy improves with more terms and with \(x\) nearer \(a\).

Convergence of Taylor series. The series need not converge, and \(f\) equals it only where it does. Analytic functions equal their Taylor series wherever the series converges; sums, products, quotients, compositions, inverses, derivatives and antiderivatives of analytic functions are analytic, and \(e^x\), \(\ln x\), \(\sin x\), \(\cos x\) are analytic. Prove equality by showing \(E_N\to0\): for \(e^x\), \(|E_N(x)|\le e^{|x|}\dfrac{|x|^{N+1}}{(N+1)!}\to0\) for every \(x\); likewise \(\sin\), \(\cos\) (all derivatives bounded by 1) — so \(R=\infty\); for \(\ln\), \(\arctan\) about their centers, \(R=1\). A truncated series can still approximate superbly even when the full series diverges (asymptotic series).

Useful approximations. For small \(x\): \(\sin x\sim x\), \(\tan x\sim x\), \(1-\cos x\sim x^2/2\), \(e^x\sim1+x\), \(\ln(1+x)\sim x\), \((1+x)^p\sim1+px\). Limit via dominant terms or series: for \(0/0\), expand numerator and denominator and cancel — often one step where L'Hôpital needs several.