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MATH 101: What to Memorize Before the Exam

Only the essentials: the things you can't work out quickly in an exam. In this course log⁡\log means the natural log, ln⁡\ln. Practice them as flashcards.

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The Definite Integral & Riemann Sums

  • ∑i=1ni=n(n+1)2\sum_{i=1}^{n} i = \dfrac{n(n+1)}{2}
  • ∑i=1ni2=n(n+1)(2n+1)6\sum_{i=1}^{n} i^2 = \dfrac{n(n+1)(2n+1)}{6}

The Fundamental Theorem of Calculus

  • ddx∫g(x)h(x)f(t) dt=f(h(x)) h′(x)−f(g(x)) g′(x)\dfrac{d}{dx}\int_{g(x)}^{h(x)} f(t)\,dt = f(h(x))\,h'(x) - f(g(x))\,g'(x)

Integration by Substitution

  • ∫abf(g(x)) g′(x) dx=∫g(a)g(b)f(u) du\int_a^b f(g(x))\,g'(x)\,dx = \int_{g(a)}^{g(b)} f(u)\,du
    u=g(x)u = g(x): change the limits too.

Integration by Parts

  • ∫u dv=uv−∫v du\int u\,dv = uv - \int v\,du
    Pick uu so it gets simpler when you differentiate it.

Trigonometric Integrals

  • ∫sec⁡x dx=log⁡∣sec⁡x+tan⁡x∣+C\int\sec x\,dx = \log|\sec x + \tan x| + C
  • ∫tan⁡x dx=log⁡∣sec⁡x∣+C\int\tan x\,dx = \log|\sec x| + C
  • ddxsec⁡x=sec⁡xtan⁡x\dfrac{d}{dx}\sec x = \sec x\tan x
    So with u=sec⁡xu = \sec x, du=sec⁡xtan⁡x dxdu = \sec x\tan x\,dx.
  • sin⁡2x=12(1−cos⁡2x)\sin^2 x = \tfrac12(1 - \cos 2x)
  • cos⁡2x=12(1+cos⁡2x)\cos^2 x = \tfrac12(1 + \cos 2x)
  • tan⁡2x=sec⁡2x−1\tan^2 x = \sec^2 x - 1
    Divide sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1 by cos⁡2x\cos^2 x.

Trigonometric Substitution

  • ∫sec⁡x dx=log⁡∣sec⁡x+tan⁡x∣+C\int\sec x\,dx = \log|\sec x + \tan x| + C
  • ∫dxx2+a2=1aarctan⁡xa+C\int\dfrac{dx}{x^2 + a^2} = \dfrac{1}{a}\arctan\dfrac{x}{a} + C
    Gives the arctan part of an in partial fractions.
  • ∫dxa2−x2=arcsin⁡xa+C\int \dfrac{dx}{\sqrt{a^2 - x^2}} = \arcsin\dfrac{x}{a} + C
    For a>0a > 0.
  • sin⁡2x=12(1−cos⁡2x)\sin^2 x = \tfrac12(1 - \cos 2x)
  • cos⁡2x=12(1+cos⁡2x)\cos^2 x = \tfrac12(1 + \cos 2x)
  • tan⁡2x=sec⁡2x−1\tan^2 x = \sec^2 x - 1
    Divide sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1 by cos⁡2x\cos^2 x.
  • For a2−x2\text{For } \sqrt{a^2 - x^2}
    x=asin⁡θx = a\sin\theta
    The root becomes acos⁡θa\cos\theta.
  • For a2+x2\text{For } \sqrt{a^2 + x^2}
    x=atan⁡θx = a\tan\theta
    The root becomes asec⁡θa\sec\theta.
  • For x2−a2\text{For } \sqrt{x^2 - a^2}
    x=asec⁡θx = a\sec\theta
    The root becomes atan⁡θa\tan\theta.

Partial Fractions

  • ∫dxx2+a2=1aarctan⁡xa+C\int\dfrac{dx}{x^2 + a^2} = \dfrac{1}{a}\arctan\dfrac{x}{a} + C
    Gives the arctan part of an in partial fractions.

Numerical Integration: Midpoint, Trapezoidal & Simpson's Rules

  • Mn=Δx[f(xˉ1)+⋯+f(xˉn)]M_n = \Delta x\big[f(\bar{x}_1) + \dots + f(\bar{x}_n)\big]
    xˉj\bar{x}_j is the midpoint of the jj-th piece.
  • Tn=Δx[12f(x0)+f(x1)+⋯+f(xn−1)+12f(xn)]T_n = \Delta x\left[\tfrac{1}{2}f(x_0) + f(x_1) + \dots + f(x_{n-1}) + \tfrac{1}{2}f(x_n)\right]
  • Sn=Δx3[f(x0)+4f(x1)+2f(x2)+⋯+4f(xn−1)+f(xn)]S_n = \dfrac{\Delta x}{3}\big[f(x_0) + 4f(x_1) + 2f(x_2) + \dots + 4f(x_{n-1}) + f(x_n)\big]
    nn must be even.
  • Midpoint rule error\text{Midpoint rule error}
    ∣EM∣≤M(b−a)324n2|E_M| \le \dfrac{M(b-a)^3}{24n^2}
    ∣f′′(x)∣≤M|f''(x)| \le M on [a,b][a, b].
  • Trapezoidal rule error\text{Trapezoidal rule error}
    ∣ET∣≤M(b−a)312n2|E_T| \le \dfrac{M(b-a)^3}{12n^2}
    ∣f′′(x)∣≤M|f''(x)| \le M on [a,b][a, b].
  • Simpson’s rule error\text{Simpson's rule error}
    ∣ES∣≤L(b−a)5180n4|E_S| \le \dfrac{L(b-a)^5}{180n^4}
    ∣f(4)(x)∣≤L|f^{(4)}(x)| \le L on [a,b][a, b].

Improper Integrals

  • The p-test at infinity\text{The } p\text{-test at infinity}
    ∫1∞dxxp converges  ⟺  p>1\int_1^\infty \dfrac{dx}{x^p} \text{ converges} \iff p > 1
    When it , its value is 1p−1\frac{1}{p - 1}.
  • The p-test at 0\text{The } p\text{-test at } 0
    ∫01dxxp converges  ⟺  p<1\int_0^1 \dfrac{dx}{x^p} \text{ converges} \iff p < 1
    The opposite of the test at infinity. p=1p = 1 diverges at both ends.
  • ∫0∞e−ax dx=1a\int_0^\infty e^{-ax}\,dx = \dfrac{1}{a}
    For any constant a>0a > 0. A common integral to compare with.

Area Between Curves

  • Area between curves\text{Area between curves}
    ∫ab(top−bottom) dx\int_a^b (\text{top} - \text{bottom})\,dx
    Or ∫cd(right−left) dy\int_c^d (\text{right} - \text{left})\,dy.

Volumes by Slicing: Disks, Washers & Cross-Sections

  • Washers\text{Washers}
    V=π∫ab(R2−r2) dxV = \pi\int_a^b \big(R^2 - r^2\big)\,dx
    Measure both radii from the axis of rotation. With no hole, r=0r = 0: the disk method.
  • Known cross-sections\text{Known cross-sections}
    V=∫abA(x) dxV = \int_a^b A(x)\,dx
    Square: s2s^2. Semicircle on diameter ss: π8s2\frac{\pi}{8}s^2. Equilateral triangle: 34s2\frac{\sqrt{3}}{4}s^2.

Work

  • Hooke’s law\text{Hooke's law}
    F=kxF = kx
    xx is the stretch beyond natural length, in meters.
  • Weight of a slice of liquid\text{Weight of a slice of liquid}
    ρg A(y) dy\rho g\,A(y)\,dy
    Water: ρ=1000\rho = 1000 kg/m3^3, g=9.8g = 9.8 m/s2^2.

Average Value of a Function

  • fˉ on [a,b]=1b−a∫abf(x) dx\bar f \text{ on } [a, b] = \dfrac{1}{b - a}\int_a^b f(x)\,dx

Center of Mass & Centroids

  • xˉ (rod)=∫abx ρ(x) dx∫abρ(x) dx\bar x\ \text{(rod)} = \dfrac{\int_a^b x\,\rho(x)\,dx}{\int_a^b \rho(x)\,dx}
    Point masses: the same with sums.
  • xˉ (region)=1A∫abx (T−B) dx\bar x\ \text{(region)} = \dfrac{1}{A}\int_a^b x\,(T - B)\,dx
  • yˉ (region)=12A∫ab(T2−B2)dx\bar y\ \text{(region)} = \dfrac{1}{2A}\int_a^b \left(T^2 - B^2\right)dx
    T2−B2T^2 - B^2, not (T−B)2(T - B)^2.

Separable Differential Equations

  • Separable: dydx=f(x) g(y)\text{Separable: } \dfrac{dy}{dx} = f(x)\,g(y)
    ∫dyg(y)=∫f(x) dx+C\int \dfrac{dy}{g(y)} = \int f(x)\,dx + C
  • Solution of dydt=ky\text{Solution of } \dfrac{dy}{dt} = ky
    y=y(0) ekty = y(0)\,e^{kt}
    "Rate proportional to the amount." Decay has k<0k < 0.

Limits of Sequences

  • Who wins as n→∞\text{Who wins as } n \to \infty
    log⁡n≪np≪an≪n!\log n \ll n^p \ll a^n \ll n!
    For p>0p > 0 and a>1a > 1.
  • lim⁡n→∞rn={0∣r∣<11r=1divergesotherwise\lim_{n \to \infty} r^n = \begin{cases} 0 & |r| < 1 \\ 1 & r = 1 \\ \text{diverges} & \text{otherwise} \end{cases}
    r=−1r = -1 gives −1,1,−1,…-1, 1, -1, \ldots, which has no limit.
  • lim⁡n→∞(1+an)n=ea\lim_{n \to \infty}\left(1 + \dfrac{a}{n}\right)^n = e^a

Geometric & Telescoping Series

  • ∑n=0∞arn=a1−r(∣r∣<1)\sum_{n=0}^{\infty} a r^{n} = \dfrac{a}{1 - r} \quad (|r| < 1)
    Diverges when ∣r∣≥1|r| \geq 1. In general: first term over 1−r1 - r.

Convergence Tests

  • The p-series\text{The } p\text{-series}
    ∑n=1∞1np converges  ⟺  p>1\sum_{n=1}^{\infty} \dfrac{1}{n^{p}} \text{ converges} \iff p > 1
    The harmonic series (p=1p = 1) diverges.
  • Divergence test\text{Divergence test}
    lim⁡n→∞an≠0  ⟹  ∑an diverges\lim_{n\to\infty} a_n \neq 0 \;\Longrightarrow\; \sum a_n \text{ diverges}
    A limit of 00 proves nothing.
  • Ratio test: L=lim⁡n→∞∣an+1an∣\text{Ratio test: } L = \lim_{n\to\infty}\left|\dfrac{a_{n+1}}{a_n}\right|
    L<1 converges, L>1 divergesL < 1 \text{ converges},\ L > 1 \text{ diverges}
    L=1L = 1: no information.
  • Alternating series error\text{Alternating series error}
    ∣S−SN∣≤bN+1|S - S_N| \leq b_{N+1}
    For an alternating series with bnb_n decreasing to 00.

Power Series

  • Ratio test: L=lim⁡n→∞∣an+1an∣\text{Ratio test: } L = \lim_{n\to\infty}\left|\dfrac{a_{n+1}}{a_n}\right|
    L<1 converges, L>1 divergesL < 1 \text{ converges},\ L > 1 \text{ diverges}
    L=1L = 1: no information.
  • 11−x=∑n=0∞xn\dfrac{1}{1 - x} = \sum_{n=0}^{\infty} x^{n}
    For ∣x∣<1|x| < 1.

Taylor & Maclaurin Series

  • Taylor series at x=a\text{Taylor series at } x = a
    ∑n=0∞f(n)(a)n!(x−a)n\sum_{n=0}^{\infty} \dfrac{f^{(n)}(a)}{n!}(x - a)^{n}
    A Maclaurin series is the case a=0a = 0.
  • 11−x=∑n=0∞xn\dfrac{1}{1 - x} = \sum_{n=0}^{\infty} x^{n}
    For ∣x∣<1|x| < 1.
  • ex=∑n=0∞xnn!e^{x} = \sum_{n=0}^{\infty} \dfrac{x^{n}}{n!}
    For every xx.
  • sin⁡x=∑n=0∞(−1)nx2n+1(2n+1)!\sin x = \sum_{n=0}^{\infty} \dfrac{(-1)^{n}x^{2n+1}}{(2n+1)!}
    For every xx. Odd powers only.
  • cos⁡x=∑n=0∞(−1)nx2n(2n)!\cos x = \sum_{n=0}^{\infty} \dfrac{(-1)^{n}x^{2n}}{(2n)!}
    For every xx. Even powers only.
  • log⁡(1+x)=∑n=1∞(−1)n+1xnn\log(1 + x) = \sum_{n=1}^{\infty} \dfrac{(-1)^{n+1}x^{n}}{n}
    For −1<x≤1-1 < x \leq 1. Starts at n=1n = 1.
  • arctan⁡x=∑n=0∞(−1)nx2n+12n+1\arctan x = \sum_{n=0}^{\infty} \dfrac{(-1)^{n}x^{2n+1}}{2n+1}
    For −1≤x≤1-1 \leq x \leq 1.
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