MonasteryMathematics A

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Mathematics A: Formulary

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Annuities · Page 1
$$C\frac{(1+i)^n-1}{i}$$

Equal end-of-period payments; i>-1, i!=0; limit nC at i=0

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Annuities · Page 1
$$C\frac{1-(1+i)^{-n}}{i}$$

Equal end-of-period payments; i>-1, i!=0; limit nC at i=0

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Annuities · Page 1
$$(1+i)C\frac{(1+i)^n-1}{i}$$

Beginning-of-period payments; same valuation date as ordinary future value

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Annuities · Page 1
$$(1+i)C\frac{1-(1+i)^{-n}}{i}$$

Beginning-of-period payments; i>-1, i!=0

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Series · Page 1
$$a_1\frac{1-q^n}{1-q}$$

q!=1; n>=1 integer; q=1 handled by na1

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Series · Page 1
$$\frac{a_1}{1-q}$$

|q|<1

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Euler number · Page 1
$$\lim_{n\to\infty}\left(1+\frac{p}{n}\right)^n=e^p$$

Real p; n positive integer tending to infinity

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Exponential series · Page 1
$$e^x=\sum_{k=0}^{\infty}\frac{x^k}{k!}$$

Real x

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Trigonometry · Page 1
$$\sin^2 x+\cos^2 x=1$$

Real x

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Trigonometry · Page 1
$$\sin(-x)=-\sin x$$

Real x

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Trigonometry · Page 1
$$\cos(-x)=\cos x$$

Real x

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Trigonometry · Page 1
$$\sin(k\pi)=0$$

Integer k; radians

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Trigonometry · Page 1
$$\cos(\pi/2+k\pi)=0$$

Integer k; radians

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Trigonometry · Page 1
$$\sin(x+\pi/2)=\cos x$$

Radians

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Trigonometry · Page 1
$$\cos(x+\pi/2)=-\sin x$$

Radians

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Trigonometry · Page 1
$$\sin(x\pm y)=\sin x\cos y\pm\cos x\sin y$$

Real x,y

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Trigonometry · Page 1
$$\cos(x+y)=\cos x\cos y-\sin x\sin y$$

Real x,y

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Trigonometry · Page 1
$$\cos(x-y)=\cos x\cos y+\sin x\sin y$$

Real x,y

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Conics · Page 2
$$(x-u)^2+(y-v)^2=r^2$$

r>0; centre (u,v)

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Conics · Page 2
$$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$$

a,b>0

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Conics · Page 2
$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$

a,b real nonzero; opens left/right

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Conics · Page 2
$$\frac{y^2}{b^2}-\frac{x^2}{a^2}=1$$

a,b real nonzero; opens upwards/downwards

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Conics · Page 2
$$y=ax^2+c$$

a>0 upwards, a<0 downwards; c real

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Conics · Page 2
$$x=ay^2+c$$

a<0 left, a>0 right; c real

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Special limits · Page 2
$$\lim_{x\to0}\frac{\sin x}{x}=1$$

Radians

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Special limits · Page 2
$$\lim_{x\to\infty}\frac{x^k}{e^x}=0$$

Integer k, as printed

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Special limits · Page 2
$$\lim_{x\to\infty}\frac{\ln x}{x^a}=0$$

a>0

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Special limits · Page 2
$$\lim_{x\to0}\frac{e^x-1}{x}=1$$

Real x tending to zero

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Specified derivatives · Page 2
$$(\tan x)\prime=\frac{1}{\cos^2 x}$$

cos x!=0; radians

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Specified derivatives · Page 2
$$(a^x)\prime=a^x\ln a$$

a>0

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Specified derivatives · Page 2
$$(\log_a x)\prime=\frac{1}{x\ln a}$$

x>0, a>0, a!=1

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Local change · Page 2
$$df=f\prime(x)\,dx$$

f differentiable; approximates finite change

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Local change · Page 2
$$\rho_f(x)=\frac{f\prime(x)}{f(x)}$$

f differentiable, f(x)!=0

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Local change · Page 2
$$\varepsilon_f(x)=\frac{xf\prime(x)}{f(x)}$$

f differentiable, f(x)!=0; relative input interpretation needs x!=0

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Taylor remainder · Page 2
$$R_n(x)=\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-a)^{n+1}$$

Sufficient differentiability on joining interval; xi between a and x

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Tangent plane · Page 2
$$z=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)$$

f differentiable at (a,b)

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Descriptive statistics · Page 2
$$\bar x=\frac{1}{n}\sum_{j=1}^n x_j$$

n>=1

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Descriptive statistics · Page 2
$$\operatorname{med}=\begin{cases}x_{((n+1)/2)}&n\text{ odd}\\(x_{(n/2)}+x_{(n/2+1)})/2&n\text{ even}\end{cases}$$

Sorted observations; n>=1

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Descriptive statistics · Page 2
$$R=x_{(n)}-x_{(1)}$$

Sorted observations; n>=1

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Descriptive statistics · Page 2
$$s^2=\frac{1}{n-1}\sum_{j=1}^n(x_j-\bar x)^2$$

n>=2; unbiased sample variance

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Descriptive statistics · Page 2
$$s=\sqrt{s^2}$$

Nonnegative sample SD

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Descriptive statistics · Page 2
$$\mathrm{CV}=\frac{s}{\bar x}$$

Mean nonzero; printed as a ratio, not percentage

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Probability · Page 3
$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$

Events in one probability space

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Probability · Page 3
$$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$$

P(B)>0

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Probability · Page 3
$$P(A)=P(A\mid B)P(B)+P(A\mid\overline B)P(\overline B)$$

Binary partition; displayed conditionals require 0<P(B)<1

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Discrete random variables · Page 3
$$E[X]=\sum_j x_jP(X=x_j)$$

Absolutely convergent for finite expectation

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Discrete random variables · Page 3
$$\operatorname{Var}(X)=E[(X-E[X])^2]$$

Finite second moment

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Discrete random variables · Page 3
$$\sigma_X=\sqrt{\operatorname{Var}(X)}$$

Finite variance

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Discrete random variables · Page 3
$$E[X^k]=\sum_jx_j^kP(X=x_j)$$

Moment exists; k positive integer

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Additional study formulas

These are not printed as standalone formulas on the supplied sheet. Learn their reasoning and when they apply.

Additional study formulas
$$A=P(1+i)^n$$

i>-1; matching periods; no intermediate flows

Additional study formulas
$$\operatorname{NPV}(i)=\sum_{k=0}^n\frac{C_k}{(1+i)^k}$$

i>-1; signed matching-period cash flows

Additional study formulas
$$\operatorname{NPV}(r)=0$$

r>-1; roots need not exist or be unique

Additional study formulas
$$(uv)\prime=u\prime v+uv\prime$$

Both functions differentiable

Additional study formulas
$$(u\circ v)\prime=u\prime(v)v\prime$$

v differentiable and u differentiable at v(x)

Additional study formulas
$$(u/v)\prime=\frac{u\prime v-uv\prime}{v^2}$$

Differentiable u,v; v!=0

Additional study formulas
$$P_n(x)=\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k$$

Derivative values through order n at a; approximation needs remainder control

Additional study formulas
$$(x^p)\prime=px^{p-1}$$

Real p on x>0; domain can extend for specified integer powers

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