Difference between revisions of "Formula sheet"

Latest revision as of 19:21, 23 September 2018

A selection of formulas are provided for both midterms as well as the final exam. You can rely on having the following formulas available but note not all of them may be provided on the midterms as the corresponding course material may not (yet) have been covered.

Formula Sheet for Math 103 Exams

Summation

$$\sum_{k=1}^N k = \frac{N(N+1)}2$$ $$\sum_{k=1}^N k^2 = \frac{N(N+1)(2N+1)}6$$ $$\sum_{k=1}^N k^3 = \left(\frac{N(N+1)}2\right)^2$$ $$\sum_{k=0}^N r^k = \frac{1-r^{N+1}}{1-r}$$

Trigonometric identities

$$\sin(\alpha+\beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta;\qquad\mbox{for $\alpha=\beta$:}\quad \sin(2\alpha) = 2\sin \alpha \cos \alpha$$ $$\cos(\alpha+\beta) = \cos\alpha \cos\beta - \sin\alpha \sin\beta;\qquad\mbox{for $\alpha=\beta$:}\quad \cos(2\alpha) = 2\cos^2 \alpha-1$$ $$\cos^2(\alpha)=\frac{1+\cos(2\alpha)}{2}; \ \sin^2(\alpha)=\frac{1-\cos(2\alpha)}{2}$$ $$\sin^2\alpha+\cos^2\alpha = 1$$ $$\tan^2\alpha+1=\sec^2\alpha=\frac1{\cos^2\alpha}$$

Some useful trigonometric values

$$\sin(0)=0,\quad\sin\left(\frac\pi 6\right)=\frac 12,\quad\sin\left(\frac\pi 4\right)=\frac{\sqrt{2}} 2,\quad\sin\left(\frac\pi 3\right)=\frac{\sqrt{3}} 2,\quad\sin\left(\frac\pi 2\right)=1,\quad\sin(\pi)=0$$ $$\cos(0)=1,\quad \cos\left(\frac\pi 6\right)=\frac{\sqrt{3}} 2,\quad\cos\left(\frac\pi 4\right)=\frac{\sqrt{2}} 2,\quad\cos\left(\frac\pi 3\right)=\frac 1 2,\quad\cos\left(\frac\pi 2\right)=0,\quad\cos(\pi)=-1$$

Derivatives

$$\frac d{dx} \arcsin x = \frac 1{\sqrt{1-x^2}}$$ $$\frac d{dx} \arccos x = -\frac 1{\sqrt{1-x^2}}$$ $$\frac d{dx} \arctan x = \frac 1{1+x^2}$$ $$\frac d{dx} \tan x = \sec^2 x$$

Taylor Series

$$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}\, (x-a)^n \qquad\text{(centered at $a$ with $f^{(n)}(a)$ denoting the $n^\text{th}$-derivative evaluated at $a$)}$$ $$e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$$ $$\sin(x) = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}\, x^{2n+1}$$ $$\cos(x) = \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}\, x^{2n} $$

Probability

$$Var(X) = M_2-M_1^2\qquad\text{where $M_1, M_2$ denote the first and second moment}$$