MATH 1226 - Test 2

$$\frac00$$

Indeterminate

$$\frac{\pm\infty}{\pm\infty}$$

Indeterminate

$$1^{\infty}$$

Indeterminate

$$\infty^0$$

Indeterminate

$$0\cdot\infty$$

Indeterminate

$$\infty-\infty$$

Indeterminate

$$0^0$$

Indeterminate

$$\infty$$

Determinate

$$\infty+\infty$$

Determinate

$$-\infty-\infty$$

Determinate

$$\frac{C}{0}\left\lbrack C\ne0\right\rbrack$$

Determinate

$$\frac{0}{C}\left\lbrack C\ne0\right\rbrack$$

Determinate

$M_{x}$ (Total Moment of the whole system about the x-axis)

$$M_{x}=m_1y_1+m_2y_2+\cdots+m_{n}y_{n}$$

M_{x}=\frac{\rho}{2}\int_{a}^{b}f\left(x\right)^2\differentialD x

$M_{y}$ (Total Moment of the whole system about the y-axis)

$$M_{y}=m_1x_1+m_2x_2+\cdots+m_{n}x_{n}$$

M_{y}=\rho\int_{a}^{b}xf\left(x\right)\differentialD x

M (Sum of masses)

M $=m_1+m_2+\cdots+m_{n}$

The centroid

$$\left(\overline{x},\overline{y}\right)$$

$$\overline{x}$$

$$\overline{x}=\frac{M_{y}}{\char"39C }$$

\overline{x}=\frac{\int_{a}^{b}\!xf\left(x\right)\,dx}{\int_{a}^{b}f\left(x\right)\differentialD x}

$$\overline{y}$$

$$\overline{y}=\frac{M_{x}}{\char"39C }$$

\overline{y}=\frac{\int_{a}^{b}f\left(x\right)^2\differentialD x}{2\int_{a}^{b}f\left(x\right)\differentialD x}

If f is continuous on [a, ∞), then

\int_{a}^{\infty}f\left(x\right)\differentialD x=\lim_{t\to\infty}\int_{a}^{t}f\left(x\right)\differentialD x

If f is continuous on (-∞, b], then

\int_{-\infty}^{b}f\left(x\right)\differentialD x=\lim_{t\to\infty}\int_{t}^{b}f\left(x\right)\differentialD x

If f is continuous on (-∞, ∞), then

\int_{-\infty}^{\infty}f\left(x\right)\differentialD x=\int_{-\infty}^{c}f\left(x\right)\differentialD x+\int_{c}^{\infty}f\left(x\right)\differentialD x

For p > 1

CONVERGES

For p ≤ 1

DIVERGES

If f is continuous on (a, b], but not continuous at a, then:

\int_{a}^{b}f\left(x\right)\differentialD x=\lim_{t\to a^{+}}\int_{t}^{b}f\left(x\right)\differentialD x

If f is continuous on [a, b), but not continuous on b, then:

\int_{a}^{b}f\left(x\right)\differentialD x=\lim_{t\to b^{-}}\int_{a}^{t}f\left(x\right)\differentialD x

If f is continuous on [a, c) ∪ (c, b], but not continuous on c in (a, b), then:

\int_{a}^{b}f\left(x\right)\differentialD x=\int_{a}^{c}f\left(x\right)\differentialD x+\int_{c}^{b}f\left(x\right)\differentialD x

Every continuous random variable has a probability density function, f(x), satisfying (2):

1. $$f\left(x\right)\ge0$$

2. \int_{-\infty}^{\infty}f\left(x\right)\differentialD x=1

The probability that a continous random variable X lies b/w x=a and x=b is given by:

P\left(a\le x\le b\right)=\int_{a}^{b}f\left(x\right)\differentialD x

Big formula for probability density function

0\le\int_{a}^{b}f\left(x\right)\differentialD x\le\int_{-\infty}^{\infty}f\left(x\right)\differentialD x=1

Direct Comparison Test

If f(x) ≥ g(x) ≥ 0, then:

\int_{a}^{b}f\left(x\right)\differentialD x\ge\int_{a}^{b}g\left(x\right)\differentialD x\ge0

$\int_1^{\infty}\frac{1}{x^{p}}$; CONVERGES if $p$>1, DIVERGES if $p$≤1

Simpson’s Rule Formula

$$S_{n}=\left\lbrack f\left(x_0\right)+4f\left(x_1)+2f\left(x_2\right)+4f\left(x_3\right)+\cdots+2f\left(x_{n-2}\right)+4f\left(x_{n-1}\right)+f\left(x_{n}\right)\right\rbrack\right.$$

$$\int^{}\frac{du}{a^2+u^2}$$

$$\frac{1}{a}\arctan\left(\frac{u}{a}\right)+C$$

$a$ = constant

$u$ = variable

$$\int_{}^{}\frac{du}{\sqrt{a^2-u^2}}$$

$$\arcsin\left(\frac{u}{a}\right)+C$$

$a$ = constant

$u$ = variable

$$\int_{}^{}\frac{du}{u\sqrt{u^2-a^2}}$$

$$\frac{1}{a}\operatorname{arcsec}\left(\frac{\left\vert u\right\vert}{a}\right)+C$$

$a$ = constant

$u$ = variable

Condition for balance

m₁d₁ = m₂d₂

_{m = weight of mass}

_{d = distance from the fulcrum (point where bar is being balanced on)}

y = sinx (graph)

y = cosx (graph)

y = tanx (graph)

y = cotx (graph)

y = ln(x) (graph)

y = e^x (graph)

y = e^-x (graph)

y = arctanx (graph)

y = secx (graph)

y = cscx (graph)

y = arcsecx (graph)

y = arccosx (graph)

y = arcsinx (graph)

Mean

$\mu=\int_{-\infty}^{\infty}\!f\left(x\right)\cdot x\,dx$, where $x$ is the weight of each probability

Median

\int_0^{M}f\left(x\right)\differentialD x=\frac12, where $M$ = $Median$

\int_{}^{}\sqrt{a^2-x^2}\differentialD x

$$x=a\sin\left(\theta\right)$$

\int_{}^{}\sqrt{a^2+x^2}\differentialD x

$$x=a\tan\left(\theta\right)$$

\int_{}^{}\sqrt{x^2-a^2}\differentialD x

$$x=a\sec\left(\theta\right)$$