Algebra
1.
ar +as =ar+s
(ar )s =ars
ar \(a^{r\ }\cdot\ a^{-s}=a^{r-s}\)
\(\left(a\cdot b\right)^r=a^r\cdot b^r\)
\(\left(\frac{a}{b}\right)^r=\frac{a^r}{b^r}=a^r\cdot b^{-r}\)
2.
\(S_o=4\pi r^2\)
\(S_1=4\pi\left(4r\right)^2\)
\(S_1=4\left(4\pi r^2\right)\)
\(S_1=4\left(S_o\right)\)
3.
\(PV\ =\ \frac{FV}{\left(a+r\right)^t}\)
\(3,000\ =\ \frac{1000000}{\left(1+r\right)^{40}}\)
r = 15.63%
Logic, Proofs, and set theory
1.
= for all
= an element of
= not an element of
= subset
= sum
= implies
= if and only if
= such that
2.
The proof is asking if \(\frac{1}{n}\Sigma_{i=1}^nX_i\sim u_x=0\)
where \(u_x\) is equal to the mean and \(\frac{1}{n}\Sigma_{i=1}^nXi\) is equal to the sum of all the numbers multiplied by one over the total amount of numbers. These are equal because that is the formula for the mean expressed in a different way.
\(\frac{1}{n}\Sigma_{i=1}^nX_i=u_i\)
\(=>\) \(\frac{1}{n}\left(X+X_1+X_2+...+X_n\right)=\frac{\left(X+X_1+X_2+...+Xn\right)}{n}\)
3.
\(\left(a+b\right)^m=a^m+\left(\frac{m}{1}\right)a^{m-1}b+...+\left(\frac{m}{\left(m-1\right)}\right)ab^{m-1}+\left(\frac{m}{m}\right)b^m\)
\(\left(a+b\right)^7=a^7+\left(\frac{7}{1}\right)a^6b+\)\(\left(\frac{7}{2}\right)a^5b^2+\left(\frac{7}{3}\right)a^4b^3+\left(\frac{7}{4}\right)a^3b^4+\left(\frac{7}{\left(5\right)}\right)a^2b^5\)\(+\left(\frac{7}{6}\right)ab^6+\left(\frac{7}{7}\right)b^7\)
\(\left(a+b\right)^7=a^7+7a^6b+21a^5b^2+35a^4b^3+35a^3b^4+21a^2b^5+7ab^6+b^7\)
4.
Necessary Condition: Living in America is a necessary condition for living in Fairfax
P = Living in Fairfax
Q = Living in America
P => Q
Sufficient Condition: Living in Fairfax is a sufficient condition for living in America
P = Living in Fairfax
Q = Living in America
P => Q
Necessary / Sufficient: Being a father is a sufficient condition for being a male, and being a male is a necessary condition for being a father
P= Being a father
Q = Being a male
P<=> Q
Equations
1.
2.
3.
a.
\(3x^2\sim5x+15=0\)
\(x\ =\ \frac{5\left(+\sim\right)\sqrt{25\sim180}}{6}\)
\(x\ =\ \frac{5\left(+\sim\right)i\sqrt{155}}{6}\)
b.
\(4x^2=1+3x\)
\(\sim4x^2+3x+1=0\)\(x=\sim\frac{3\left(+\sim\right)\sqrt{9+16}}{8}\)
\(\sim\frac{3\left(+\sim\right)5}{8}\)
= \(\frac{1}{4}\) and -1
c.
\(\left(2Q\sim14Q^2\right)\)
\(2Q\left(1\sim7Q\right)=0\)
\(Q\ =\ 0\ and\ \frac{1}{7}\)
4.
a.
\(P=a-bQ\ \ \ \ P=\alpha-\beta Q\)
\(a-\alpha-bQ=\beta Q\)
\(a-\alpha=Q\left(b+\beta\right)\)
\(Q=\frac{\left(a-\alpha\right)}{b-\beta}\)
P= \(a-b\left(\frac{\left(a-\alpha\right)}{b+\beta}\right)\)
b.
\(3x+2y=3\ \ \ \ \ \ 6y+6x=-1\)
\(2y=3-3x\)
\(y=\frac{\left(3-3x\right)}{2}\)
\(6\left(\frac{\left(3-3x\right)}{2}\right)+6x=-1\)
\(x=\frac{10}{3}\)
\(3\left(\frac{10}{3}\right)+2y=3\)
\(y=-3.5\)
Uni-variate Functions
1.
a.
y+12=-4(x-2)
b.
y=.7(x-1)
2.