Arctanx.tk

in the notes, he says ${\displaystyle P={\textrm {Re}}\left\{{\frac {V_{s}^{2}\left(R_{L}+jX_{L}\right)}{(R_{S}+R_{L})^{2}+(X_{S}+X_{L})^{2}}}\right\}}$ but in the derivation the denominator is ${\displaystyle \left[(R_{S}+R_{L})+j(X_{S}+X_{L})\right]^{2}}$ which doesn't expand out. I'm wondering if you could have a look at your notes to see if they match

Evaulate ${\displaystyle \int _{0}^{2}(x^{2}+1)dx}$ by calculating ${\displaystyle \lim _{||P||\rightarrow 0}\sum _{i=1}^{n}f(x_{i}^{*})\Delta x_{i}}$ , where ${\displaystyle P}$ is the partition of ${\displaystyle [0,2]}$ into ${\displaystyle n}$ equal intervals and ${\displaystyle x_{i}^{*}}$ is the right end of the ${\displaystyle i}$ th. subinterval.

(Hint: ${\displaystyle \sum _{i=1}^{n}i^{2}={\frac {n(n+1)(2n+1)}{6}}}$ )

${\displaystyle \sum _{x=1}^{n}f(x)}$ means that you take ${\displaystyle f(1)}$ , ${\displaystyle f(2)}$ , ${\displaystyle f(3)}$ , etc. up to ${\displaystyle f(n)}$ , and add them all together.

For example,

${\displaystyle \sum _{x=1}^{5}2x=2(1)+2(2)+2(3)+2(4)+2(5)=30}$

Note that:

${\displaystyle \sum _{x=1}^{5}2x=2\sum _{x=1}^{5}x}$

and that a rule exists that

${\displaystyle \sum _{x=1}^{n}x={\frac {n(n+1)}{2}}}$

so

${\displaystyle \sum _{x=1}^{5}2x=2\sum _{x=1}^{5}x=2{\frac {5(5+1)}{2}}=2{\frac {30}{2}}=30}$