0.3 Review of pre-calculus

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Formulas from geometry

$A=\text{area},$ $V=\text{Volume},\phantom{\rule{0.2em}{0ex}}\text{and}$ $S=\text{lateral surface area}$

Laws of exponents

$\begin{array}{ccccccccccccc}\hfill {x}^{m}{x}^{n}& =\hfill & {x}^{m+n}\hfill & & & \hfill \frac{{x}^{m}}{{x}^{n}}& =\hfill & {x}^{m-n}\hfill & & & \hfill {\left({x}^{m}\right)}^{n}& =\hfill & {x}^{mn}\hfill \\ \hfill {x}^{\text{−}n}& =\hfill & \frac{1}{{x}^{n}}\hfill & & & \hfill {\left(xy\right)}^{n}& =\hfill & {x}^{n}{y}^{n}\hfill & & & \hfill {\left(\frac{x}{y}\right)}^{n}& =\hfill & \frac{{x}^{n}}{{y}^{n}}\hfill \\ \hfill {x}^{1\text{/}n}& =\hfill & \sqrt[n]{x}\hfill & & & \hfill \sqrt[n]{xy}& =\hfill & \sqrt[n]{x}\sqrt[n]{y}\hfill & & & \hfill \sqrt[n]{\frac{x}{y}}& =\hfill & \frac{\sqrt[n]{x}}{\sqrt[n]{y}}\hfill \\ \hfill {x}^{m\text{/}n}& =\hfill & \sqrt[n]{{x}^{m}}={\left(\sqrt[n]{x}\right)}^{m}\hfill & & & & & & & & & & \end{array}$

Special factorizations

$\begin{array}{ccc}\hfill {x}^{2}-{y}^{2}& =\hfill & \left(x+y\right)\left(x-y\right)\hfill \\ \hfill {x}^{3}+{y}^{3}& =\hfill & \left(x+y\right)\left({x}^{2}-xy+{y}^{2}\right)\hfill \\ \hfill {x}^{3}-{y}^{3}& =\hfill & \left(x-y\right)\left({x}^{2}+xy+{y}^{2}\right)\hfill \end{array}$

If $a{x}^{2}+bx+c=0,$ then $x=\frac{\text{−}b±\sqrt{{b}^{2}-4ca}}{2a}.$

Binomial theorem

${\left(a+b\right)}^{n}={a}^{n}+\left(\begin{array}{l}n\\ 1\end{array}\right){a}^{n-1}b+\left(\begin{array}{l}n\\ 2\end{array}\right){a}^{n-2}{b}^{2}+\cdots +\left(\begin{array}{c}n\\ n-1\end{array}\right)a{b}^{n-1}+{b}^{n},$

where $\left(\begin{array}{l}n\\ k\end{array}\right)=\frac{n\left(n-1\right)\left(n-2\right)\cdots \left(n-k+1\right)}{k\left(k-1\right)\left(k-2\right)\cdots 3\cdot 2\cdot 1}=\frac{n!}{k!\left(n-k\right)!}$

Right-angle trigonometry

$\begin{array}{cccc}\text{sin}\phantom{\rule{0.1em}{0ex}}\theta =\frac{\text{opp}}{\text{hyp}}\hfill & & & \text{csc}\phantom{\rule{0.1em}{0ex}}\theta =\frac{\text{hyp}}{\text{opp}}\hfill \\ \text{cos}\phantom{\rule{0.1em}{0ex}}\theta =\frac{\text{adj}}{\text{hyp}}\hfill & & & \text{sec}\phantom{\rule{0.1em}{0ex}}\theta =\frac{\text{hyp}}{\text{adj}}\hfill \\ \text{tan}\phantom{\rule{0.1em}{0ex}}\theta =\frac{\text{opp}}{\text{adj}}\hfill & & & \text{cot}\phantom{\rule{0.1em}{0ex}}\theta =\frac{\text{adj}}{\text{opp}}\hfill \end{array}$

Trigonometric functions of important angles

 $\theta$ $\text{Radians}$ $\text{sin}\phantom{\rule{0.1em}{0ex}}\theta$ $\text{cos}\phantom{\rule{0.1em}{0ex}}\theta$ $\text{tan}\phantom{\rule{0.1em}{0ex}}\theta$ $0\text{°}$ $0$ $0$ $1$ $0$ $30\text{°}$ $\text{π}\text{/}\text{6}$ $1\text{/}2$ $\sqrt{3}\text{/}2$ $\sqrt{3}\text{/}3$ $45\text{°}$ $\text{π}\text{/}\text{4}$ $\sqrt{2}\text{/}2$ $\sqrt{2}\text{/}2$ $1$ $60\text{°}$ $\text{π}\text{/}\text{3}$ $\sqrt{3}\text{/}2$ $1\text{/}2$ $\sqrt{3}$ $90\text{°}$ $\text{π}\text{/}2$ $1$ $0$ —

Fundamental identities

$\begin{array}{cccccccc}\hfill {\text{sin}}^{2}\theta +{\text{cos}}^{2}\theta & =\hfill & 1\hfill & & & \hfill \text{sin}\left(\text{−}\phantom{\rule{0.1em}{0ex}}\theta \right)& =\hfill & \text{−}\text{sin}\phantom{\rule{0.1em}{0ex}}\theta \hfill \\ \hfill 1+{\text{tan}}^{2}\theta & =\hfill & {\text{sec}}^{2}\theta \hfill & & & \hfill \text{cos}\left(\text{−}\phantom{\rule{0.1em}{0ex}}\theta \right)& =\hfill & \text{cos}\phantom{\rule{0.1em}{0ex}}\theta \hfill \\ \hfill 1+{\text{cot}}^{2}\theta & =\hfill & {\text{csc}}^{2}\theta \hfill & & & \hfill \text{tan}\left(\text{−}\phantom{\rule{0.1em}{0ex}}\theta \right)& =\hfill & \text{−}\text{tan}\phantom{\rule{0.1em}{0ex}}\theta \hfill \\ \hfill \text{sin}\left(\frac{\pi }{2}-\theta \right)& =\hfill & \text{cos}\phantom{\rule{0.1em}{0ex}}\theta \hfill & & & \hfill \text{sin}\left(\theta +2\pi \right)& =\hfill & \text{sin}\phantom{\rule{0.1em}{0ex}}\theta \hfill \\ \hfill \text{cos}\left(\frac{\pi }{2}-\theta \right)& =\hfill & \text{sin}\phantom{\rule{0.1em}{0ex}}\theta \hfill & & & \hfill \text{cos}\left(\theta +2\pi \right)& =\hfill & \text{cos}\phantom{\rule{0.1em}{0ex}}\theta \hfill \\ \hfill \text{tan}\left(\frac{\pi }{2}-\theta \right)& =\hfill & \text{cot}\phantom{\rule{0.1em}{0ex}}\theta \hfill & & & \hfill \text{tan}\left(\theta +\pi \right)& =\hfill & \text{tan}\phantom{\rule{0.1em}{0ex}}\theta \hfill \end{array}$

Law of sines

$\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}A}{a}=\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}B}{b}=\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}C}{c}$

Law of cosines

$\begin{array}{ccc}\hfill {a}^{2}& =\hfill & {b}^{2}+{c}^{2}-2bc\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}A\hfill \\ \hfill {b}^{2}& =\hfill & {a}^{2}+{c}^{2}-2ac\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}B\hfill \\ \hfill {c}^{2}& =\hfill & {a}^{2}+{b}^{2}-2ab\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}C\hfill \end{array}$

Addition and subtraction formulas

$\begin{array}{ccc}\hfill \text{sin}\phantom{\rule{0.2em}{0ex}}\left(x+y\right)& =\hfill & \text{sin}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}y+\text{cos}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}y\hfill \\ \hfill \text{sin}\phantom{\rule{0.1em}{0ex}}\left(x-y\right)& =\hfill & \text{sin}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}y-\text{cos}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}y\hfill \\ \hfill \text{cos}\phantom{\rule{0.1em}{0ex}}\left(x+y\right)& =\hfill & \text{cos}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}y-\text{sin}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}y\hfill \\ \hfill \text{cos}\phantom{\rule{0.1em}{0ex}}\left(x-y\right)& =\hfill & \text{cos}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}y+\text{sin}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}y\hfill \\ \hfill \text{tan}\phantom{\rule{0.1em}{0ex}}\left(x+y\right)& =\hfill & \frac{\text{tan}\phantom{\rule{0.2em}{0ex}}x+\text{tan}y}{1-\text{tan}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{tan}y}\hfill \\ \hfill \text{tan}\left(x-y\right)& =\hfill & \frac{\text{tan}\phantom{\rule{0.2em}{0ex}}x-\text{tan}y}{1+\text{tan}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{tan}y}\hfill \end{array}$

Double-angle formulas

$\begin{array}{ccc}\hfill \text{sin}\phantom{\rule{0.2em}{0ex}}2x& =\hfill & 2\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}x\hfill \\ \hfill \text{cos}\phantom{\rule{0.2em}{0ex}}2x& =\hfill & {\text{cos}}^{2}x-{\text{sin}}^{2}x=2\phantom{\rule{0.1em}{0ex}}{\text{cos}}^{2}x-1=1-2\phantom{\rule{0.1em}{0ex}}{\text{sin}}^{2}x\hfill \\ \hfill \text{tan}\phantom{\rule{0.2em}{0ex}}2x& =\hfill & \frac{2\phantom{\rule{0.1em}{0ex}}\text{tan}\phantom{\rule{0.2em}{0ex}}x}{1-{\text{tan}}^{2}x}\hfill \end{array}$

Half-angle formulas

$\begin{array}{ccc}\hfill {\text{sin}}^{2}x& =\hfill & \frac{1-\text{cos}\phantom{\rule{0.2em}{0ex}}2x}{2}\hfill \\ \hfill {\text{cos}}^{2}x& =\hfill & \frac{1+\text{cos}\phantom{\rule{0.2em}{0ex}}2x}{2}\hfill \end{array}$

can you give me a problem for function. a trigonometric one
state and prove L hospital rule
I want to know about hospital rule
Faysal
If you tell me how can I Know about engineering math 1( sugh as any lecture or tutorial)
Faysal
I don't know either i am also new,first year college ,taking computer engineer,and.trying to advance learning
Amor
if you want some help on l hospital rule ask me
it's spelled hopital
Connor
hi
BERNANDINO
you are correct Connor Angeli, the L'Hospital was the old one but the modern way to say is L 'Hôpital.
Leo
I had no clue this was an online app
Connor
Total online shopping during the Christmas holidays has increased dramatically during the past 5 years. In 2012 (t=0), total online holiday sales were $42.3 billion, whereas in 2013 they were$48.1 billion. Find a linear function S that estimates the total online holiday sales in the year t . Interpret the slope of the graph of S . Use part a. to predict the year when online shopping during Christmas will reach \$60 billion?
what is the derivative of x= Arc sin (x)^1/2
y^2 = arcsin(x)
Pitior
x = sin (y^2)
Pitior
differentiate implicitly
Pitior
then solve for dy/dx
Pitior
thank you it was very helpful
morfling
questions solve y=sin x
Solve it for what?
Tim
you have to apply the function arcsin in both sides and you get arcsin y = acrsin (sin x) the the function arcsin and function sin cancel each other so the ecuation becomes arcsin y = x you can also write x= arcsin y
Ioana
what is the question ? what is the answer?
Suman
there is an equation that should be solve for x
Ioana
ok solve it
Suman
are you saying y is of sin(x) y=sin(x)/sin of both sides to solve for x... therefore y/sin =x
Tyron
or solve for sin(x) via the unit circle
Tyron
what is unit circle
Suman
a circle whose radius is 1.
Darnell
the unit circle is covered in pre cal...and or trigonometry. it is the multipcation table of upper level mathematics.
Tyron
what is function?
A set of points in which every x value (domain) corresponds to exactly one y value (range)
Tim
what is lim (x,y)~(0,0) (x/y)
limited of x,y at 0,0 is nt defined
Alswell
But using L'Hopitals rule is x=1 is defined
Alswell
Could U explain better boss?
emmanuel
value of (x/y) as (x,y) tends to (0,0) also whats the value of (x+y)/(x^2+y^2) as (x,y) tends to (0,0)
NIKI
can we apply l hospitals rule for function of two variables
NIKI
why n does not equal -1
ask a complete question if you want a complete answer.
Andrew
I agree with Andrew
Bg
f (x) = a is a function. It's a constant function.
proof the formula integration of udv=uv-integration of vdu.?
Find derivative (2x^3+6xy-4y^2)^2
no x=2 is not a function, as there is nothing that's changing.
are you sure sir? please make it sure and reply please. thanks a lot sir I'm grateful.
The
i mean can we replace the roles of x and y and call x=2 as function
The
if x =y and x = 800 what is y
y=800
800
Bg
how do u factor the numerator?
Nonsense, you factor numbers
Antonio
You can factorize the numerator of an expression. What's the problem there? here's an example. f(x)=((x^2)-(y^2))/2 Then numerator is x squared minus y squared. It's factorized as (x+y)(x-y). so the overall function becomes : ((x+y)(x-y))/2
The
The problem is the question, is not a problem where it is, but what it is
Antonio
I think you should first know the basics man: PS
Vishal
Yes, what factorization is
Antonio
Antonio bro is x=2 a function?
The
Yes, and no.... Its a function if for every x, y=2.... If not is a single value constant
Antonio
you could define it as a constant function if you wanted where a function of "y" defines x f(y) = 2 no real use to doing that though
zach
Why y, if domain its usually defined as x, bro, so you creates confusion
Antonio
Its f(x) =y=2 for every x
Antonio
Yes but he said could you put x = 2 as a function you put y = 2 as a function
zach
F(y) in this case is not a function since for every value of y you have not a single point but many ones, so there is not f(y)
Antonio
x = 2 defined as a function of f(y) = 2 says for every y x will equal 2 this silly creates a vertical line and is equivalent to saying x = 2 just in a function notation as the user above asked. you put f(x) = 2 this means for every x y is 2 this creates a horizontal line and is not equivalent
zach
The said x=2 and that 2 is y
Antonio
that 2 is not y, y is a variable 2 is a constant
zach
So 2 is defined as f(x) =2
Antonio
No y its constant =2
Antonio
what variable does that function define
zach
the function f(x) =2 takes every input of x within it's domain and gives 2 if for instance f:x -> y then for every x, y =2 giving a horizontal line this is NOT equivalent to the expression x = 2
zach
Yes true, y=2 its a constant, so a line parallel to y axix as function of y
Antonio
Sorry x=2
Antonio
And you are right, but os not a function of x, its a function of y
Antonio
As function of x is meaningless, is not a finction
Antonio
yeah you mean what I said in my first post, smh
zach
I mean (0xY) +x = 2 so y can be as you want, the result its 2 every time
Antonio
OK you can call this "function" on a set {2}, but its a single value function, a constant
Antonio
well as long as you got there eventually
zach
2x^3+6xy-4y^2)^2 solve this
femi
follow algebraic method. look under factoring numerator from Khan academy
moe
volume between cone z=√(x^2+y^2) and plane z=2
Fatima
It's an integral easy
Antonio
V=1/3 h π (R^2+r2+ r*R(
Antonio