okay solve for x
this is a quadratic in disguises
so we did ones like this before
so we see this and multiply by e to the x
that becomes e to the 2x
so we get 2x e to the x squared
minus 12
equals 4e to the x
now subtarct 4e to the x so you get e tot he 2x
mijus 4e to the x
minus 12
equals 0
you cqn do a substitution at this point if
e to the x bothers you
youj have y equals e to the x
which makes this y squared
minus 4y
minus 12
everyone see that okay?
okay that factors very nicely
y minus 6
y plus 2
an thats
means that y
=6
y=-2 but y is really e to the x
so e tot he x equals 6
or e to the x equals -2, e to the x
always positive so you can throw that out
and of e to the x equals 6
x equals natural log of 6
thats not very hard right?
well do you think thats a minimum competence or thats a part 2?
who votes for part 2? there you go its a part 2
lets try something similar
what if i had
what about that
this factors
into sinx-3
sinx-1 again you can do distribution
if you have y equals sinx
here sinx equals 3
the sin will equla 1
sinx cant equal 3
x is between 1 and negative 1, including 1 and negative 1
you can throw that out
they just need to know where does x equal 1 on this interval
you are suppose to know
x equals pi/2
the review sheet i posted on
a couple days ago was actually for the 126 class
so must of that also applies to this final and stuff
feel free to look at that
i put that uo on
saturday, friday
i forget
lets do another ond
lets do another thing you guys should be able to do
ill think of another one, in the meantime ill give you something else to do
alright a half life problem
its a half life problem so what about setting up the half life equation
anything thats got exponential growth
you write it a y equals a times b to the x
so it says initially we have 40 grams so thats when we know that aequlas 40
or if you want to show it
that means that 40
equals a times b to the 0
so b to the 0 is 1
so this is just 40 to the a
so now we can go to our equation
and you can substitute in
40 for a
now we just to find out b, with a full equation we can find the half life
so i know that 6 hours later i have 30 grams
30 equals
40
b to the 6
you divide both sides by 40
you get 3/4
b to the 6
you need to solve b
b equals 3/4
to the 1/6
that means i can now plug this in here and y equals 40
times 3/4\
to the x/6
now i just have to find the half life
whats the half life, divide by 1.6
times x
so half life, i want to get 20
20 equals 40 times 3/4 to the x/6
divide both sides by 40 you get a half
3/4
to the x/6
i take the log of both sides
take the log of both sides i can take any log i want
the natural log i can do common log
log base anything
bring the x/6 in front
you get log 1/2
equlas x/6
log of 3/4
do a little algebra
and solve for x
you get 6log of 1/2
over log of 3/4
howd we do on that?
wy is b to the 0, initially the time is 0
next question
no lets do something else
if you has f of x equals 6x squared minus 24x minus 30
and the bottom you have
2x squared minus 2x minus 12
find the vertical asymptote, the horizontal asymptote and the x intercept
y intercept
lets find the various pieces of this thing
we need the verticle asymptote
the horizontal asymptote, the vertical asymptote
is also known as the
the curve and acorss
the horizontal asymptte is the end behavior
thats what happens when x goes to infinity
you can cross the vertical asymptote but you cant cross the horizontal asymptote
the x intercept which are the values
where you cross the x axis
and the y intercept
is your values
when you cross the y axis
well if you want to do this the first thing youy want to do wis demostrate
something by facoring
that becomes 6 times x minus 5
x plus 1
this is 2 times x-3
x plus 2
ots not my job to tell you how to factor
suppose to know that
so far so good?
vertical asymptote
look for when the denominator is 0
the denominator is 0 in 2 places
x=3
and x=-2
the horizontal asymptote
well now we are looking for the end behavior, looks at what happens to thei
as x goes out to infinity
so you look at the power on top, the degree of the top is x squared
the degree of the botto is x squared
so this is 6x squared and this is 2x squared
so 6/2 is 3
so this is going to approach
y=3
remember thesE?
x intercepts
are where the function is going to equal 0
or easily where the numerator is equal to 0
so the numerator will equal to 0 at x equals 5
or x=-1 or you can write that as
coordinates
okay you can do either way
doesnt matter
and y intercept is what you get when you plug in 0
so when you plug in 0 on top what do you get?
-30
when you plug in 0 in the bottom you get -12
so its -30/-12
you dont need to simplify that if you dont want to
but if you do you get 5/2
we would not ask you to graph that
how do i find the y intercept? y intercept is when x equals 0
i plug in 0 on top i get 0,0,-30
and the bottom you get 0,0, -12
so its -30/-12
good lets do some other type of stiff
okay lets do the domains for a minute
whats the domain of sinx
all real numbers, you get sin anything you want
how bout cosine x
also all reals
to give a funtion
to have sinx or cosinx in the numerator
or theres not denominator
then that
part you can plug in anything you want
what about e to the x
dont confuse domain and range
you can take e to any value you want
you can do e to the anything
is the domain is all real
but an important thing to remember about e
is the range is only positive numbers so
e to the someting is always positive
sin and cosin range from -1 to 1
e to the x always comes out a positive number and you plug in anything you want
lets see whats the domain of log of x, any log it doesnt matter
the domain of log of x is positive numbers
also known as x is greater then 0, remember the log
is the inverse of e so log always comes out a positive number
so when you are doing natural log you can only plug in a positive number
you cannot take the log of 0 and you cannot take the log of a negative number
what about 1/x
the domain of x connot be 0
because you cant have 1/0
in fact it can be any number on top, any positive
and last what about square root of x
x has to be greater then or equal to 0
because you cannot take a square root of a negative number
alright now lets put it all together
and lets say i have f of x
is
whats the domain
so if you want find the domain of this
this is what you should pay attention to
when you take the square root you take the square root of a positive number
so x+5
has to be greater then or equal to 0
so x has to be greater then or equal to 5
and x squared - 3 cant be 0
x cannot equal plus or minus the square root of 3
and thats good enough
lets do another one
suppose i had
this is a little tricky
first look at the numerator
the numerator can do anything it wants
im not to concern about the numerator so the key is just
then you look at the denominator and say well i have a log
so x-2 has to be greater then 0
x has to be greater then 2
but we have another problem
the problem is you can have 0 in the denominator
so this
cant come out 0
so when is log x-2 equal to 0
well when you do the log of 1
the log of 1 is 0
so that would be when x-2 equals 1
x cant equal 3
it can be greater then 2 but ca be 3
thats probably a little over tricky
but i thought id throw that in tomake sure you guys know all those things
alright lets do some other kind of stuff
im sort of going back and fourth for part 1 and part 2 stuff
just so youre prepared for everything
alright thats not to hard
so if you want to find the equation of a line
you need a point and the slope but we already have
2 points
so we basically want
use the form y-y1
equals the slope times x-x1
we already have a point
so you can either use 5,10 ad 12,7 either it doesnt matter
now youre just gonna find slope
the slope is 10-7
over 5-12
so this -3/7
3/=7
so the equation is
y-10
=3/7
times x-5
or you could use y-7
and -3/7 times x-12
doesnt matter
if you simplify you can arrange and end up in the same spot
so you also have to find the equation of a line
so we need the equation of a circle
with the center of h,k and the radius of r
x-h squared
plus y minus k squared
equals r squared
so for example
the centers at
-3,2
the radius is 5
and it will be x+3
squared
y-2
squared equals 5 squared
that wasnt very hard
how can we make this harder? lets see
suppose i gave you
suppose i gave you that
whats the center? whats the radius?
i recommend you factor it
dont need to complete the square you just need to factor it
x squared plus 8x plus 16
is x plus 4
squared
right thats x plus 4 times x plus 4
how do you know its the square root of this and the square root of this
y squared minus 6y plus 9
is y minus 3
squared
cause its the square root of y
and square root of 9
so now you know that the cente
is at -4,3
and the radius is at 10
lets do something slightly messier
well you have the equation
the equation either this or this
so if i said whats the center and whats the radius
when you see it in this form
you have to be able to transform it into this form
so if you had something like
x squared
plus 3x
plus 225
how do you know how to factor that
well this is a perfect square and this is a perfect square
and this number in the middle
is doubled this number
then it factors into this
the key is the middle number has to be
double that square root
or generally
x squared plus 2ax plus a squared
the factors of x plus a
squared
so square root of a
double that number is 2a
and if you have a minus, so if you had
square root of 36 is 6
12 is double 36
so it owuld be x minus 6
squared
if you have that pattern okay
binomial
thats the square root
how about some inverse syuff
you like the inverse
alright lets find the inverse of this
y=8x-4
over 7
you switch x and y
x is 8y cubed
minus 4/7
you dont have to switch the x and y you cans witch it at the end
doesnt really matter
e is going to isolate the other variable now
what do we do? you multiply both sides by 7
you get 7x
8y cubed minus 4
and we add 4
divide by 8
and so on so 7x
plus 4
is 8x cubed
divide by, 8y sorry
divide by 8
and take the cubed root
and heres something very important
you get so excited you get to this part
youd be so happy
youd be so happy you then forget to write f
inverse of x
is the cubed root of
7x plus 4
alright dont forget the last step okay
all you did hear was find y you didnt demonstrate you know the inverse
were we able to find the inverse
okay lets move onto the next stuff
lets see you do those 3
things
f of cubed x
we take g of x
and we plug it inside of x
so everywhere i see an x i put in an x/7
and you can sort of leave that alone
i dont care if you simplify that but if you really wanted to
you get x squared
over 49
minus 4x
over 67
but if you really want to show how great you are
x squared
minus 28x
over 49
but you know they are all the same to me
you demonstrate which ones goes inside which one
what about the last part, f of x plus h
well what is f of x plus h
f of x plus h is
x plus h
squared minus 4 times
x plus h
what is x plus h squared?
i put that on the instagram
thats x squared
plus 2xh
plus h squared
i think i put that on that, i recommend you remember that
if you cant just foil it out, i recommend you remember it
minus 4x minus 4h
so if i wanted to find f of x plus h minus f of x
f of x plus h
minus f of x
is
this
minus
f of x which is this
thats x squared plus 2xh
plus h squared minus 4x
minus 4h
thats f of x plus h, minus
x squared minus 4x
the clue
whenever you have a polynomial
when you have f of x plus h minus f of x
all the polynomial terms will cancel
the terms of the left side, the f of x plus h
so everything that will be left will have an h in it
so you would cancel and get 2x
h plus h squared
minus 4h, how did we do on that
able to do that part?
thats a good sign
that makes you very happy because thats what part 1 stuff
lets do something else
was that hard?
now we are going to give you
on the test
the sin of a plus b
and the cosin of a plus b ill write that down for you
pay attention its on the cover sheet
the sin of a plus b
is sin a
cosinb
plus cosina
sinb
and cosin of a plus b
is cosina
cosinb
minus sin a
sin b
you dont know this youre not gonna know the others
if you can figure out the sin plus b you can find out sin minus b
alright so lets do these
we have a couple of stages with this so first\
you just have to figure out missing pieces
sin of a is 9/10
the sin of a is 9/10
and here in quadrant 2
so that means this is 9
and this is 10
and you use pythagorean theorem
and thats the square root of 19 and negative
you should just know that in the 2nd quadrant cosin and sin are negative
the cosin of b
is negative 7/8
and were in the third quadrant
thats negative 7 and thats 8
thats the square root of 8 square minus negative 7 squared
which equals minus 49
thats the square root of 15 and thats gonna be negative
found in the 3rd quadrant
so we want to find sina
cosin b
and this is minus so its gonna be minus
cosina
sinb
well sina is 9/10
cosb is -7/8
cosa
is
- radical 19/10
and sinb
is -7/8
i dont really care to simplify
how do you do cisn2b
again you dont have to know the double angle formula
cuase if we tell you
cosina plu b
is cosacosb minus sinasinb
to find the cosin of 2b
you can write this like b plus b
so cosb
cosb minus sinb
sinb
thats all you need to do
so to find the cos2b
its gonna be -7/8
-7/8
minus negative radical 15/8
negative radical 15
over
8
couple other things
how do we feel about these?
good?
im gonna erase this whole mess
do this one
something like this is very straight forward
basically you do the two parts seperately
so where does 5x squared minus 2x equal to 0
well 5x plus 2x squaed
5/2 equals x squared
x is plus or minus squared root of 5/2
but that is the piece for when x has to be 0
so you can throw out the positive answer
so you can only use x equals negative
square root of 5/2
you look at the other part of the function and set it equal to 0
3x plus 6 equals 0
or x equals -2
but thats in the branch where x has to be greater then or equal to 0
so you throw that out
the only solution is negative square root of 5/2