some of you the competent questions are pretty straight forward, be careful you dont psych yourself out
and turn it into harder questions then they are
theres no graphing on the exam
theres no ellipses on the test
I know that makes you happy
cant use calculators, what else would people like to know?
the part one we have the whole take it over in the testing center 2 weeks from now
at the end on the final there will be a part one and a part two
part one on the final, thats what you really need to make sure to pass
thats the real minimum competent part, obbviously
if you pass that at then end you pass the minimum competent course even if you didnt pass the first few times around
thats where you demonstrate to us that you can move up to calculus
theyre not teaching 125 , im sorry
wasnt up to me
other questions
well i think it would probably be the same as last time, 10 out of 16
im pretty sure, well from memory
im pretty sure its 16 points and again you need 10 again
thats about the number, you got to get some of them right
other questions
you sure? okay so heres what im gonna do
im gonna put a bunch of questions on the board that are going to be similar to the minimum competent questions on the test
if you can do these
you are minimally competence
be really happy
so i will give you these four and youll get the basic idea
everybody do that one
okay theres some to start off
so if you cant read these well because of my beautiful hand writing
represent log base 8 of 128 as a fraction, interger or radical
whats the largest domain on which log base 5 to the 5-4x is defined
solve for x. 16 to the 3x-5 equals 32 to the x+!
find the equation of a line from 8,pi and 4,1
dont be scared on pi, pi is just a number
log base a is 20, log base b is 30, simplify log square root of a/ b cubed, i dont know why i said simplify
write the equation of a circle with center at 8,-4 and radius 10
represent log base 10 of 1000 as an interger, radical or fraction
represent log base 8 of 128, as a interger, fraction or radical
so you take log base 8 of 128
you say thats equal to something so we call it x
you guys can see that over there?
that means that 8 to the x
is 128
so now
you go how am i suppose to solve 8 to the x, you could solve it
by doing logs again but thats not what were looking for
this is 2 cubed
so 128 must be 2 to the something
who wants to give a gues
7
yea work it out, thats what you have scrap paper for
so now that we know that 2 to the 3x equals 2 to the 7
then 3x must equal 7
x
is 7/3
okay, how do we feel about this one
okay if you can do this one, its 2 points
if you can practice these
i posted one like this a couple days ago
theres some other practice problems
make sure we do this
8 to what exponent is 128
7/3
cause 8 squared is 64
and 8 cubed is 512, so somewhere between 2 and 3
what is the largest domain of this log base 5
5-4x
is defined
now it can be log any base
it could be natural log
it can be any base and the rules of the log are the same
it happens, when you take the log it has to be a positive number
so whether that says ln
or that say log, log base whatever it doesnt matter,=. this
must be greater than 0, 5 minus 4x
is greater then 0
so 5 has to be greater then 4x
or 5/4
greater then x. or you can flip that around
x is less then 5/4, you dont need to use interval notation
you can write it as, any number less then 5/4
okay you get that? 2 more points
youre up to 4
you can call your mom and say maybe i will be a doctor
well she was hoping, cbecause you have to pay for your time
got to move to florida
i told you i want the good doctors from this class
and if one of you become a billionaire thats fine too
we can talk
so this is not very different from that
16 to the 3x-5
is 32 to the x+!
so lets get a common base, what base would you like to use?
7, i mean 7 would were but thers another
we can do 2? well 4, does 4 work?
no, 8, no 2
so this is 2 to the 4
to the 3x-5
equals 2 to the 5
to the x+1
so that means that 4
times 3x-5
has to equal 5
times x+!
hope that works
so then you get 1x
-20
is 5x
+5
so 7x is 25
x is 25/7
6 points
all you got to do is stuff like this
this is minimum
thats why they are minimum competent
got to impress us to demostrate your confidence
whats incompetent, incompetent is for example two outs in the bottom of the 9th and you give up a homerun
thats it
so when i find the equation of a line
see if some one were to watch that video 20 years from now it will make no sense
so what do we need? we need a slope and a point. remember the equation of a line
y-y1
equals m times x-x1
i know some of you really want to use y=mx+b because you were taught that
its just so nice and easy to remember
but its actually not a good way to find the equation of a line
if you have this kind of information
in fact i dont know why we teach that
its not a good formula of a line, its not a good formula at all
we need m, m is the slope how bout m is
pi-1
over 8-4
or pi-1/4
so if you have 1-pi/-4, thats the same thing
and then brace yourself, pick either point it doesnt matter
y-1
pi-1/4
times x-1, and your done, 4 points
you can pick either point
you dont have to simplify, pi is not 3.14
pi is pi
e is e, pi is pi
these are numbers
so far so good?
you feeling better?
you think your stomach is getting a little better?
8-4
pi-1, 8-4
gives you 1-pi and 4-8
then you would have 1-pi over -4
same thing
doesnt matter which point you use doesnt matter which way you find the slope
if you work it out
youll end up in the same place
you put it in the mx+b formula its not the same
wait theres more
oh my gosh
we ready? how we doing on these?
so far so good?
maybe its not as bad as you thought it would be
dont worry i havent done the harder problems yet
but thats okay this is just the minimum competent stuff
if you can do the minimum competent stuff, youre okay
if get it right youll get at least a c in the class
on the exam, not the class, the exam
log of a is 20
log base b is 30
whats the log of square root of a/b cubed
well lets see
that is he same as the log square root of a
minus
the log of b cubed
so far so good
square root of a, is a to what power?
1/2, is everyone clear it has to be to the 1/2 powe
you sure? okay
that means you can now take the powers in front and make this 1/2log a
-3logb
thats the same as saying a to 1/2
is 1/2, cubed root of a is 1/3
4th root of a is a to the 4th, thats why you use the word 4th
you put the 1/2 in front and you get a 1/2 log a
-3logb, now youre gonna substitue
log a is 20
so this is 1/2 of 20
-3 times 30
comes out -8
you get -80, yeA?
whered the half come from?
square root of a is a to what power
the square root of a
is the same as a to the 1/2
so this is the same as log a to the 1/2
and our rule is you put the power in front
1/2 log a
did you get -80?
you are doing the minimum competent at this point
write the equation of a circle with a center 8,-4
and radius 10, remember the formula for a circle?
its x-h squared
+y-k
squared
equals r squared where
hk
is the center
and r is the readius
so this one is very straight forward
its just gonna be x-8
squared
+y+4
squared
eqauls 10 squared
you do not have to simplifythat
you dont have to multiply it out or anaything, you dont have to make 10 squared into 100
so another words if you have h-h
be okay try not to do that
well put 100 in for 10 squared isnt accidental
but its perfectly fine, yes
so you can square root a number out or not, another words
you dont have to do more work then that
its the minimum amount of work
all you have to do is say x minus
the whole thing squared
plus y minus the other thing squared
equals the radius squared, this is just the pythagorean theorem
so you can turn it to 100 if you want but its not necessary
so far so good?
log base 10 of 1000
log base 10 of 1000 equals x
that means 10 to the x is 1000
10 to the what is 1000
3 if you notice in scientific notation
just count the zeros
the log base 10
1 followed by how many zero is the number of zeros
you doing good so far? yes?
you can just write 3
write a nice big 3
you ready for some harder stuff?
there you go, that would keep you busy for a minute
first when you see this, first you want to do is factor that
i say wow can i factor that? oh sure
x suqraed times 4x+3
pull out a 3
and thats gonna factor into
x-3
x-1 on top
and x+5
x+1
on the bottom
you do the factoring okay?
you gonna ask how i factored that?
what is the largest domain, another words
domain can be anything you want, you can say its 7 if you felt like it
so the question is whats the biggest
section you can do, another words whats the only section you can exclude?
domain can be x to the anything except
-5
or -1, because when you plug those in you make the denominator equal to zero
the domain is the denominator cant equal t zero
because you cant have zero on the bottom of a fraction
have you ever seen a fraction that looks like that?
i dont think so
never seen a zero on the bottom before
you also probably seen better hand writting then that
okay that was a zero, whatever
what value, if any does y approach as x approaches infinity
this is sort of asking for the horizontal asymptotes
the problem with using the word asymptote is
its not really an asymptote its end behavior
so we want to be mathematically correct
so we just say what is this approach
as x goes towards infinity
so remember the rules
you look at the power on top
the highest power is 3x squared
look at the powers on the bottom, the highest power is x squared
since the powers are the same y
is going to approach
3
because its 3/1
so far so good?
now if you have a higher power on top
then it wouldnt approach anything, youd just say
you can either write infinity or it does not exist
okay but the number does not exist
what dont you understand?
so you look at the highest power for each
the numerator and denominator
the highest power on top is 3x squared
the highst power on the bottom is x squared
the power, thats the power
the powers are the same
the coefficient of this term
and the coefficient of this term
3/1
if the bottom is a bigger power, then its just 0
if the top is a bigger power, then it doesnt exist
and i put this on the instagram the other day
so agin you look at the powers
match them up
if its the same on top and same on bottom its coefficient over coefficient
x-1 and x+1 dont cancel
x-1, and x-1 would cancel
remember your rules
ill right it down in a second
what if any are the zeros of fx
the zeros is where the numerator equals 0
so where does the numerator equal 0
at x=3
x=1
and what value does the x cross the x axis?
the y axis sorry, let x equal 0
when you plug 0 in what are you gonna get on top?
what do you get on top if you plug in zero?
what do you get on the bottom when you plug in 0
so it is 9/5
one more time
at what value does f of cross the x axis?
that means your gonna let x=0
okay and when x equals 0
y
get 9 on top
and 5 on bottom, we plug 0 on top
youre gonna get 3 times 0 squared
+_
12 times0
plus 9 so thats just 9
and on the bottom you get
0 squared
plus 5
look what we get 9/5
whered i get 3 and 1 again?
you look at the numerator
the numerator is equal to 0 when this is 0
or thats 0
you set x-3=0
x=3
you set x-1=0
x=1
you can write 9/5 either 9/5 or you can write 0,9/5
either one is okay
this is where end behavioral rule again
so you have some rational expression
you have something times x to the m plus
smaller terms on top
and something plus x to the n smaller terms on the bottom
so x to the m is the highest term on top
x ot the n is the highest term on the bottome
we go that part?
now if the top power is bigger
then it doesnt change anything
the top power is bigger
it just goes towards infinity
the bottom power is bigger
then the y value approaches 0
and if the two powers are the same
then you get a/b
so here the 2 powers are the same, x squared and x squared
so you get 3/1 thats equal to 3
you can write infinity you can write no value
you can write dne, any of of those things
just make clear you know
that it just keeps growing
not that you have no idea what you are doing
even if you do have know idea what you are doing
okay so far so good?
polynomial division
im actually gonna put that on the middle board because some sides wont see it very well
time to erase
what you want to do is 2x-3
is gonna go into
2x to the 4th
+7x to the 3rd
plus 16x
plus 29
x
+21
alright i dont have my glasses on
so far so good?
you look at the left term, you get 2x
what do i have to multiply 2x by
to get 2x to the 4th
i take 2x and multiply what to get 2x to the fourth
x to the 3rd
x to the 3 times 2x is x to the 4th
x to the 3rd times 3
is 3x to the 3rd
so far so good?
you subtract
2x to the 4th times 2x to the 4th answers
7-3=4 so you get 4x cubed
and you bring down the next term
so far so good?
now we do it again
2x times what is 4x cubed
so 2x squared times 2x is 4x cubed
plus 6x squared
16x squared -6x squared is 10x squared
bring down the next term
2x times what is 10x square
5x
you get 10x squared
plus 15x
subract
and bring down the 14x
bring down the 21
a little crocked
and what do i have to multiply 2x to get 14x, i have to mulitply by 7
and when i subtract
there is no remainder
thats very good theres no remainder, what if there is a remainder?
then i would just add that number
divide it by 2x+3
if i had 8, it would be plus 2x+3
i have a quarter of you paying attention, thats okay
howd we do on this one?
so suppose the ramainder
suppose the remainder was 10
then you would have, the next term would be 10/
2x+3
if you had a remainder of 10
you can have a remainder
then you can just ignore it
you just add so you would have plus
something over 2x+3
that would be your remainder
whatevers left okay
want to try another problem
i know how much you want to do this
how bout that? solve that for x
i noticed many of you made the same mistake on this
so first thing youre gonna do is take the log of both sides
then you put the 3x+2 infront
so 3x+2
times log 6
equals x
log 9
many of you forgot the perenthesis
that wwould get you a wrong answer
remember the 3x+2 is multiplied by the log 6, not just the log 2
distribute the log 6, and you get 3x
log 6
plus 2 log 6
equals x
log 9
okay remember what you do next
you put all the terms that containx on one side
and all the terms that do not contain x on the other side
so we can get 2 log 6
is x log 9
minus 3x
log 6
so its always the same thing, first you distribute
group all the terms that contain x on one sides
and all the terms that do not contain x on the other side
factor out the x you get 2 log 6
is x
times log 9
minus 3log6
and divide you get 2log
6/
log9 minus 3
log 6
some of you will have -2log6
and you would have the bottom terms reverse, thats okay
so go through it again
take the log of both sides
take the power in front
that power goes in perenthesises
and distribute
to get 3x, times the log
plus 2 times the log
thats this term
everything with x goes on one side
everything without and x goes on the other side
then you factor out x
then you divide
how out word problem?
we love word problems
you can use yeast in beer, in alochol if your over 21
or bread
or all sorts of things
okay lets see so a(t)
the amount
at time
t
how
much
will you have
after 24 hours
2 part question
were gonna use growth formula
y=a times b to the x
this is any
this is any exppnentail growth
and we use this equation
of course were gonna use t instead of x
and use a of t instead of y, but well fix that at the end
so what do we know? we know that initially
theres 20 grams
5 hours later
theres 30 grams
the question is how much is there
24 hours later
okay
so at time 0 were gonna have 20, so 20
a times b to the 0
anything to the 0 is 1 except for 0
so a is 20
you can fill in one of the variables
in our equation instead of having a times b to the x, you have 20 times b to the x
now we also know
when x is 5 y is 30
you have 30
is 20
times b to the 5
so 3/2
b to the 5
b is 3/2
to the 1/5
the equation because y=20
times 3/2
1/5 to the x or x/5
since i asked for a of t
this would be a of t
is 20
times 3/5
t/5
3/2
we just changed the letters
do we understand that? the first one helps figure out a
the second one helps us firgure out b
and now we just plug in
and now if i want to find out how many there are after 24 hours
say a of 24 hours
20
times 3/2
24/5
howd we do on these?
doing okay?
you have to write a of 24? no
you can write a
we just need to know what you are doing
one last quick one
can i cover this?
ill uncover it in a minute
if you deposit 2000 dollars
in the bank or a user fund, something
at 8% compound continuously
how much will we have after 10 years?
remember the formula is just the future amount
present amount
e to the rt
plu in the future amount
2000 dollars
e
.08
is r
t is 10
you can stop that what we need to do
thats all
how would you figure that out without a calculator?
or a log table
thats the answer