heres a fun practice problem
you originally have 500 grams of an element
8 hours later you have 400 grams
if you split this up you get 250 grams, so whats half life
how are we gonna find that, well
the easiest
you set up basic exponetial equation
y=a times b to the x, now put the thought in here
the original amount will be a
if youre not sure you can always figure that out
you can say, when x is 0, i have 500
so 500
is a times b to the 0
b to the 0 is 1
anything to the zero is one besides 0
a is 500
so now we know a, you can take this equation
and prove it
y=500b to the x
now you say okay
and after 8 hours i have 400
so 400
equals
500
times b to the 8
so you divide by 400 you get 4/5
so b
to the 8th root
of 4
5
or write it in a better way, 4/5
to the 1/8
understand how im getting there?
you start with your two pieces of information
at time 0
you have 500 at time 8
you have 400
so first you can use the 500
and 0 to find a is 500
and then you plug in a and do it again
i know when x is 8 y is 400
now i can get for b
that means my equationis really
y is 500
times
4/5
to the 1/8
to the x
you are just saying that 500
times 4/5 to the x/8
so far so good?
last step
now that i know that y is 500
times 4/5
to the x/8
i want to find the half life so i want to find when y=250
cause 250 is half of 500
so 250
is 500
times 4/5
x/8
divide 250/500 you get 1/2
and now you just have to solve for x
the easiest way to solve for x is with logs
you can use natural logs, you can use common logs you can use any log youd like
log of 1/2
is log of 4/5
and x/8
3x/8
then you just solve for x
you get 8
times a log of 1/2
over the log of 4/5
equals x, howd we do on the half life problem
good, bad
you okay?
practice one more?
lets practice one more
ill leave that up there
ill make up another one
you originally have a 1000 grams of an element
6 hours later you have 200 grams, the half life is less then 6 hours
after 6 hours you only have a 5th
so what is its half life
so we have y is
a times b tot he x
and you should know that a is 1000, but just in case
start off with 1000
x is 0
and anything to 0 is 1
a is 1000
we rewrite this equation as 1000
yes
further more we would have 200 after 6 hours
so 200
1000
b to the 6
divide it by 1000 you get 1/5
b to the 6
b is 1/5
to the 1/6
the 6th root
so far so good?
easy, alright now we can take our equation
you get y=1000
times 1/5
to the x/5
you take the x
and multiply by 1/6
you get that
and then you substitute in a you solve for
you have 500 grams
the half life
so 500
is 1000
times 1/5
to the x/6
divided by a 1000 you get 1/2
1/5
to the x/6
take the log of both sides
you get log of 1/2
log of
1/5
to the x/6
bring the x of 6 in front
and solve for x and you get 6
log of 1/2
over log of 1/5
equals x howd we do?
you get it? yay
alright
second set of things
tan of a is 9/13
and cosine of b is -5/11
where a is between pi and 3pi/2
and b is between pi/2 and pi
find the sin of a plus b
cos2a
and tan of a plus b
this is mostly formulas and figure out where the angles go
so i tell you that a is between pi and 3pi/2
that means that a
somewhere here in the third quadrant
this is pi
this is 3pi/2
the tangent of a is 9/13
sohcahtoa
thats 9 and 13
you do pythagorean theorem and find that this is 250
square root of 250
howd we do on that
b is between pi/2 and pi
so this is pi/2
and this is pi
cosine is -5/11
little pythagorean theorem
and then this is 96 and the square root of 96
so far so good
so now we just have to remmeber the formulas we will not give them on the test
we might give you like one of them
to help jog your memory
what can i say
i suggest you keep it in your head until youre done with math
otherwise youre just gonna have to memorize it again
sin of a plus b
sin of a cos of b
plus cos a
sin of b
if im motivate ill put the formula on instagram
if im motivated
its a two hour train trip either way well
lets see what i can get
the sin of a
is -9/ the square root of 250
how do we know its negative?
because were in the 3rd quadrant
the cosine of b
why it told you what cosb is its -5/11
cosA
is -13/ square root of 250
the sin of b
is square root of 96/11
and you can leave it right now
i dont think you have to really simplify from there
we dont really care if we multiply out
now i have to fins cos2a
well the cosin of 2a
is cosin squared a
minus sin square a
you also can do cosa plus a
you can memorize a couple of these formulas to figure out the rest
the cos if a
is negative 13
over the square root of 250
squared
and the sin of a
is -9/ square root of 250
and you dont need to simplify that
so 9
minus 81
over square root of 250
which is 88/square root of 250
i certainly dont expect you to rationalize
and simplify from there its not important to me
oh yea 250, thank you
now we have to find the tqangent of a plus b
to find the tangent of a plus b, the simple way is you already have the sin of a plus b
find the cosin of a plus b and divide
the cosin of a plus b
is
cosin a
cosin b
minus
sinasinb
so what are they? cosa is
is -13/square root of 250
cosb
is minus 5/11
and sin of a
is -9/square root of 250
and the sin of b
is the square root of 96
over 11
and you take the top of sin and divde it by the cos and get the tangent
so its messy
now we have the cosin
is we wanted to find the tangent
you just take the sin
which is on this board
take the sin and divide it by the cosin
the sin divided by the cos is tangent
you can write the unsimplified expression over the other unsimplified expression
that would be fine with me
okay yes
many professors dont care
if you get to this step its usually good enough
you have to remind the tas of that one
lets do another problem
you have two ships
ship a leaves port at noon and sails 8 mph
ship b leaves port at 6:30
and sails about 10 mph
if the angle between the ships is 60 degrees how far apart are they at 2pm
and next semester youll learn how fast
but not yet
going this way, going that way
60 degrees thats
so how can we find c
well we can use the law of cosine
c sqyared
is a squared plus b squared
minus 2ab
cos c
and what else do we know?
we know that a is how far the ship as gone
at 2pm
also 8mph in 2 hours
so thats 16
b
is going 10 mph
for an hour and a half
thats 15
so far so good?
you know where those numbers come from?
8mph for 2 hours is
16 miles
so this is 16
10 mph
per hour and 1/2
is 15 miles
10 mph for 1 hour is 10 miles
plus an hour and 1/2
you get another 5 miles
you get 15 miles
minus 2
times
2 times 16 imes 15
cos60 degrees
that is 256
plus 225
-480 times cos60 which is a 1/2
2 times 16 times 15 is 480
cos60 is 1/2
481
minus 240
c squared
c squared is
241 and c is
square root of 241
that would be 15.7
maybe 15.8
howd we do on this one?
so once again
where do the 16 and 15 come from
you go 8 mph
for 2 hours
so 8 miles per hour for 1 hour is 8 miles
2 hours at another 8 is 16 miles
10 mph for an hour and a half
so 10 to the first hour
five for the next hour is 15
do we know the law of cosin or should we do another of those
one more? alright
airplane a and airplane b
are initially at the same point
a flies at 500 mph and b flies at 300mph
the annually pi/6
how far are they after 3 hours?
alright well c squared is well lets see
airplane a
is going 200 mph
for 200 mph
for 3 hours
is 600 miles
everyone see where the 600 comes from?
200 mph for 3 hours
b
b goes 300 mph for 3 hours so b is going 900 miles
minus 2 times 600
times 900
cos150 degrees
do scientific notation
and you get 360,000+810,000
minus 1,080,000
times negative radical 3/2
arent those fun numbers?
why is it negative radical 3/2
cos150 is negative degrees
youre in the second quadrant
its the same as cos30
but where on the other side
so you add those together
and since square root of something awful like that
i dont really care what numbers come out
whats another kind of thing you need to use
we can do one of those
everyone set with this?
piecewise function
x squared minus 4 is greater then or equal to zero
and 5 plus 3x is less then 0
find all values of x
where f of x equals 0
this is very straight forward it will only take two or three minutes
if we wanna know where f of x equals 0
f of x is x squared minus 4
x is positive 5 plus 3x s is negative
so you shouold know where both of these parts equal to zero
x squared minus 4 equals 0
well when x squared equals 4
s is plus or minus 2
you would have to throw out the x is -2 becaues it has to be x is greater then 0
so you have to throw one of those out
so you only get x=2
now for the second equation you find out what is 0
so 5
=3x equals 0
and x is negative 5/3
negative 5/3 is less than 0, so thats also true
alright find the domain
the numerator can be anything it wants
we dont care whats in the numerator
denominator what about the cubed root
you can get a negative number
sure
the only problem in the denominator is 0
the domain would be all reals
except x is 5
wen x is 5 the denominator is 0
what if instead
i gave you
6-2x over the squrae root of x-5
so once again
the numerator can be anything it wants
but the denominator, you cant have a negative number
the denominator has to be greater then 0
and that the denominator is 0
x is greater then y
interval notation, you want to d that again?
thats 5 and infinity, thats 5 and
negative infinity and 5
and 5, infinity
what i i turn this upside down
and do h of x
square root of x-5
over 6 minus 2x
interval notation is nice but not necessary
only necassary when theres graph stuff
now
now we do care about the numerator
because you cant do the square root of a negative number
so the numerator
x has to be greater than or equal to 5, whats the difference
between here and here
here x cant be 5 because
because you have 5 in the denominator
you put in 5
and denominator you put 0
thats bad
of the numerator is 0, thats fine of the numerator is 0
and
6-2x cant be 0
so x cannot equal 3
you plug in 3 you get 0 in the denominator
square root of x-5 in the numerator so you
you can have 0 in the numerator
x could be equla to 5
or the middle one, you cant have 0 in the denominator so x cannot be 5
got the idea
yes no?
how bout something like that
sin cos negative 1 (3/7
first of all you notice its a positive number
and we are in the first quadrant
that means we have some number
the cos of this angle is 3/7
and now were just trying to find the sin of that angle
you just want to find the sin of x
use the pythagorean theorem and thats square root of 40
sinx is square root of 40/7
that wasnt so bad
one last one of these
cos now we do negative, where do we do inverse cos negative
the second quadrant
we have sin x
cos of x
is negative 2/9
so we use pythagorean theorem
and that comes out
square root 7 over 7
9 square minus 2 squared
this is gonna be the tangent of x
which is
negative square root of 77 over 2