Math 1a Final
When limit is approaching negative infinity always remember to view terms like
#/#^infinity this goes to zero!
$60 to cents
6000 cents
At approx what time is the length of the line the longest? always served vs line accumulated
At the point where rate starts to go negative At the point where the rate starts to go negative
CP ALWAYS REMEMBER when discontinuity
CHECK IF UNDEFINED THEY SET YOU UP FOR THAT CRITICAL POINT
True or false If f is differentiable and f'(c) = 0, then f must have either a local maximum or a local minimum at c. If true, explain why; if false, provide a counterexample
False -- just because it is a critical point doesn't mean it has to a local max or min ex: x^3
Amount of time a person arriving at t must wait to be served when you serve 30/hr
S r(t) dt - 30 (change in t) ----------------------------- 30
Volume of a cylinder
V=πr²h
At approx what time is the length of the line increasing most rapidly ? always served vs line accumulated
new graph is now main graph so highest point where the graph jumps because the line was only accumulating and no one was being served
Compute lim to 0 (sin(1/x)-3)/x^2
oscillations love !! = neg inf
arccos(0)
pi/2 think flipped x and y
cost =
restrain cost + # sold(how much it costs to make)
True or false If limx→∞ f(x) exists, then limn→∞ f(n) also exists and is equal to limx→∞ f(x).
true
When all numbers aren't displayed on graph what do you do for rate in rate out quest
use point slope form and plug in the x that you want to see what the y is
When k= 1 what is the extra thing you have to do When k=0 what is the extra thing you have to do
you have to start at x0 to get the first part of the interval you have to see what the actual endpoint is because always goes from x0 to xn
0^-inf lim
= inf because translates to 1/0
• arctan x def
= the angle in the interval −π 2 , π 2 whose tangent is x tany=x
Simplify arcsin[sin(10pi/3)]
Don't fall for the trick of just cancelling out the inverses because arcsin is only in the domain of [-pi/2,pi/2] 1. you should find the value of sin(10pi/3) by ref angle pi/3 and then figuring out the sign by ASTC 2. then you solve arcsin(-rad(3)/2) which would be -pi/3 can also think of it by having the inverses cancel out and then using the reference angle but keeping in mind where the angle falls in ASTC
How do lim to inf sin(pi*n) and sin(pi*x) differ
For every integer n, sin(πn) = 0, so limn→∞ sin(πn) = 0. However, limx→∞ sin(πx) does not exist because the graph of sin(πx) oscillates between −1 and 1 no matter how far to the right we go.
When you do L'hopital always write
L'H over it
logs: make them
NICER
Domain of arcsinx Range of arcsinx
[-1,1] [-pi/2, pi/2]
How to see if a critical point is a max or min from f''
by the sign concave up is a min concave down is a max
How to find the width of the rectangles in riemann sums
change in interval/ number of rectangles
limit definition of a derivative is with ex f'(6) what happens as h goes to 0 in terms of tangent and secant line
the slope of the secant line between (6,f(6)) and (6+h,f(6+h)) as h gets closer to 0 secant line between (6,f(6)) and (6+h,f(6+h)) approaches the tangent line
Circumference of a sector
theta/2pi * 2pir simplifies to theta*r
Area of a sector
theta/2pi * pir^2
HA'(s) of x/(4x^2+1)^1/2 can make sure by
to inf 1/2 to neg inf -1/2 checking if even or odd function but also know that a sqrt in the denominator calls for plus or minus
If you have multiple critical points and want to find abs max/min you should and then
use sign line take limit (at the open interval) to see HA and to see what the end behavior is to actually find the absolute compared to the local
Suppose w'(70) = 8. Which is the following is the most reasonable conclusion? (strategy)
usually range answer and it always has estimating language !!! and make sure everything matches, easy process of elim
When you have the limits graph and you have to solve for a definition of a derivative function make sure you figure out ex: lim h to 0 f(1+h)-f(1)/h
what x is in f'(x) solve for f'(1)
What does lim to infinity xe^sinx oscillate between
x/e and ex
Graphing Trigonometric Functions
y = Asin(Bx - C) + D A- (max-min)/2 B- Period2pi/b= ______ is x value of a full curve thing C- phase shift D- middle of max and min
Simplify e^(2x)-1
(e^x+1)(e^x-1)
lim x to infinity of 70^(x/2)-1,00,000/2^3x+3^2x+10^100
(rad(70)/3^2)^x = 0
oscillations for cos(pi/x)
- 1 and 1 !!
Derivative of arccos(x)
-1/√(1-x^2)
if f(x) grows slower than g(x) what would the limit approaching infinity be f(x)/g(x)
0 since g(x)>>f(x)
If someone know that the derivative of arctanx is 1/1+x^2 but doesn't remember why prove it by using what you know about proving derivative of tanx
1. we know that arctanx = y is tany=x 2. so use this and do implicit differentiation in terms of x 3. solve for dy/dx-you get 1/(secx)^2 but you should make it (cosx)^2 4. then use the lovely triangle and do the cosine of it to get 1/rad(x^2+1) 5. dont forget to square it :)))))) and you've proved the derivativeee
Often, interest on loans is compounded daily (for example, this is usually the case for credit card debt). Suppose you have taken out a $10,000 loan with a nominal annual interest rate of 15%. If the interest on the loan is compounded daily and you make no payments, how much will you owe after 1 year (365 days)? Please write an exact expression, and then use a calculator to evaluate it basic set up Continuous compounding is defined to be the limit as n → ∞ of having n compounding periods a year. (Daily compounding corresponds to n = 365.) A reasonable question is whether there's a big difference between compounding continuously and compounding daily (after all, daily is already very frequent). To figure this out, write down a limit that gives the amount you'd owe after 1 year if interest had been compounded continuously rather than daily. Evaluate the limit you wrote down in (c).
10000(1 +0.15/365)^365 P(1+r/n)^nt lim n to inf 10,000(1+.15/n)^nt 10,000e^.15(1) e^ln form -- also take out the t
Area of the rectangle within a cylinder
2pirh
Area of a cylinder
2πr^2 + 2πrh
Volume of a Sphere Formula so semi?
4/3(pi)r^3 2/3pir^3
lim x--> 0 cos(1/x)
DNE due to erratical oscillations
arctanx domain and range
Domain: (-inf, inf) Range: (-pi/2, pi/2)
Can you tell from an f' graph where f is continuous where might it be pictured in an f' graph though
NOOO in a jump discontinuity
Revenue =
Price x Quantity
Defining e
We know that the derivative of an exponential is proportional to the exponential function ex: if f(x)=b^x then f'(x)=f'(0)*b^x when the number e is the base of our exponential ***f'(0)=1 so e^x*f'(0) = f' being e^x
When people start getting served/when ship has water that has entered etc... What point would this be
Where both equations intersect at first.
Where should you start your integral for rate in rate out
Where the rate in is more than the rate out. The other parts of the graph are unnecessary
domain of t f(x) = arcsin(4x^2 ) ****
[-1/2,1/2]tweak the [-1,1] domain by setting the inside eq equal to 1 and -1
When a certain number makes an equation undefined and the limit exists it is
a removable discontinuity
if you get a limit approaching #/0 and you're not given 0+ or 0- think ex with lim x to 0 arccosx/cosx-1
about the graph arccosx heads to pi/2when looking at graph cosx-1 x would be negative
Rate in Rate out graph difference between always serving and accumulating a line and then serving
always serving: do not include the portion where the number of people being served is less than how much people can be served at that certain point (so start integrals from where # served intersects the curve) but shift the rest of the graph down the amount you're serving per time serve after accumulating a line: Include everything before and after the ceiling because people weren't being served in the beginning
When ask when is the line increasing most quickly or where the water level is rising most quickly where should you look(2)
at where the r(t) is steepest or look at where there's the biggest change between rate in and rate out
simplify 2^x*9^(x/2)-x not a fraction
become 6^x-x
solve lim x to 0+ x^1/x solve lim x to 0+ [ln(x)-ln(tan(x))]
both 0
When trying to see if two integral equations are a shift up or down what should you do
check the sign of F(x)-G(x) --combine them by subtracting bounds
With limits (first) ALWAYS
direct sub !!!!
When deriving arcs remember
do not distribute the squareds because they're a part of the arcs derivative, not the chain rule one.
If f'(x) is even f(x)...
does not have to be odd because it needs to be at origin to be odd
How to travel through [-pi/4, pi] for cos(2x)=0 draw!
domain is -pi/2 to 2pi
dy/dt=
dy/dx(dx/dt)
if f'(x) is odd what is f(x)
even
Relative rates as x approaches negative infinity for exponentials logs polynomials
exponentials: depends on if the inside term is less than or greater than 1 i.e. if (1/2)^-inf it equals infinity if 1^-inf it equals 0 logs: doesn't make sense polynomials: depends on if the exponent is even orodd if even the answer will be infinity if odd the answer will be negative infinity
If lim x to infinity f(x)/g(x) doesn't equal 0 or infinity we say
f and g grow at the same rate
abs val bars inside the integral: abs val bars outside the integral:
flip the neg y to pos y you solve the integral and then make it pos if the answer is negative
If there's one critical point in either an open or close interval and it is a local max/min must be but only if function is Explanation:
global max/min continuous Since there's only one critical point and it is a local it must be the global because (mention inc/dec)
On a closed & continuous interval [a,b] theres always a this is
global max/min extreme value theorem
cos(2x) compared to cos(x)
half the width of cosx
When told to put several integrals in order be sure to check
if its going from a to b or from b to a !!! if opposite it is negative
WIth logs check
if solution is in domain
lim x to inf 1.1^x + 7x^2/1.05^x
inf because 1.1^x>>1.05^x when f(x) >> g(x) lim = inf
Lim x to 0 x^lnx ***
infinity solve using function to a function -inf * -inf = inf plug back into e and its still inf
always remember #/0 is ALSO
infinity it DNE!!!!! even though its an ans
f(n) includes f(x) includes
integers only all real numbers
Find domain of f(x)=ln(x)^ln(ln(x)) (2)
it is (1,infinity) 1. we first start by knowing that this function will always be greater than 0 because it is a function to a function (can't ln a negative) 2. Then we take a look at the exponent function to see what would work. we can't plug in 0 because its not restricted in the domain, we then see that we can't plug in 1 because ln(1) is 0 and the exponent would end up with an ln(0) which doesnt work (composition function)! be mindful of compositions
At what time between t = 0 and t = 8 is the amount of water in the pond increasing most quickly? vs At what time between t = 0 and t = 8 is the rate of water in the pond increasing most quickly?
just look at highest rate val look at rate of the rate (slope or diff)
Definition of Derivative (2)
lim h->0 f(x+h)-f(x)/h or f(x)-f(#)/x-#
Approximating the number of people that went to the clinic which case?
look at entire original curve and calculate the area under it for both cases
if a lim it going to 0+ in the denominator and you want to make it negative what can you do
make the denominators function -x
Domain of a function to a function
must be greater than 0 because you cant ln a negative function
If asked if t=0 or t=8 has more amount of something you have to look at if
net change if positive net change more at 8 if negative net change more at 0
if f(x)>>g(x) limit approaching infinity would bef(x)/g(x)
plus or minus infinity
When theres a sqrt in the denominator of a function and you're doing the VA (infinity analyzation) zero from the left is (sign)
positive
When finding integral of 1/tan or tan what should you do
put it in terms of sin and cos and do a u sub
When doing l'h and see that one function is 1/x what do you do
put it the denominator automatically
solve lim x to inf 2+sinx/x
separate & you get 0 view with oscillations
When asked for where a composition function is undefined or 0 what do you do
set the inside -x- of the prime fuction where the value is undefined same for 0
When asked about absolute max/min on an open interval always do a because you must think of
sign chart end behavior
Dorothy says "arcsinx is the inverse function of sinx" Why isn't this right
sin(x) has no inverse function, you must restrict it to make it one. arcsin(x) is the inverse function of sin(x) on the interval [-pi/2,pi/2]
When deriving something like f(t)=3t^-2t
take the constant out to the front and derive to get the rest and then multiply by that number (3) TAKE IT COMPLETELY OUT and then multiply it afterwards ans -6t^-2t(lnt+1)
When solving a limit for something like f(t)=3t^-2t lim x to 0+
take the constant out to the front and derive to get the rest and then multiply by that number (3) and in this case remember to plug in BOTH e and that constant !!!! 3
if theres no cp from setting f' = 0 but theres a discontinuity make sure
thats the cp bc cp is 0 or DNE !! use it for both 1st and 2nd deriv test
If f'(x) or f"(x) cannot equal 0 when finding critical points this implies that
the answers are only positive or negative so plug in values in the domain to see which sign it is
When is log diff useful for crazy eq?
when multiplied and divided not when added because you want to use ln rules
When to use closed interval endpoint test
when theres more than one critical point
When you have e times a random quadratic where can the equation equal 0 When doing a sign line and have a function to a function when you set the equation equal to zero what is the sign of that portion
whenever the random quadratic equals zero because e^x can never equal 0 ALWAYS POSITIVE BECAUSE OF ITS DOMAINx > 0
When domain is (0,infinity) how do you explain
write r cannot equal 0 because it doesn't work in the equation (undefined) goes to infinity because no matter how big r is you can always make h really small to make it equal the volume
When dealing with arc the implicit dif is in terms of
x
arccos(x) def
x = the angle in the interval [0,π] whose cosine is x cosy=x
