anonymous
  • anonymous
Lim as x approaches pi/6 of (sin(x)-sin(pi/6))/x-pi/6
Mathematics
schrodinger
  • schrodinger
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myininaya
  • myininaya
compare to definition of derivative
myininaya
  • myininaya
\[\lim_{x \rightarrow a}\frac{f(x)-f(a)}{x-a}=f'(a)\]
anonymous
  • anonymous
So the limit would = 1?

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myininaya
  • myininaya
no
myininaya
  • myininaya
what is f?
anonymous
  • anonymous
...?
myininaya
  • myininaya
what is f(x) and what is a?
anonymous
  • anonymous
F(x)=sin(x), f(a)=sin(pi/6)
myininaya
  • myininaya
so what is f'?
anonymous
  • anonymous
Cos(x)-cos(pi/6)?
myininaya
  • myininaya
:( if f=sin(x), then f'=?
anonymous
  • anonymous
Cos(x)
myininaya
  • myininaya
right now plug in a to get your answer
anonymous
  • anonymous
A=sin(pi/6)?
myininaya
  • myininaya
\[\lim_{x \rightarrow a}\frac{f(x)-f(a)}{x-a}=f'(a) \text{ or we could say it equals } =(f(x))'|_{x=a}\]
myininaya
  • myininaya
\[\lim_{x \rightarrow \frac{\pi}{6}}\frac{\sin(x)-\sin(\frac{\pi}{6})}{x-\frac{\pi}{6}}=(\sin(x))'|_{x=\frac{\pi}{6}} \]
myininaya
  • myininaya
do you see all i'm trying to get you to do is use the definition of derivative
myininaya
  • myininaya
you are suppose to determine f and a then find f' then plug a into f' to get the result you are searching for
anonymous
  • anonymous
Sqrt(3)/2?
myininaya
  • myininaya
yep
anonymous
  • anonymous
Thank you
anonymous
  • anonymous
Any chance you can help with critical numbers?
myininaya
  • myininaya
sure
anonymous
  • anonymous
F(x)=4x/x^2+1
anonymous
  • anonymous
I know i need the derivative
myininaya
  • myininaya
do you mean 4x/(x^2+1)? Also what did you get for the derivative?
anonymous
  • anonymous
Yes. F'(x)=(-4x^2+4)/(x^2+1)^2
myininaya
  • myininaya
thats great now find when F'=0 Also we don't have to worry when F' DNE because x^2+1 is always positive. so F'=0 when top=0 so when -4x^2+4=0 solve for x now
anonymous
  • anonymous
Sqrt(-1)
myininaya
  • myininaya
I think you should retry solving -4x^2+4=0 I will give you a (you will get two real solutions)
anonymous
  • anonymous
...+or-sqrt(-1)
anonymous
  • anonymous
+or-sqrt(1)
myininaya
  • myininaya
right sqrt(1)=1 so x=1 or x=-1
myininaya
  • myininaya
those are the critical numbers for F.
anonymous
  • anonymous
Wow...
anonymous
  • anonymous
Find the intervals where f(x)=4/(x^+1) is increasing or decreasing.
myininaya
  • myininaya
is that f(x)=4/(x+1)? like you weren't trying to write an exponent right?
anonymous
  • anonymous
I Find the derivative...then set the denominator equal to zero...put those numbers on a number line and plug in numbers to find the intervals?
anonymous
  • anonymous
I was writing an exponent
anonymous
  • anonymous
I apologize for missing the exponent
anonymous
  • anonymous
It should have said 4/(x^2+1)
myininaya
  • myininaya
well I don't know what f you are looking at exactly. but yeah you definitely need to find f' because f' tells you if f is decreasing or increasing or neither f'=0 means the function is resting f'>0 means f is increasing f'<0 means f is decreasing I first find when f' =0 and when f' does not exist and then test each interval around those numbers to see if the function is increasing there or decreasing there
myininaya
  • myininaya
so what is f' first?
anonymous
  • anonymous
-8x/(x^4+1)
anonymous
  • anonymous
X=-1 and x=1
myininaya
  • myininaya
(x^2+1)^2 does not equal x^4+1
myininaya
  • myininaya
just leave is at (x^2+1)^2
anonymous
  • anonymous
Increasing from negative infinity to 0 and decreasing from 0 to positive infinity
myininaya
  • myininaya
also x^2+1 is always positive so f' and f exist everywhere so you just need to find when f'=0 which is when -8x=0 which is when x=?
myininaya
  • myininaya
ok yep
anonymous
  • anonymous
F(x)=3x^2-12x=5 over the closed interval 0,3. Find absolute extrema.
anonymous
  • anonymous
Do i again begin with taking the derivative?
myininaya
  • myininaya
see if there any critical numbers between 0 and 3 first
myininaya
  • myininaya
which means you need to find f' which all means you need find when f'=0 this looks like you meant to write a polynomial so we don't need to worry when f' dne
anonymous
  • anonymous
X=2
myininaya
  • myininaya
ok do F(0) and F(2) and F(3) which ever one of these are the lowest value is the absolute min value which ever one of these are the highest value is the absolute max value
anonymous
  • anonymous
Ok so you use the tow numbers in the interval and the critical number?
myininaya
  • myininaya
also I don't know what your function is exactly critical numbers in between the critical numbers and any included endpoints just plug them back into the original function
myininaya
  • myininaya
you have two equal signs
myininaya
  • myininaya
i'm guessing that second equal sign is a + or a -
myininaya
  • myininaya
can't decide which one
anonymous
  • anonymous
Should be +5
myininaya
  • myininaya
ok do you want me to check your answers for this one or are you just going through problems and getting ideas on how to do them so you can pass your final
myininaya
  • myininaya
like it seems like you are studying
anonymous
  • anonymous
I just need ideas on how to pass my final...i struggle with a few areas.
myininaya
  • myininaya
i would like to help you more but...
myininaya
  • myininaya
I have to go do some house stuff good luck on your final
myininaya
  • myininaya
i hope i was helpful
anonymous
  • anonymous
Ok thanks

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