Lim as x approaches pi/6 of (sin(x)-sin(pi/6))/x-pi/6

- anonymous

Lim as x approaches pi/6 of (sin(x)-sin(pi/6))/x-pi/6

- schrodinger

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- myininaya

compare to definition of derivative

- myininaya

\[\lim_{x \rightarrow a}\frac{f(x)-f(a)}{x-a}=f'(a)\]

- anonymous

So the limit would = 1?

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## More answers

- myininaya

no

- myininaya

what is f?

- anonymous

...?

- myininaya

what is f(x) and what is a?

- anonymous

F(x)=sin(x), f(a)=sin(pi/6)

- myininaya

so what is f'?

- anonymous

Cos(x)-cos(pi/6)?

- myininaya

:(
if f=sin(x), then f'=?

- anonymous

Cos(x)

- myininaya

right
now plug in a to get your answer

- anonymous

A=sin(pi/6)?

- 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

\[\lim_{x \rightarrow \frac{\pi}{6}}\frac{\sin(x)-\sin(\frac{\pi}{6})}{x-\frac{\pi}{6}}=(\sin(x))'|_{x=\frac{\pi}{6}} \]

- myininaya

do you see all i'm trying to get you to do is use the definition of derivative

- 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

Sqrt(3)/2?

- myininaya

yep

- anonymous

Thank you

- anonymous

Any chance you can help with critical numbers?

- myininaya

sure

- anonymous

F(x)=4x/x^2+1

- anonymous

I know i need the derivative

- myininaya

do you mean 4x/(x^2+1)?
Also what did you get for the derivative?

- anonymous

Yes.
F'(x)=(-4x^2+4)/(x^2+1)^2

- 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

Sqrt(-1)

- myininaya

I think you should retry solving -4x^2+4=0
I will give you a (you will get two real solutions)

- anonymous

...+or-sqrt(-1)

- anonymous

+or-sqrt(1)

- myininaya

right sqrt(1)=1
so x=1 or x=-1

- myininaya

those are the critical numbers for F.

- anonymous

Wow...

- anonymous

Find the intervals where f(x)=4/(x^+1) is increasing or decreasing.

- myininaya

is that f(x)=4/(x+1)?
like you weren't trying to write an exponent right?

- 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

I was writing an exponent

- anonymous

I apologize for missing the exponent

- anonymous

It should have said 4/(x^2+1)

- 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

so what is f' first?

- anonymous

-8x/(x^4+1)

- anonymous

X=-1 and x=1

- myininaya

(x^2+1)^2 does not equal x^4+1

- myininaya

just leave is at (x^2+1)^2

- anonymous

Increasing from negative infinity to 0 and decreasing from 0 to positive infinity

- 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

ok yep

- anonymous

F(x)=3x^2-12x=5 over the closed interval 0,3. Find absolute extrema.

- anonymous

Do i again begin with taking the derivative?

- myininaya

see if there any critical numbers between 0 and 3 first

- 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

X=2

- 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

Ok so you use the tow numbers in the interval and the critical number?

- 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

you have two equal signs

- myininaya

i'm guessing that second equal sign is a + or a -

- myininaya

can't decide which one

- anonymous

Should be +5

- 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

like it seems like you are studying

- anonymous

I just need ideas on how to pass my final...i struggle with a few areas.

- myininaya

i would like to help you more but...

- myininaya

I have to go do some house stuff
good luck on your final

- myininaya

i hope i was helpful

- anonymous

Ok thanks

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