Question

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

Answer 1

compare to definition of derivative

Answer 2

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

Answer 3

So the limit would = 1?

Answer 4

no

Answer 5

what is f?

Answer 6

...?

Answer 7

what is f(x) and what is a?

Answer 8

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

Answer 9

so what is f'?

Answer 10

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

Answer 11

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

Answer 12

Cos(x)

Answer 13

right
now plug in a to get your answer

Answer 14

A=sin(pi/6)?

Answer 15

\[\lim_{x \rightarrow a}\frac{f(x)-f(a)}{x-a}=f'(a) \text{ or we could say it equals } =(f(x))'|_{x=a}\]

Answer 16

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

Answer 17

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

Answer 18

you are suppose to determine f and a
then find f'
then plug a into f' to get the result you are searching for

Answer 19

Sqrt(3)/2?

Answer 20

yep

Answer 21

Thank you

Answer 22

Any chance you can help with critical numbers?

Answer 23

sure

Answer 24

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

Answer 25

I know i need the derivative

Answer 26

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

Answer 27

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

Answer 28

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

Answer 29

Sqrt(-1)

Answer 30

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

Answer 31

...+or-sqrt(-1)

Answer 32

+or-sqrt(1)

Answer 33

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

Answer 34

those are the critical numbers for F.

Answer 35

Wow...

Answer 36

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

Answer 37

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

Answer 38

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?

Answer 39

I was writing an exponent

Answer 40

I apologize for missing the exponent

Answer 41

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

Answer 42

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

Answer 43

so what is f' first?

Answer 44

-8x/(x^4+1)

Answer 45

X=-1 and x=1

Answer 46

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

Answer 47

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

Answer 48

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

Answer 49

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=?

Answer 50

ok yep

Answer 51

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

Answer 52

Do i again begin with taking the derivative?

Answer 53

see if there any critical numbers between 0 and 3 first

Answer 54

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

Answer 55

X=2

Answer 56

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

Answer 57

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

Answer 58

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

Answer 59

you have two equal signs

Answer 60

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

Answer 61

can't decide which one

Answer 62

Should be +5

Answer 63

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

Answer 64

like it seems like you are studying

Answer 65

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

Answer 66

i would like to help you more but...

Answer 67

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

Answer 68

i hope i was helpful

Answer 69

Ok thanks