yeah i just came to say that i got it..anyway thanks.....

sorry the editor isn't working..

Let f:R->R be a differentiable function & f(1)=4,then the value of: lim(x->1) (integral) (from f(x) to 4) (2t dt)/(-1+x) is:

A) f'(1)

b)4f'(1)

c)2f'(1)

d)8f'(1)

see for graphs you should consider the points where thefunction takes its maximas/minimas,asymptotes, and also at points where you know the value of the function to get an overall picture...

yeah see for the first onw, you differentiate and het f'(x)=1-2cosx.

Now ,you equate it to 0 and you get value of cos x=1/2. therefore x=pi/3. or 2n(pi)+/-(pi/3),Thus the number line is divided as (-infinity,pi/3) and (pi/3,infinity). Thus,in the first interval the function is negative and hence decreasing and increasing for the second interval.

for the second oasne,f'(x)=1-e^x.

Wwhen we equate it to zero,we geet e^x=1.,whichr gives x=0.Thus for less tahn 0,we get an increasing function and for greater than zero,we get a decreasing function.

for eg. if x=-1(<0):

1-(e^(-1))=1-(1/e)=1-(a number less than 1),hence >0.therefore an increasing function.This can be applied to x>0to get the result for the decreasing interval.

TMH series..

H C Verma for basics its a very good book will help you clear your concepts

2giridhar g:

no we will not do v+b because we are considering the volume which is empty, that is the total space free of the molecules hence assuning that volume of all molecules taken together is b, the emptyvolume for n moles is v-nb

and you don not need nickels to ask the community only to ask experts you need nickels...

edited

sorry i got the wrong question! im sorry!

FOR EXAMPLE,

f(x)=x²-4x+6

On differentiating you get: f'(x)=2x-4

now on equating to 0,x=2.

Now 2 divides the number line in two intervals (-infinity,2) and (2,infinity)

In the first interval f'(x)<0.Therefore the function is decreasing in (-infinity,2) and for x>2,f'(x)>0,therefore it is increasing in (2,infinity)

If there are multiple intervals then you proceed as above taking the sign of the function just before and just after every point.

But if the sign of the function does not change then it is not a monotonic function,i.e. neither strictly decreasing nor stricly increasing

Hoping taht this has helped you..

Nudge if you have any questions!

to check the monotonicity of a given function f(x),you must differentiate it and check whether f'(x) is grester than zero or less than zero.

If it is less than zero,then the fuction is a decreasing function and if grester than zero,then it is an increasing function.For this you must factorise f'(x) equate it to zero and get the critical points and check for the sign of the function just before the point and just after..

You can also check it using graphs..

i think we should give it a try first..

yes sir...how do you approach this kind of a question?