# 59 - Calculus Questions Answers

no of solutions of  e-x^2/2 - x2 = 0 are

Joshi sir comment

two intersecting points means two solutions

lim x»infinity( ((x+1)(x+2)(x+3)(x+4))^¼ - x )

Joshi sir comment

multiply with the conjugate in numerator again and again you will get

limx->((x+1)(x+2)(x+3)(x+4) - x4)/((x+1)(x+2)(x+3)(x+4))1/4 + x )*((x+1)(x+2)(x+3)(x+4))1/2 + x2 )

after solving the numerator get a 3 power expression of x then take 3 power of x common, similarly take x and 2 power of x from the first and second denominator terms

Maths >> Calculus >> Functions IIT JEE

best calculus books for iitjee.

Joshi sir comment

its I. E. Maron

lim n--> ∞  4^n/n!

Joshi sir comment

limn->∞ 4n/ n!

= limn->∞ [4/1][4/2][4/3][4/4][4/5][4/6][4/7]................................[4/n]

besides first four bracket, all brackets are real numbers less than 1 so their product will be zero for n tends to infinite

Sir,is the function (2x+5) is differentiable everywhere in its domain set. If yes the what is the case with l2x+5l.
Joshi sir comment

since first one is a straight line so at every point only one tangent is possible so it is differentiable

but second one which is modulus has a sharp turn at x = -5/2 so two tangents at x = -5/2 are possible so is not differentiable.

A window frame is shaped like a rectangle with an arch forming the top ( ie; a box with a semicircle on the top)

The vertical straight side is Y, the width at the base and at the widest part of the arch (Diam) is X.

I know that the perimeter is 4.

Find X if the area of the frame is at a maximum?

Joshi sir comment

perimeter = 4 so 2Y + X + πX/2 = 4

and area = XY + [π(X/2)2]/2

put Y from 1st equation to 2nd we get A = X[4-X{1+(π/2)}]/2 + πX2/8

now calculate dA/dX and then compare it to zero

X will be 8/(4+π)

y=px+a/p

Joshi sir comment

this is a special type of differential equation in which p = dy/dx

its solution will be y = cx+(a/c)  here c is a constant

y-xp=x+yp

Joshi sir comment

on applying p = dy/dx

(ydx-xdy)/dx = (xdx+ydy)/dx

or ydx-xdy = xdx+ydy

or dy/dx = (y-x)/(y+x)

now it is homogeneous

Importance High!!!!!!!!!!!!!!!!!

Myself Rajeev Shrivastava, I am putting a signs series in front of you which based on numbers from 01 to 100. Actually I want to know that what reason of behind Approx equability of signs is

(++), (+-), (--) & (-+) at end of the month or year between the below mentioned.

Let's define a kind of mapping:
0↔1
2↔3
4↔5
6↔7
8↔9

(+) means: an Even number
(-) means: an Odd number

I simply say: for any EO (+-) it exists an OE (-+) which is paired to.
And that's true since each E is mapped to a O and vice versa (1 to 1 relation where "f(f(x))=x" ) !
E.g.: 29 ↔ 83

There is 25 numbers in the EO (+-) group, so 25 in OE (+-).
There is 25 numbers in the EE (++) group, so 25 in OO (--).

Picking a number at random between 01 and 100 inclusive is choosing equitably in EO, OE, EE or OO group. (25% for each)

So, I really need of your help to solve a puzzle please. I just am trying to make you understand what type of solution I need:

There are some results for your review:

 MONTH DATE DAY FIRST_SERIES SECOND_SERIES FIRST_SERIES SECOND_SERIES APR 1 SUN + - - + +- -+ APR 2 MON + + - + ++ -+ APR 3 TUE + - + - +- +- APR 4 WED + + + + ++ ++ APR 5 THU - + - + -+ -+ APR 6 FRI + + + - ++ +- APR 7 SAT + - - + +- -+ APR 8 SUN - + + - -+ +- APR 9 MON - + - + -+ -+ APR 10 TUE + + + + ++ ++ APR 11 WED + + - - ++ -- APR 12 THU - + - - -+ -- APR 13 FRI + + + + ++ ++ APR 14 SAT + - + - +- +- APR 15 SUN + - + - +- +- APR 16 MON - - + + -- ++ APR 17 TUE - - + + -- ++ APR 18 WED APR 19 THU APR 20 FRI APR 21 SAT APR 22 APR 23

(Complete sheets for the year 2011 and 2012 are attached with this mail)

You can see that there are two times falls in a day of pair make by even (+) and odd (-) signs, i.e. (++), (+-), (--) & (-+) as above said.

The total of pair combination signs become approx equal to each other at the end of the month or year (every year- so, I am sending you attachment to go through the year’s result).

By this analysis:

1-     I want to know that what is the relation between current falling signs and past fell signs.

2-     How can I come to know, what type of combination of sign would be any particular date or day.

3-     As you can seen on 22nd and 23rd April’2012 I don’t know what combination of sign is.

4-     There is surety that there is some relation between current First fall of combination of signs and past fall or First fall of combination of signs and Second fall of combination of signs.

Please help me to how come to know what type of fall may be for next day by reviewing past or first combination of signs. So reply or elaborate me with an example.

I would be very grateful to you till entire life.

Regards

RAJEEV SHRIVASTAVA

Importance High!!!!!!!!!!!!!!!!!

Myself Rajeev Shrivastava, I am putting a signs series in front of you which based on numbers from 01 to 100. Actually I want to know that what reason of behind Approx equability of signs is

(++), (+-), (--) & (-+) at end of the month or year between the below mentioned.

Let's define a kind of mapping:
0↔1
2↔3
4↔5
6↔7
8↔9

(+) means: an Even number
(-) means: an Odd number

I simply say: for any EO (+-) it exists an OE (-+) which is paired to.
And that's true since each E is mapped to a O and vice versa (1 to 1 relation where "f(f(x))=x" ) !
E.g.: 29 ↔ 83

There is 25 numbers in the EO (+-) group, so 25 in OE (+-).
There is 25 numbers in the EE (++) group, so 25 in OO (--).

Picking a number at random between 01 and 100 inclusive is choosing equitably in EO, OE, EE or OO group. (25% for each)

So, I really need of your help to solve a puzzle please. I just am trying to make you understand what type of solution I need:

There are some results for your review:

 MONTH DATE DAY FIRST_SERIES SECOND_SERIES FIRST_SERIES SECOND_SERIES APR 1 SUN + - - + +- -+ APR 2 MON + + - + ++ -+ APR 3 TUE + - + - +- +- APR 4 WED + + + + ++ ++ APR 5 THU - + - + -+ -+ APR 6 FRI + + + - ++ +- APR 7 SAT + - - + +- -+ APR 8 SUN - + + - -+ +- APR 9 MON - + - + -+ -+ APR 10 TUE + + + + ++ ++ APR 11 WED + + - - ++ -- APR 12 THU - + - - -+ -- APR 13 FRI + + + + ++ ++ APR 14 SAT + - + - +- +- APR 15 SUN + - + - +- +- APR 16 MON - - + + -- ++ APR 17 TUE - - + + -- ++ APR 18 WED APR 19 THU APR 20 FRI APR 21 SAT APR 22 APR 23

(Complete sheets for the year 2011 and 2012 are attached with this mail)

You can see that there are two times falls in a day of pair make by even (+) and odd (-) signs, i.e. (++), (+-), (--) & (-+) as above said.

The total of pair combination signs become approx equal to each other at the end of the month or year (every year- so, I am sending you attachment to go through the year’s result).

By this analysis:

1-     I want to know that what is the relation between current falling signs and past fell signs.

2-     How can I come to know, what type of combination of sign would be any particular date or day.

3-     As you can seen on 22nd and 23rd April’2012 I don’t know what combination of sign is.

4-     There is surety that there is some relation between current First fall of combination of signs and past fall or First fall of combination of signs and Second fall of combination of signs.

Please help me to how come to know what type of fall may be for next day by reviewing past or first combination of signs. So reply or elaborate me with an example.

I would be very grateful to you till entire life.

Regards

RAJEEV SHRIVASTAVA

Joshi sir comment

If you are making the sequence on the basis of dates then you should take signs accordingly then you can make this sequence for many years

for example if you take 23 june then first sign series will be +-, + for even number and - for odd number , now as conversion rule as you gave in your explanation 23 will be converted to 32 so second series will be -+

similarly you can make it for any date

for example 29 december first series will be +- and its conversion is 38 so next series will be -+

if you are thinking something else then explain your question correctly

how to find differential eqn. of all conics whose axes coincide with coordinate axes??   tell me the eqn. of tht. conic

Joshi sir comment

Parabola   eq.  y2 = 4ax ,

on differentiating this eq. with respect to x we get 2y(dy/dx) = 4a

on taking this value of 4a in eq. y2= 4ax we will get the differential eq. of parabola.

similarly for ellipse    x2/a2  +   y2/b2  =   1

on differentiaing this eq. twice we will get two additional eq.

by solving these eq. get an eq. free from a and b, this will be the differential eq. of the ellipse.

similarly for hyperbola