59 - Calculus Questions Answers

Maths >> Calculus >> Functions IIT JEE

Let f(x) = x135 + x125 - x115 + x5 +1. If f(x) is divided by x3-x then the remainder is some function of x say g(x). Find the value of g(10)

Asked By: VAIBHAV GUPTA
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Joshi sir comment

for getting reminder put x3= x so

x135 + x125 - x115 + x5 +1 will give

 x45 + x41*x2 - x38*x +x*x2 +1

x15 + x13*x2*x2 - x12*x2*x +x*x2 +1

 x17 + x3 +1

x5*x2 + x + 1

x+ x + 1

x2*x + x +1 

x3 + x +1 

x+x+1

2x+1

now put x = 10

How to inegrate ( In t) dt /(t-1)

Asked By: PARTHASARATHY
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Joshi sir comment

let lnt = x so dt/t = dx so dt = tdx = exdx

now ∫(ln t)dt/(t-1) = ∫xexdx/(ex-1) = ∫x(ex-1+1)dx/(ex-1)

now solve 

Maths >> Calculus >> Integrations IIT JEE

∫√tanx

Asked By: BONEY HAVELIWALA
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Joshi sir comment

let  √tanx = t

so sec2xdx/2√tanx = dt

so (1+t4)dx/2t = dt

so dx = 2tdt/(1+t4)

now solve

Maths >> Calculus >> Functions AIEEE

if ƒ:R−{0} -> R,  2ƒ(x) − 3ƒ(1⁄x) = x²  then ƒ(3)= ?

Asked By: BONEY HAVELIWALA
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Joshi sir comment

2ƒ(x) − 3ƒ(1⁄x) = x²   (1)

so 2ƒ(1/x) − 3ƒ(x) = 1/x²  (2)

multiply (1) to 2 and (2) to 3, then add the two equations, you will get f(x) then calculate f(3)  

Maths >> Calculus >> Functions Others

Sir/Madam,

                    Suddenly a question struck on my mind:   0×∞= 1 or 0?? as 1⁄0=∞.

     

Asked By: BONEY HAVELIWALA
is this question helpfull: 1 2 read solutions ( 2 ) | submit your answer
Joshi sir comment

it will be 0, assume it by considering that ∞ is a big number 

for ex.   0*(1111111111111111111111111) = 0

Maths >> Calculus >> Functions IIT JEE

FIND DOMAIN OF f(x)=log4log3 log2x WHERE 4,3,2 ARE BASES ?

Asked By: SIDDHANT
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Joshi sir comment

f(x)= log4log3log2

so 0 < log3log2x < 

so 1 < log2x < ∞ 

so 2 < x < 

Maths >> Calculus >> Differential Equations Engineering Exam
Differential form of ol conics whose axes coincide with the co-ordinate axes??
Asked By: NEHA GUPTA
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Joshi sir comment

let us consider the case of ellipse with x and y axes as their axes

eq. is x2/a2 + y2/b2= 1

on differentiating we get 2x/a2 + 2yy'/b2 = 0                                y'is first differential

so yy'/x = -b2/a2

again differentiate and get answer. 

You should remember that you should differentiate as many times as the number of constants.

for ex. in the case of parabola only first diffrentiation is sufficient.

now complete it for all conics

using the transformation x=r cosθ and y=r sinθ,find the singular solutionof the differential equation x+py=(x-y)(p2+1)½           where p=dy/dx

Asked By: DEBANJAN GHOSHAL
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Joshi sir comment

x=rcosθ and y=rsinθ

so dx=-rsinθdθ  and dy = rcosθdθ

so p = -cotθ

so given eq. will become x-cotθy = (x-y)cosecθ 

on putting values of x and y we get 

0 = r(cosθ-sinθ)/sinθ

so tanθ = 1  so θ = nπ+π/4

so x = rcos(nπ+π/4) and y = rsin(nπ+π/4)

show that the family of parabolas y2=4a(x+a) is self orthogonal. 

Asked By: DEBANJAN GHOSHAL
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Joshi sir comment

y2= 4a(x+a)

so 2yy' = 4a so a = yy'/2

on putting a we get y2= 4yy'/2(x+yy'/2)      

so y2 = yy' (2x+yy')  or y = 2xy' + yy'2    (1)

now on putting -1/y' in the place of y'

we get y2 = -y/y'[2x-y/y']

so -yy'2 = 2xy' - y (2)

similarity of (1) and (2) shows that the given curve is self orthogonal 

Sir , i have solved   limx-->0    (sin-1x  - tan-1x )/x. plz have a look on others.

Asked By: VISHAL PHOGAT
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Joshi sir comment

limx-->0    (sin-1x  - tan-1x )/x3

use D L Hospital rule or expansion of sin-1x and tan-1x to solve it

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