Recently Active Calculus Questions

Let a<b<c be reals such that a+b+c=6 and ab+bc+ca=9 Then prove that 0<a<1<b<3<c<4 ...

if the function 0∫x f(t) dt->5 as |x|->1,then the value of (a) so that the equation 2x + 0∫x f(t) dt =a has atleast 2 roots of opposite sign in (-1,1) is (a) a ε (0,1) (b) a ε (0,3) (c) a ε (-1,∞) (d) a ε (3,â ...

These are some easy ones on limit... Please explain them and help me finish limits. Q1) \lim_{x\rightarrow 0} \frac{e-(1+x)^{1/x}}{tan x} Q2) \lim_{x\rightarrow 0} \left(\frac{(1+x)^{1/x}}{e} \right)^{1/x} Q3) {P{n}}= \frac{2 ...

Q.1) \lim_{x\rightarrow 0}(\left[f(x) \right]+x^2)^1^/^\left\{f(x) \right\} , where f(x) = (tanx/x) and \left\{f(x) \right\} denotes fractional part of f(x) and [f(x)] denotes the greatest integer function of f(x). is equal t ...

If α, β are the roots of the equation x2 + bx + c = 0 then find lim (1 - cos(x2 + bx + c))/(β - x)2 x->β ...

the value of ∫x2+2 /[(x4+5x2+4)tan-1(x2+2)/x dx ............. ...

\int_{0}^{\propto}\frac{sinx}{x}dx ===??? ...

\lim_{n\rightarrow \infty }\prod_{r=2}^{r=n}{\frac{r^{3}-1}{r^{3}+1}} ...

Prove 2<e<3 ... ...

if I(n)=\int \frac{x^{n}}{(ax^{2}+bx+c)^{1/2}} dx n belongs to N find I(n+1) in terms of I(n) and I(n-1) ...

Q1. [] represents the GINT. Then the value of the infinite series [ 2008/2 ] + [ 2009/4 ] + [ 2011/8 ] + [ 2015/16 ] + [ 2023/32 ] + ...... is? I need a short method for this.. All i could do was calculate tr = [ 2007+2r-1/2r ...

Ques1) Find the solution of dy/ dx = (x+y) 2 /(x+2) (y-2) Ques2) Find the solution of dy/dx = (x-1) 2 + (y-2) 2 tan -1 ( y-2/x-1 )/(xy - 2x - y+2) tan -1 ( y-2/x-1 ) ...

∫ dx/x+√x2-x+1 ...

A function f(x) = max.(sinx, cosx, 1-cosx) is not derivable for some x in [0, 2Ï€] which may lie in the interval....... A) [0,Ï€/2) B) [Ï€,3Ï€/2) C) [Ï€/2,Ï€) D) [3Ï€/2,2Ï€] (More than one options may be correct) ...

P(x) be a polynomial of degree at most 5 which leaves -1 and 1 upon divison by (x-1)3 and (x+1)3 respectively then 1)no of real roots of p(x)=0 a)1 b)3 c)5 d)2 2)the maximum value of y=p"(x) can be obtained at x= a)-1/ 3 b)0 ...

the value of \int_{1}^{e}{} (1+x2logx)/(x+x2logx)dx is not giving the choices please verify the answer. ...

let f(x)=x^{3}+x^{2}+100x+7sinx then the eq \frac{1}{y-f(1)}+ \frac{2}{y-f(2)}+ \frac{3}{y-f(3)}=0 has 1)exactly one root lying in (f(1),f(2)) 2)booth roots lying in (f(2),f(3)) 3)exactly one root lying in (-∞,f(1)) 4)exact ...

If ... \lim_{x \rightarrow \infty} \frac{ a \left(2x^3 -x^2 \right) +b \left(x^3+5x^2-1 \right) -c \left(3x^3+x^2 \right)}{a\left(5x^4-x \right)-bx^4 +c \left(4x^4+1 \right) +2x^2 + 5x} = 1 , then the value of (a+b+c) can be ...

Q 1) \int\frac{x-1}{(x+1)\sqrt{x^{3}+x^{2}+x}}dx Q 2) \int{\left({\tan^{-1}\sqrt{(\sqrt x-1)}}\right)}dx Q 3) \int{(x^{1/3}+(\tan x)^{1/3})}dx Q 4) \int_{1}^{e}\frac{Inx}{x(\sqrt{1-Inx}+\sqrt{Inx+1})}dx Q 5) \int_{\frac{\pi } ...

f(x) is a periodic fn with pd λ and f(x-\frac{\lambda }{2})=-f(x) ,then show thta g(x+\lambda)=g(x) where g(x)=\int_{0}^{x}{f(t)dt} ...

sininverse((2x+2/( 4x2+8x+13 ))) ...

1/(sinx+cosx) ...

the estimated value of 1/1000 + 1/1001 + 1/1002 +.............. 1/2000 ...

Ques) Find the solution of x + y dy/dx/y - x dy/dx = x sin 2 (x2+y2)/y3 . ...

If f(x+y+z)=f(x)+f(y)+f(z) with f(1)=1,f(2)=2 and x,y,z εR,thne evaluate \lim_{n\rightarrow\infty }\frac{\sum_{r=1}^{n}{(4r).f(3r)}}{n^3} ...

*Image* ...

let f(x) , x ε [0,∞) be a non negative continuous function. If f '(x) cosx ≤ f(x) sinx for all x≥0 then value of f(2π) is- a)0 b)1 c)Πd)none ...

P(x) be a polynomial of degree at the most 5 which leaves the reaminders -1 and 1 upon division by (x-1)3 and (x+1)3 1)....find the no of real roots of p(X)=0 2)....the max value of y=P''(x) can be obtained at x 3).....the su ...

prove that e^{x}+\sqrt{1+e^{2x}}>(1+x)+\sqrt{2+2x+x^{2}} for x belongs to R(note its ≥ not >) ...

Define na! for positve integers n and a na!=n(n-a)(n-2a)(n-3a)......(n-ka) where k is the greatest integer for which n>ka then find the quotient of 728!/182! ...