In Differential Calculus by Arihant Publication Pg.no. 88 - Q.no. 2,3,5Pg.no. 90 - Q.no. 15,17Pg.no. 91 - Q.no. 29FOR THOSE WHO DONT HAVE THIS BOOK:NOTE: D is same as (d/dx)f(-1)(x) is same as f inverse of x^3 means 'power 3''pi' is same as 22/7 or 3.14'sqrt.' is same as square root of 1) Let f(x) = (2x-pi)^3 + 2x - cosx. The value of D(f(-1)(x)) at x = pi is:   a) 3 pi^2 + 2  b) -2  c) 1/(3pi^2 + 2)  d) 1/32) A function f:R->R satisfies   sinx cosy (f(2x+2y) - f(2x-2y)) = cosx siny (f(2x+2y) + f(2x-   2y)).  If f'(0)=1/2, then:  a) f''(x) - f'(x) =0  b) 4f''(x) + f(x) =0  c) f''(x) + f(x) =0  d) 4f''(x) - f(x) =03. If y= sec(-1)((x-1)/(x+1)) + sin(-1)((x-1)/(x+1)), then Dx=?  a) 1                     b) 0  c) (x-1)/(x+1)       d) (x+1)/(x-1)4. If sqrt.(x^2 + y^2) = a e^(tan(-1)(y/x)), a>0 then y''(0) is:   a)(a/2) e^(-pi/2)               b) a e^(pi/2)   c)(-2 e^(-pi/2))/a              d) does not exist5. If P(x) is a polynomial such that P(x^2+1)={P(x)}^2 + 1 and P(0)=0 then P'(0) is equal to:   a) 1                  b)0   c) -1                 d)none of these6)Let f:R->R be a differential function satisfying           f(y).f(x-y)=f(x) for all x,y and f'(5)=q and p.f(0)=f'(0) then f(5) is:  a) p^2/q               b) p/q  c) q/p                   d) q.Please help me askIITians.

Aman Bansal
592 Points
12 years ago

Dear Abhishek,

1) It's "pi". (2) pi is _not_ 22/7

Now that that's over with, first find

df(x)/dx = 3(2x - pi)^2 * 2 + 2 + sin(x)

Let x = pi to get 3*pi^2 * 2 + 2 + 0 = 6pi^2 + 2

So the answer is | 1 / (6pi^2 + 2) | = 1 / (6pi^2 + 2)

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Aman Bansal