Home
Class 12
MATHS
If f(x)=(1+x)^2, then the value of f(x0)...

If `f(x)=(1+x)^2,` then the value of `f(x0)+f^(prime)(0)` `+(f^(0))/(2!)+(f^(0))/(3!)+(f^n(0))/(n !)dot`

Text Solution

Verified by Experts

The correct Answer is:
`2^(n)`
`f(x)=(1+x)^(n),f(0)=1`
`therefore" "f'(x)=n(1+x)^(n-1),f'(0)=n`
`therefore" "f''(x)=n(n-1)(1+x)^(n-2),f''(0)=n(n-1)`
Similarly, proceeding, we have
`f'''(0)=n(n-1)(n-2)`
`f''''(0)=n(n-1)(n-2)(n-3)` and so on.
`f^(n)(0)=n(n-1)(n-2)(n-3)...1`
`"or "f(0)+f'(0)+(f''(0))/(2!)+(f'''(0))/(3!)+...(f^(n)(0))/(n!)+`
`=1+n+(n(n-1))/(2!)+(n(n-1)(n-2))/(3!)+...+(n(n-1)(n-2)...1)/(n!)+...`
`=""^(n)C_(0)+""^(n)C_(1)+""^(n)C_(2)...+""^(n)C_(n-1)+""^(n)C_(n)=2^(n)`
Promotional Banner

Topper's Solved these Questions

  • DIFFERENTIATION

    CENGAGE ENGLISH|Exercise Concept Application 3.9|14 Videos

  • DIFFERENTIATION

    CENGAGE ENGLISH|Exercise Execrises|137 Videos

  • DIFFERENTIATION

    CENGAGE ENGLISH|Exercise Concept Application 3.7|6 Videos

  • DIFFERENTIAL EQUATIONS

    CENGAGE ENGLISH|Exercise Matrix Match Type|5 Videos

  • DOT PRODUCT

    CENGAGE ENGLISH|Exercise DPP 2.1|15 Videos

Similar Questions

Explore conceptually related problems

If f(x)=(1-x)^n, then the value of f(0)+f^(prime)(0)+(f^('')(0))/(2!)+(f^(''')(0))/(3!)+......(f^n(0))/(n !)dot

if f(x)=e^(-1/x^2),x!=0 and f (0)=0 then f'(0) is

If f(x)=m x+ca n df(0)=f^(prime)(0)=1. What is f(2)?

If f(x)a n dg(x) are differentiable functions for 0lt=xlt=1 such that f(0)=10 ,g(0)=2,f(1)=2,g(1)=4, then in the interval (0,1)dot (a) f^(prime)(x)=0 for a l lx (b) f^(prime)(x)+4g^(prime)(x)=0 for at least one x (c) f(x)=2g'(x) for at most one x (d) none of these

If f(x)a n dg(x) are differentiable functions for 0lt=xlt=1 such that f(0)=10 ,g(0)=2,f(1)=2,g(1)=4, then in the interval (0,1)dot (a) f^(prime)(x)=0 for a l lx (b) f^(prime)(x)+4g^(prime)(x)=0 for at least one x (c) f(x)=2g'(x) for at most one x (d) none of these

if f(x)=3e^(x^2) then f'(x)-2xf(x)+1/3f(0)-f'(0)

For x in R , x ne0, 1, let f_(0)(x)=(1)/(1-x) and f_(n+1)(x)=f_(0)(f_(n)(x)),n=0,1,2….. Then the value of f_(100)(3)+f_(1)((2)/(3))+f_(2)((3)/(2)) is equal to

A function f: R->R satisfies sinxcosy(f(2x+2y)-f(2x-2y)=cosxsiny(f(2x+2y)+f(2x-2y))dot If f^(prime)(0)=1/2,t h e n (a) f^(primeprime)(x)=f(x)=0 (b) 4f^(primeprime)(x)+f(x)=0 (c) f^(primeprime)(x)+f(x)=0 (d) 4f^(primeprime)(x)-f(x)=0

If f(x)=(4+x)^(n),"n" epsilonN and f^(r)(0) represents the r^(th) derivative of f(x) at x=0 , then the value of sum_(r=0)^(oo)((f^(r)(0)))/(r!) is equal to

If f'(x)=f(x)+ int_(0)^(1)f(x)dx , given f(0)=1 , then the value of f(log_(e)2) is