If `x` is so small that `x^3` and higher powers of `x` may be neglectd, then `((1+x)^(3//2)-(1+1/2x)^3)/((1-x)^(1//2))` may be approximated as a. `3x+3/8x^2` b. `1-3/8x^2` c. `x/2-3/xx^2` d. `-3/8x^2`

Options

A. `(x)/(2) - (3)/(8) x^(2)`
B. `- (3)/(8) x^(2)`
C. `3x + (3)/(8) x^(2)`
D. `1 - (3)/(8) x^(2)`

Answer

b

Explanation

We have,
` (1 + x)^(n) = 1+ nx + (n(n-1))/(2)x^(2)`
`therefore (1 + x)^(3//2) - (1 + (x)/(2))^(3)`
` ={( 1 + (3)/(2) x + (3)/(8) x^(2)) - (1 + (3)/(2) x + (3)/(4) x^(2))} {1 +(1)/(2) x + (3)/(8) x^(2)}`
`= - (3)/(8) x^(2) (1 + (x )/(2) + (3)/(8) x^(2)) = - (3)/(8) x^(2)`