Basic Mathematics Problems

These problems are some of very famous and chosen by me for my own interview preparations.

(Just for someone interested and ignorant.. the best site for detailed solutions would be wolfram-alpha)

First Problem

Differentiate x^x

Answer: We do it by chain rule assume it as fx^gx so we apply chain rule on this function [d fx^gx/d(fx)]x d(fx)/dx + [d fx^gx/dgx ]x d(gx)/dx

hence the answer would be (1+log(x)) x^x

^{(Just noticed another way, this one is not brute force but rather smart way)}

^{y =} x^x

^{take log}

^{log y = xlogx}

^{now differentiate}

^{dy/y = (logx + x/x)}

^{=> dy = y(logx +1)}

^{same as above ( Maths is consistent when correct :) )}

^{Second problem}

Integrate cos²(x) and cos³(x).

cos(2A) = 2cos²(x)-1

=> cos²(x) = [1+cos(2A)]/2

rest is pretty simple

for cos³(x) write it as cos(x)cos²(x) now write squared terms in form of sin and then put sin(x) as y then cos(x)dx would be dy so we will have Integral(1-Y²)dy

rest is simple for this one as well.

Third problem

What is 1.06 raised to the power 10.5

let y = 1.06^10.5

then ln y = 10.5X ln(1.06)

ln(1+x) when x tends to zero is equal to x

hence ln(1.06) can be approximated to 0.06

hence

lny = 0.63

hence y = exp(0.63) which could be approximated around 1.8

(this was the first method that came to my mind)