Question

The expansion of f(x, y)=e^x Sin(y), is

(a) x + xy + ….

(b) y + y^2 x + ….

(c) x + x^2 y + ….

(d) y + xy + ……..

The question was asked during an online interview.

This is a very interesting question from Taylor Mclaurin Series in section Differential Calculus of Engineering Mathematics

Answer 1

Correct choice is (d) y + xy + ……..

Best explanation: Now, f(x, y)=e^x Sin(y), f(0,0) = 0

Therefore,

fx (x,y) = e^x Sin(y), hence fx (0,0) = 0

fy (x,y) = e^x Cos(y), hence fy (0,0) = 1

fxx (x,y) = e^x Sin(y), hence fxx (0,0) = 0

fyy (x,y) = -e^x Sin(y), hence fyy (0,0) = 0

fxy (x,y) = e^x Cos(y), hence fxy (0,0) = 1

By taylor expansion,

f(x,y) = f(0,0)+\([x \frac{∂f}{∂x}+y \frac{∂f}{∂y}]+\frac{1}{2!} [x^2 \frac{∂^2 f}{∂x^2}+2xy \frac{∂^2 f}{∂x∂y}+y^2 \frac{∂^2 f}{∂y^2}]+…\)

f(x,y) = 0 + 0 + y + \(\frac{1}{2!}\) [0 + 2xy + 0] +…..

f(x,y) = y + xy + ……..