Question

Given f (x,y)=sin⁡xy, what is the value of the  third degree first term in Taylor’s series near (1,-\(\frac{π}{2}\)) where it is expanded in increasing order of degree & by following algebraic identity rule?

(a) \(\frac{π^3}{8}\)

(b) \(\frac{π^3}{8} \frac{(x-1)(y+\frac{π}{2})}{3!}\)

(c) 0

(d) \(-\frac{π^3}{8} \frac{(x-1)^3}{3!}\)

The question was asked in exam.

I'd like to ask this question from Taylor’s Theorem Two Variables topic in division Maxima and Minima of Engineering Mathematics

Answer 1

Right answer is (c) 0

To elaborate: Third degree first term in Taylor’s series is given by \(\frac{(x-a)^3 f_{xxx} (x,y)}{3!}\) Where a=1                                                                                                                   \(b=-\frac{π}{2},  f_{xxx} (x,y)=\frac{∂^3 f(x,y)}{∂x^3} \,i.e\, \frac{∂^3  sin⁡xy}{∂x^3} = -y^3  cos⁡xy\)…… (partial differentiating f (x,y) w.r.t x only)

at \(a=1, b=-\frac{π}{2}, \frac{∂^3  sin⁡xy}{∂x^3} = -\frac{π^3  cos-\frac{⁡π}{2}}{8}=0\) hence third degree first term is given by \(-\frac{π^3}{8} \frac{(x-1)^3}{3!}.0 = 0.\)