Consider a function $w=f(x,y,z)$. If $x$ increases by $1$ unit $w$ increases by $15$ units, if $y$ falls by $2$ units $w$ increases by $5$ units and if $z$ falls by $6$ units $w$ falls by $1$ unit. Now suppose that in a given situation $x,y$ and $z$ all increase by $3$ units, $20$ units and $2$ units respectively. What is the net change in $w$?
The options are $-11, 1, 101$ and $89$. I calculated the changes separately and added them to get $-14/3$ which does not match with any of the options.
Given w= f(x,y,z), $dw= f_x dx+ f_ydy+ f_zdz$. Saying that "If x increases by 1 unit w increases by 15 units" means that $f_x= 15$, "if y falls by 2 units w increases by 5 units" means that $f_y= -5/2$ and "if z falls by 6 units w falls by 1 unit" means that $f_z= 1/6$. so $dw= 15dx- \frac{5}{2}dy+ \frac{1}{6}dz$.
"Now suppose that in a given situation x,y and z all increase by 3 units, 20 units and 2 units respectively" Then dx= 3, dy= 20, and dz= 2 so $dw= 15(3)- \frac{5}{2}(20)+ \frac{1}{6}(2)= 45- 50+ \frac{1}{3}= -5+ \frac{1}{3}= \frac{-15+ 1}{3}= -\frac{14}{3}$.