Question

5. Which vector points in the direction in which one should travel in order to experience the greatest rate of change of the function g(x,y,z) = xy" + 5yz’if one is currently located at the point (3,2, 4) ? a)8î +116) +80 b) 2î - 129 + c) -71 -5j + 46k d)841+ -51 e) none of these

Answer 1

Maximum rate if change is along the gradient of the function

Let us find gradient of the function at given point

(3,2,4)

$\large \bigtriangledown g(x,y,z) =i\frac{\partial g}{\partial x}+j\frac{\partial g}{\partial y}+k\frac{\partial g}{\partial z}$

$\large \bigtriangledown g(x,y,z) =i\frac{\partial (xy^{3}+5yz^{2})}{\partial x}+j\frac{\partial (xy^{3}+5yz^{2}) }{\partial y}+k\frac{\partial (xy^{3}+5yz^{2})}{\partial z}$

= i(y) + j(3xy² + 5z2) + k(10yz)

$\large =i (2^{3})+j (3*3*2^{2}+5*4^{2})+k(10*2*4)$

$\large =8i+j (36+80)+80k$

$\large =8i+116j+80k$

So a is the right answer