Question

by about how much will f(x,y,z) = In root x^2 +y^2 +z^2 change if the point P(x, y, z) moves from P0(2,2,6) a distance of ds = .1 units in the direction of 3i+6j-2k?

Answer 1

Answer:

$\frac{1}{44} \approx0.0227\hspace{3}units$

Step-by-step explanation:

In order to find the answer we need to calculate the directional derivative:

$D_uf(x,y,z)=\nabla f\bullet u$

$\nabla f =The\hspace{3}gradient\hspace{3}of\hspace{3}f\\u=Unit\hspace{3}vector$

The gradient of f is:

$\nabla f(x,y,z)=\langle\frac{x}{x^2+y^2+z^2}, \frac{y}{x^2+y^2+z^2},\frac{z}{x^2+y^2+z^2}\rangle$

The unit vector can be found as:

$u=\frac{v}{|v|} =\frac{3i+6j-2k}{\sqrt{3^2+6^2+(-2^2)} } =\langle\frac{3}{7} ,\frac{6}{7}, \frac{-2}{7} \rangle$

The directional derivative at the point is:

$D_uf(2,2,6)=\langle\frac{1}{22}, \frac{1}{22} ,\frac{3}{22} \rangle \bullet \langle \frac{3}{7} ,\frac{6}{7} ,\frac{-2}{7} \rangle =\frac{5}{22} \bullet \langle \frac{3}{7} ,\frac{6}{7} ,\frac{-2}{7} \rangle =\frac{5}{22}$

Finally, the problem tell us that the point moves a distance of 0.1 units. Hence f(x,y,z) will move:

$\frac{5}{22} * 0.1=\frac{1}{44}\approx 0.0227\hspace{3}units$