Solve the problem.
-About how much will $f ( x , y , z ) = 2 x - 4 y + 10 z$ change if the point $( x , y , z )$ moves from $( 10 , - 7,10 )$ a distance of $\mathrm { ds } = \frac { 1 } { 10 }$ unit in the direction $2 \mathbf { i } - 3 \mathbf { j } + 6 \mathbf { k }$ ?
A) $\frac { 36 } { 35 }$
B) $\frac { 38 } { 35 }$
C) $\frac { 8 } { 7 }$
D) $\frac { 6 } { 5 }$

Question

Quizplus · Accepted Answer

The change in $f(x, y, z)$ can be approximated using the directional derivative formula: $\nabla f \cdot \mathbf{u}$ds, where $\nabla f$ is the gradient of $f$, $\mathbf{u}$ is the unit direction vector, and ds is the distance moved. The gradient of $f(x, y, z) = 2x - 4y + 10z$ is $\nabla f = (2, -4, 10)$. The given direction is $2\mathbf{i} - 3\mathbf{j} + 6\mathbf{k}$, so the unit direction vector $\mathbf{u}$ is $\frac{1}{\sqrt{2^2 + (-3)^2 + 6^2}}(2, -3, 6) = \frac{1}{7}(2, -3, 6)$. The change in $f$ is then $\nabla f \cdot \mathbf{u} \times \text{ds} = (2, -4, 10) \cdot \frac{1}{7}(2, -3, 6) \times \frac{1}{10} = \frac{1}{70}(4 + 12 + 60) = \frac{76}{70} = \frac{38}{35}$.

Solve the Problem f(x,y,z)=2x−4y+10zf ( x , y , z ) = 2 x - 4 y + 10 zf(x,y,z)=2x−4y+10z

Solve the Problem $f ( x , y , z ) = 2 x - 4 y + 10 z$