Find the linearization of the function at the given point.
-$\mathrm { f } ( \mathrm { x } , \mathrm { y } , \mathrm { z } ) = \mathrm { e } ^ { 5 \mathrm { x } - 8 \mathrm { y } + 3 \mathrm { z } }$ at $( 0,0,0 )$
A) $L ( x , y , z ) = \frac { 5 } { 2 } x - 4 y + \frac { 3 } { 2 } z + 1$
B) $L ( x , y , z ) = 5 x - 8 y + 3 z$
C) $L ( x , y , z ) = 5 x - 8 y + 3 z + 1$
D) $L ( x , y , z ) = \frac { 5 } { 2 } x - 4 y + \frac { 3 } { 2 } z$

Question

Quizplus · Accepted Answer

The linearization of a function at a point is given by:
$$L(x,y,z) = f(a,b,c) + \frac{\partial f}{\partial x}\bigg\vert_{(a,b,c)}(x-a) + \frac{\partial f}{\partial y}\bigg\vert_{(a,b,c)}(y-b) + \frac{\partial f}{\partial z}\bigg\vert_{(a,b,c)}(z-c)$$
In this case, we have $f(x,y,z) = e^{5x-8y+3z}$ and $(a,b,c) = (0,0,0)$. Therefore,
\begin{align*}
L(x,y,z) &= e^0 + \frac{\partial f}{\partial x}\bigg\vert_{(0,0,0)}x + \frac{\partial f}{\partial y}\bigg\vert_{(0,0,0)}y + \frac{\partial f}{\partial z}\bigg\vert_{(0,0,0)}z \
&= 1 + (5e^{5x-8y+3z})\bigg\vert_{(0,0,0)}x + (-8e^{5x-8y+3z})\bigg\vert_{(0,0,0)}y + (3e^{5x-8y+3z})\bigg\vert_{(0,0,0)}z \
&= 1 + 5x - 8y + 3z
\end{align*}
Therefore, the answer is (C).

Find the Linearization of the Function at the Given Point f(x,y,z)=e5x−8y+3z\mathrm { f } ( \mathrm { x } , \mathrm { y } , \mathrm { z } ) = \mathrm { e } ^ { 5 \mathrm { x } - 8 \mathrm { y } + 3 \mathrm { z } }f(x,y,z)=e5x−8y+3z

Find the Linearization of the Function at the Given Point $\mathrm { f } ( \mathrm { x } , \mathrm { y } , \mathrm { z } ) = \mathrm { e } ^ { 5 \mathrm { x } - 8 \mathrm { y } + 3 \mathrm { z } }$