Question

The gradient of $f$ and a point $P$ on the level surface $f(x, y, z)=0$ are given. Find an equation for the tangent plane to the surface at the point $P.$ $$\text { grad } f=2 x \vec{i}+z^{2} \vec{j}+2 y z \vec{k}, P=(10,-10,30)$$

          The gradient of $f$ and a point $P$ on the level surface $f(x, y, z)=0$ are given. Find an equation for the tangent plane to the surface at the point $P.$
$$\text { grad } f=2 x \vec{i}+z^{2} \vec{j}+2 y z \vec{k}, P=(10,-10,30)$$

Added by Felipe B.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Zhumagali Shomanov

04/10/2022

Step 1

The gradient is given as $\text{grad } f = 2x\vec{i} + z^2\vec{j} + 2yz\vec{k}$. Substitute the values of $x=10$, $y=-10$, and $z=30$ into the gradient: $\text{grad } f = 2(10)\vec{i} + (30)^2\vec{j} + 2(-10)(30)\vec{k}$ $\text{grad } f = 20\vec{i} + 900\vec{j} - Show more…

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The gradient of $f$ and a point $P$ on the level surface $f(x, y, z)=0$ are given. Find an equation for the tangent plane to the surface at the point $P.$ $$\text { grad } f=2 x \vec{i}+z^{2} \vec{j}+2 y z \vec{k}, P=(10,-10,30)$$