Tangent plane, directional derivatives

In summary, the conversation is about finding the equation for the tangent plane of the surface F(x,y,z)=ln(x+z)-yz=0 at the point (0,0,1). The attempt involves rearranging the equation and finding the partials, with a final result of x-y+z-1=0. There is some uncertainty about the correctness of the answer and the calculation of the partials.
  • #1
Laura1321412
24
0

Homework Statement



find the equation on the tangent plane of yz=ln(x+z) at point (0, 0, 1 )


Homework Equations



Tangent plane equation...

The Attempt at a Solution



I wasn't sure how to determine the partials on this equation. My attempt was to rearange as ln(x+z)-yz=0 so Fx = 1/(x+z) Fy= -1 Fz = 1/(x+z) at the point (001) Fx= 1 Fy= -1 Fz= 1

into the plane equations and i get

x-y+z-1=0

... I am not sure if the answer is right, i think the issue is when i rearanged the equation, how can i determine the partials on this correctly?

Thank you!
 
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  • #2
Hello? :(
 
  • #3
You are finding the gradient vector to the surface F(x,y,z)=ln(x+z)-yz=0. Some of your components aren't quite correct for a general x,y,z but they are correct for (x,y,z)=(0,0,1). So yes, the normal vector is (1,-1,1) and the plane has to pass through (0,0,1) so I think your answer is correct. Can you correct some of the 'partials'?
 

FAQ: Tangent plane, directional derivatives

What is a tangent plane?

A tangent plane is a flat surface that touches a curve or surface at a single point. It is the best approximation of the curve or surface at that point.

How is a tangent plane calculated?

A tangent plane is calculated by finding the slope or gradient of a curve or surface at a specific point. This is done using the concept of directional derivatives.

What is a directional derivative?

A directional derivative is the rate of change of a function in a specific direction. It measures how much a function changes along a given direction from a particular point.

How do you find the directional derivative?

The directional derivative is found by taking the dot product of the gradient of the function and the unit vector in the desired direction. This gives the rate of change in that direction.

What is the significance of the tangent plane and directional derivatives?

The tangent plane and directional derivatives are crucial in understanding the behavior of curves and surfaces. They help us calculate the rate of change and approximate the behavior of functions at specific points, which is useful in many fields such as physics, engineering, and computer graphics.

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