Question about directional derivatives and tangent planes ?

Thread starter basenne
Start date Jan 20, 2012
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Derivatives Planes Tangent

In summary, a directional derivative is a derivative of a multivariable function in the direction of a given vector. To calculate it, you find the gradient of the function at the given point and dot it with the unit vector in the direction of interest. Tangent planes are used to approximate the behavior of a function at a point and can help us understand its slope and rate of change in different directions. To find the equation of a tangent plane, you need a point and the normal vector, which can be found using the gradient of the function at the point. Directional derivatives are closely related to tangent planes, as the directional derivative in the direction of the normal vector gives the slope of the plane in that direction and the gradient vector of a

Jan 20, 2012

basenne

Is it possible to find a directional derivative for a point on z = f(x,y) at a point (x,y) in a direction (u1,u2) using the plane tangent to z at (x,y)?

If so, how?

Thanks!

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Jan 21, 2012

mathwonk

Science Advisor

Homework Helper

take the ray in that plane lying over the direction vector and take the directed slope.

1. What is a directional derivative?

A directional derivative is a derivative of a multivariable function in the direction of a given vector. It measures the rate of change of the function in the direction of the vector.

2. How do you calculate a directional derivative?

To calculate a directional derivative, you first find the gradient of the function at the given point. Then, you dot the gradient with the unit vector in the direction of interest. This gives you the directional derivative.

3. What is the significance of tangent planes?

Tangent planes are used to approximate the behavior of a function at a given point. They can help us understand the slope and rate of change of a function in different directions.

4. How do you find the equation of a tangent plane?

To find the equation of a tangent plane, you need a point on the plane and the normal vector to the plane. The normal vector can be found by taking the gradient of the function at the point of interest. Then, you can use the point and normal vector to write the equation in the form ax + by + cz = d.

5. What is the relationship between directional derivatives and tangent planes?

Directional derivatives are closely related to tangent planes. The directional derivative in the direction of the normal vector of a tangent plane gives the slope of the plane in that direction. Additionally, the gradient vector of a function at a point is perpendicular to the tangent plane at that point.