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Section 14.5Gradients and Directional

Derivatives in Space

- Directional Derivative of Function of 3 Variables
- Let w f(x,y,z) and let the partial derivatives

of f exist at (a,b,c) in its domain. Let

be a unit vector.

Then - Now, using a similar argument to last section we

can show that - where ? is the angle between the gradient and

the unit vector

- Similar to before we have
- If ? 0, then our direction is the same as the

gradient and - If ? p, then our direction is the opposite of

the gradient and - If ? p/2, then our direction is the

perpendicular to the gradient and

- When we had a function of 2 variables, if the

directional derivative was zero we were moving in

a direction tangent to the level curve and the

gradient was perpendicular to the level curve - With a function of 3 variables, if the

directional derivative is zero we are moving in a

direction tangent to the level surface and the

gradient is perpendicular to that level surface

Example

- Find
- Find the derivative in the direction of
- Find the maximum rate of change of f at (3,4,5)
- Find the vector in the direction of the maximum

rate of change at (3,4,5)

Finding a Tangent Plane

- Determine the equation of the plane tangent to

at (1,1) - Now consider f the function of a level surface of

where w 0 - Now we know that the gradient of w at a point

(a,b,c) is perpendicular to w at that point - Therefore we have
- Now find the tangent plane using the gradient
- Lets make sure they are the same