# Section 14.5 Gradients and Directional Derivatives in Space - PowerPoint PPT Presentation

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## Section 14.5 Gradients and Directional Derivatives in Space

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### Section 14.5 Gradients 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 ... – PowerPoint PPT presentation

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Title: Section 14.5 Gradients and Directional Derivatives in Space

1
Derivatives in Space
2
• 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

3
• Similar to before we have
• If ? 0, then our direction is the same as the
• If ? p, then our direction is the opposite of
• If ? p/2, then our direction is the

4
• 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

5
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)

6
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