Unformatted text preview: rtical planes that we could use to slice
the surface. We will study these other planes in section 12.6 when we develop the directional derivative.
So are having the partials exist enough to say the function is differentiable? NO
Example: Both partials exist and equal zero at the point (0,0) A tangent plane ought to approximate the graph well in ALL directions – This is what is meant by differentiability.
In other words, for small and This can be shown to be true if the partials are continuous (Theorem 4.2)
So continuity of the partials IMPLIES Differentiability.
The common thread is that the derivative at a point serves as a linear approximation of the function at that point. Section 12.8 Maximum / Minimum Problems What is the difference between Absolute Extrema and Relative Extrema?
Absolute extrema is the largest or smallest values on the entire domain of f.
Relative extrema is the largest or smallest values compared locally to the values around it.
How do you find extrema? Possible extrema happen where is zero or undefined. Our critical points. Calc 101C
How do we find extrema? Instead of using the slope of the tangent line we use the tangent plane
What is the Calc 101C version of the derivative? The gradient
So our critical points come from when or is undefined There is one other place where extrema can happen – where is that? At the boundary of the domain of f.
It was easy to check the boundary for 101A since this consisted of the endpoints of the interval.
But for 101C we have a curve in the xy-plane that represents the boundary for our domain.
There are two approaches here:
1) We will parameterize the curve
Find the absolute maximum and minimum values of the function
. on the closed disk There is no extremum in the interior of of the domain. The extremum must happen on the boundary.
Parameterize the boundary curve with So we have At
At we have the point
we have the point . Absolute Maximum Value =
. Absolute Maximum Value = Example:
Find all extrema of the function
at the point and is never undefined. Thus we have a relative maximu...
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- Spring '08