Unformatted text preview: ontinuous. Ie: The points ( 0 , y ) where y 0. 0. So we have already found some points where this function is The only other point where there is a possibility for a discontinuity is at the point ( 0 , 0 ). The function is defined there:
g(0,0) = 0 but because of the piecewise nature of the functions definition we might have a discontinuity there. To check
this out we must take the limit of the function and see if that limit is equal to 0. If yes, then the function is continuous
at (0,0). If the limit is any other number besides zero, we then have a discontinuity at (0,0).
This limit was found on page 945 using Theorem 2.1 – the limit turned out to be zero so the function is continuous at the
origin (0,0). Thus we are left with the discontinuities at the points: ( 0 , y ) where y 0. Theorem 2.2
Suppose that is continuous at Then and is continuous at the point is continuous at . . Example of Using Formal Definition of Limit: SHOW: So our goal is to get this expression: to pop up below: Thus let
The above inequality has some jumps in it that are not obvious:
Comes from the Triangle Inequality: Proof of Square Both Sides: given: Foil the Left:
Simplify: which is true for all A and B So now the official proof for the limit looks like this:
Let Given It follows
DONE Therefore: 12.4: Partial Derivatives
101A Definition of Derivative Measures Rate of Change of f(x) with respect to a change in x In the case of TWO or MORE independent variables, we are looking at the rate of change of
with respect to
a change in
. There are an infinite amount of directions to change the point
. Which direction do we
We will choose the paths parallel to the coordinate axes. This is called Partial Differentiation.
Partial Derivative with respect to x
Partial Derivative with respect to y
NOTATION: Partial Derivatives give information about functions ONLY in the direction of the Coordinate Axes. Trace of z=f(x,y) in the plane y = y0 The point (x0 , y0 ) in the xy-plane
Tangent line at the point (x0 , y0 , z0 ) The slope of the tangent line at th...
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- Spring '08