MATH 20C Lecture 24

14.8

Find the closest point on the plane $x+y+2z=1$ to the origin

, distance squared
We want to find the global min of subject to the constraint
It has to be a critical point obtained by solving


Plug into

Plug in
Since we only have one ctitical point and the global min exists, it has to be

Find the extreme values of $f(x,y)$ in the domain $D$

is bounded and closed has a global max and a global min in
Step 1: find the interior critical points of
Step 2: find the critical points on the boundary of
Step 3: compare the values of of all these points

15.1 Double Integral

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Recall

Single Integral

Divide into subintervals of width
Area over the i-th subinterval under the graph of
Riemann sum

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Let’s consider functions of 2-variables
Let be the set of points s.t.
is a rectangle
We define the double integral as the volume under the graph of over

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Now let’s define the double integral using Riemann sum

Step 1 (subdivisions):

• subdivide into subintervals of the same length
• subdivide into subintervals of the same length
  This creates an grid of subrectangles in .

Step 2:
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