Question 650829
A point is moving along the circle x² + y² = 25

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>>...x coordinate changes at the rate of 2cm/sec...<<
<pre>
The vector pointing right is {{{(dx)/(dt)}}} = 2 cm/sec (right because it's
positive. The vector pointing down is {{{(dy)/(dt)}}} = ? cm/sec.
So the point is moving clockwise, and since the point is moving downard
toward the x-axis, {{{(dy)/(dt)}}} vector is pointing downward, so we can
expect its speed to be to be negative.  

</pre>
>>how fast is its y=coordinate changing as the point passes through (3,4)?<<
<pre>
We want to know: {{{(dy)/(dt)}}} = ? when x=3 and y=4

CAUTION:
x and y are varying as the point is moving before it reaches (3,4),
so DO NOT substitute for x or y until after we have taken the derivative.
Only then do we freeze the motion at the point (3,4)

                  x² + y² = 25

            2x{{{(dx)/(dt)}}} + 2y{{{(dy)/(dt)}}} = 0

Divide through by 2

              x{{{(dx)/(dt)}}} + y{{{(dy)/(dt)}}} = 0

                     y{{{(dy)/(dt)}}} =  -x{{{(dx)/(dt)}}}

                      {{{(dy)/(dt)}}} =  {{{-x/y}}}·{{{(dx)/(dt)}}}

Now finally we can freeze the motion by substituting x=3, y=4, {{{(dx)/(dt)}}} = 2

                      {{{(dy)/(dt)}}} =  {{{-3/4}}}·(2)

                      {{{(dy)/(dt)}}} =  {{{-3/2}}}

                      {{{(dy)/(dt)}}} =  -1.5 cm/sec

That means that at the instant when the point passes through (3,4) 
it is falling toward the x-axis at a rate of 1.5 cm/sec.

Edwin</pre>