Question 1183948
<pre>
{{{drawing(300,300,-.3,5,-.3,5,
graph(300,300,-.3,5,-.3,5,x^2), line(-2,-8.25,8,21.75),
circle(1.5,2.25,.05), locate(1.6,2.3,P), locate(1,.4,Q),
circle(.75,0,.06))}}}

Suppose the line is tangent to the curve at P(h,h<sup>2</sup>). 

Then its slope is the derivative of {{{y=x^2}}} at P(h,h<sup>2</sup>).
The derivative is {{{dy/dx=2x}}}, so the slope at P(h,h<sup>2</sup>) is 2h.
Since it goes through (h,h<sup>2</sup>), its equation is:

{{{y-h^2=2h(x-h)}}}

Its x-intercept P is the x-value when y=0, so

{{{0-h^2=2h(x-h)}}}
{{{-h^2=2hx-2h^2)}}}
Divide through by h
{{{-h=2x-2h}}}
{{{h=2x}}}
{{{expr(1/2)h=x}}}

So the coordinates of Q are {{{Q( expr(1/2)h,0)}}}.

{{{drawing(300,300,-.3,5,-.3,5,
graph(300,300,-.3,5,-.3,5,x^2), line(-2,-8.25,8,21.75),
circle(1.5,2.25,.05), locate(1.6,2.5,P(h,h^2)), locate(1,.7,Q(expr(1/2)h,0)),
circle(.75,0,.06) )}}}

Now let's switch the letter h to x.  [It would have been too confusing
if we had started out with x because the equation of a line uses x).

{{{drawing(300,300,-.3,5,-.3,5,
graph(300,300,-.3,5,-.3,5,x^2), line(-2,-8.25,8,21.75),
locate(.33,0,q),
circle(1.5,2.25,.05), locate(1.6,2.5,P(x,x^2)), locate(1,.7,Q(expr(1/2)x,0)),
circle(.75,0,.06))}}}

We want to know how fast Q is moving.  Since Q is on the x-axis, Q moves the
same speed as its own x-coordinate {{{expr(1/2)x}}}.  So let the variable
q = the x-coordinate of Q.  {{{q=expr(1/2)x}}}. We want to know {{{(dq)/(dt)}}} at P(2,4) which is when x=2.

The point P moves up the curve at the rate of 2 units per second, so

{{{ds/dt = 2}}}

{{{(ds/dt)^2=(dx/dt)^2+(dy/dt)^2}}}

{{{2^2=(dx/dt)^2+(2x*expr(dx/dt))^2}}}

{{{4=(dx/dt)^2+4x^2*(dx/dt)^2}}}
   
{{{4=(dx/dt)^2(1+4x^2)}}}

{{{4^""/(1+4x^2)=(dx/dt)^2}}}

{{{dx/dt=sqrt(4^""/(1+4x^2))}}}

{{{dx/dt=2sqrt(1^""/(1+4x^2))}}}

{{{q=expr(1/2)x}}}
{{{dq/dt = expr(1/2)(dx/dt)}}}
{{{dq/dt = expr(1/2)(2sqrt(1^""/(1+4x^2)))}}}
{{{dq/dt = sqrt(1^""/(1+4x^2))}}}

Evaluating that when x=2,

{{{matrix(1,5,dq/dt,when,x=2, is, sqrt(1^""/(1+4(2)^2)))}}}{{{""=""}}}{{{1/sqrt(17)}}} units per second.

Edwin</pre>