Question 1180829
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Evaluate the function at x=-8 and x=7: f(-8)=0; f(7)=sqrt(15).<br>
Find the equation of the secant: (-8,0) to (7,sqrt(15))<br>
slope = sqrt(15)/(7-(-8)) = sqrt(15)/15<br>
{{{y-0=(sqrt(15)/15)(x+8)}}}
{{{y = (1/sqrt(15))(x+8)}}}
Graph the function and the secant:<br>
{{{graph(400,400,-10,10,-2,5,sqrt(x+8),(1/sqrt(15))(x+8))}}}<br>
Find the derivative of the function:<br>
{{{y = (x+8)^(1/2)}}}<br>
{{{dy/dx = (1/2)((x+8)^(-1/2))(1) = 1/(2sqrt(x+8))}}}<br>
Find the x value where the slope of the tangent is the same as the slope of the secant.<br>
{{{(1/(2sqrt(x+8)))=1/sqrt(15)}}}
{{{2sqrt(x+8)=sqrt(15)}}}
{{{4(x+8)=15}}}
{{{4x+32=15}}}
{{{4x=-17}}}
{{{x=-17/4}}}<br>
Graph the line with slope sqrt(15)/15 touching y=sqrt(x+8) at x=-17/4 and observe that it is parallel to the secant:<br>
Graph the function and the secant:<br>
{{{graph(400,400,-10,10,-2,5,sqrt(x+8),(1/sqrt(15))(x+8),(1/sqrt(15))(x+17/4)+sqrt(15)/2)}}}<br>
ANSWER: x = -17/4.<br>
At x=-17/4, the tangent to y=sqrt(x+8) is parallel to the secant through x=-8 and x=7.<br>