Question 1136445
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The slope of the given line is -12/7; the slope of a line perpendicular to it has a slope of 7/12.<br>
The given curve is {{{y = (7x-6)^(1/3)}}}<br>
The derivative is {{{dy/dx = (1/3)(7)(7x-6)^(-2/3) = 7/(3(7x-6)^(2/3))}}}<br>
We need to find the value of x for which the derivative (slope) is 7/12:<br>
{{{7/(3(7x-6)^(2/3)) = 7/12}}}
{{{3(7x-6)^(2/3) = 12}}}
{{{(7x-6)^(2/3) = 4}}}
{{{7x-6 = 4^(3/2) = 8}}}
{{{7x = 14}}}
{{{x = 2}}}<br>
We need to determine the y value of the curve at x=2:<br>
{{{y = (7(2)-6)^(1/3) = 8^(1/3) = 2}}}<br>
We want the equation of the line with slope 7/12 passing through (2,2):<br>
{{{2 = (7/12)2+b}}}
{{{2 = 7/6+b}}}
{{{b = 5/6}}}<br>
The equation of our tangent line is<br>
{{{y = (7/12)x+5/6}}}<br>
A graph....<br>
{{{graph(400,400,-2,6,-4,4,(7x-6)^(1/3),(7/12)x+5/6)}}}