CPM Homework Help : CALC Problem 3-154

Sketch a graph and find the equation of the line tangent to the curve $y = 3 \sqrt [ 3 ] { x - 2 }$ at $x = 3$ . Then, find another point on the curve with the same slope as the tangent.

Hint:

equation of tangent at $x = 3$ :
$y − y_1 = m(x − x_1)$

Answer:

equation of tangent at $x = 3$ :
$y − 3 = 1(x − 3)$
$y = x − 3 + 3$
$y = x$

Hint:

To find another point with the same slope, consider the symmetry of the curve.

Step 1:

$m = y^\prime(3)$
Since $y= 3\sqrt[3]{x-2}=3(x-2)^{\frac{1}{3}}$
so $y'= (x-2)^{-\frac{2}{3}}= \frac{1}{(x-2)^{\frac{2}{3}}}$ and $y^\prime(3) = 1 = m$

Step 2:

$x_1 = 3$
$y_{1} = 3\sqrt[3]{(3)-2}=3$

Use the eTool below to explore the graph of the curve.
Click the link at right for the full version of the eTool: Calc 3-154 HW eTool