CPM Homework Help : CALC Problem 3-154

Sketch a graph and find the equation of the line tangent to the curve at x = 3. Then, find another point on the curve with the same slope as the tangent. Homework Help ✎

Hint:

equation of tangent at x = 3:
y − y₁ = m(x − x₁)

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 '(3)

$\text{since }y= 3\sqrt[3]{x-2}=3(x-2)^{\frac{1}{3}}$

$\text{so }y'= (x-2)^{-\frac{2}{3}}= \frac{1}{(x-2)^{\frac{2}{3}}}$

and y '(3) = 1 = m

Step 2:

x₁ = 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