CPM Homework Help : CALC Problem 7-31

Consider the curve xe⁵^y = 3y. Homework Help ✎
1. Find .
2. Find the equation of the tangent line at (0, 0).
3. If x = 0.1, estimate y using the tangent line.
4. Using , determine if the tangent approximation is an over or under estimate. Justify your answer in words.

Hint (a):

Implicitly differentiate.

Hint (b):

Notice that the derivative you found in part (a) has both x- and y-values.

Hint (c):

Since x = 0.1 is easier to evaluate in a linear equation than in the curve shown above, use the tangent line to estimate the y-value at x = 0.1.

Hint (d):

$\text{When finding }\frac{d^{2}y}{dx^{2}}, \text{ remember to substitute your answer in part (a)}$

Hint (d):

The sign of the 2nd derivative at (0,0) will determineif the tangent line is above or below the curve.

Hint (d):

When a function is concave up, the tangent line will be below the curve.
When it is concave down, the tangent line will be above the curve.
Verify these statements by sketching a few examples.