Home > APCALC > Chapter 7 > Lesson 7.1.3 > Problem7-30

7-30.

Consider the equation $xe^{5y} = 3y$.

1. What is $\frac { d y } { d x }$?

Implicitly differentiate.

2. Write the equation of the line tangent to the curve at $(0, 0)$.

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

3. If $x = 0.1$, estimate $y$ using the tangent line.

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

4. Using $\frac { d ^ { 2 } y } { d x ^ { 2 } }$, determine if the tangent line approximation is an overestimate or an underestimate. Justify your answer in words.

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

$\text{for all }\frac{dy}{dx}\text{ terms that appear as intermediate steps.}$

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

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.