AP Calculus BC / Unit 5: Analytical Applications of Differentiation
5.12 Exploring Behaviors of Implicit Relations
Notes
For curves like $x^2+xy+y^2=7$, use $\displaystyle \frac{dy}{dx}$ from implicit differentiation:
- Horizontal tangent: numerator of $\displaystyle \frac{dy}{dx}$ is 0 (denominator not 0).
- Vertical tangent: denominator is 0 (numerator not 0).
- Second derivatives need substitution of $\displaystyle \frac{dy}{dx}$.
Practice
For $x^2+y^2=25$, where are the horizontal tangents?
$\displaystyle \frac{dy}{dx}=-\frac{x}{y}=0$ when $x=0$.
For the same circle, where are the vertical tangents?
The denominator $y=0$.
For $xy=4$, $\displaystyle \frac{dy}{dx}$ at $(2,2)$ is:
$y+xy'=0$ gives $\displaystyle y'=-\frac{y}{x}$.
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