Introduction

Welcome to AP Calculus AB/BC Topic 5.12 - Exploring Behaviors of Implicit Relations. Good news, this lesson is relatively short, and does not require learning anything "new."

Topic 5.12 will show us how we can extend the information we learned in Unit 5 to implicitly defined relations.

Implicit Relationships Follow the Same Rules

Implicit relationships (for example $x^2+y^2=1$) still follow the same rules as functions in terms of $\frac{dy}{dx}$. See below!

Increasing or Decreasing

Your implicit relation is increasing if $\frac{dy}{dx}>0$ and decreasing if $\frac{dy}{dx}<0$.

Critical Points

Your implicit relation will have a critical point if $\frac{dy}{dx}=0$ or $\frac{dy}{dx}$ does not exist.

Concavity

Your implicit relation is concave up if $\frac{d^{2}y}{d{x}^2}>0$ and concave down if $\frac{d^{2}y}{d{x}^2}<0$.

Relative Extrema

If ${\frac{dy}{dx}|}_{(x,y)}=0$ and ${\frac{d^{2}y}{d{x}^2}|}_{(x,y)}>0$, then $(x,y)$ is a relative minimum.

If ${\frac{dy}{dx}|}_{(x,y)}=0$ and ${\frac{d^{2}y}{d{x}^2}|}_{(x,y)}<0$, then $(x,y)$ is a relative maximum.

Tangent Lines

Your implicit relation will have a horizontal tangent line when $\frac{dy}{dx}=0$ and a vertical tangent line when $\frac{dy}{dx}$ does not exist.

Understanding all these relationships you should have already known is all of Topic 5.12! Let's try some practice problems.

Practice Problems

Here's how topic 5.12 can be tested on the AP Exam for both AB and BC.