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AP Calculus BC 5.12 Implicit Relations Overview

Use first and second implicit derivatives to locate critical points or vertical tangents and justify increasing, decreasing or concavity behavior.

Syllabus
Effective Fall 2025
Course
AP Calculus BC

Exam points

  • identify the governing calculus relationship in the problem
  • justify the result with symbolic, graphical or contextual evidence

Covered syllabus points

FUN-4.D—Determine critical points of implicit relationsFUN-4.E—Justify conclusions about the behavior of an implicitly defined function based on evidence from its derivatives

5.12 Exploring Behaviors of Implicit Relations question 1

[Maximum number: 1]

The curve defined by x3+xy−y2=10x^{3}+x y-y^{2}=10 has a vertical tangent line when x=

A

1.037

B

1.087

C

2.074

D

2.096

5.12 Exploring Behaviors of Implicit Relations question 2

Given the following differential equation .

(b) Find the second derivative, d2ydx2\frac{d^{2} y}{d x^{2}}, in terms of x and y. The region in

the x y-plane where all the solution curves to the differential

equation are concave down can be expressed as a linear inequality.

Find this region.

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