A closed curve is defined implicitly by .

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

A closed curve is defined implicitly by

2x^{2}+3y^{2}=12x+18

. (1) Compute

\dfrac{dy}{dx}

in terms of

x

and

y

. Present your work. (2) Compute

\dfrac{d^{2}y}{dx^{2}}

in terms of

x

and

y

. Simplify your expression. (3) The upper half of the curve (where

y>0

) can be written locally as

y=g(x)

. Determine the intervals of

x

on which this upper half is concave up and the intervals on which it is concave down. Express your final answer in interval notation and justify by analyzing the sign of

\dfrac{d^{2}y}{dx^{2}}

Related theory

Higher-Order Derivatives
Advanced Differentiation
Curve Analysis
Second order derivatives

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