An ellipse is described implicitly by

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

9y^{2}+4x^{2}=36

. Restrict attention to the upper half of the ellipse (

y>0

). (a) Derive

\dfrac{dy}{dx}

for the ellipse in terms of

x

and

y

. (b) Obtain a simplified formula for

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

in terms of

x

and

y

. (c) Using the result from part (b), determine all

x

-values on the upper half of the ellipse for which the curve is concave down. Justify each step of your analysis. (d) The point

\left(\,1,\dfrac{\sqrt{27}}{3}\right)

lies on the ellipse. Evaluate

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

at this point and explain what the value indicates about the shape of the ellipse there.

Related theory

Curve Analysis
Advanced Differentiation
Implicit and parametric differentiation
Derivatives of implicit functions
Continuity and Differentiability

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