The ellipse is drawn on a coordinate plane.

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

Problem

The ellipse

x^{2}+2y^{2}=8

is drawn on a coordinate plane. (i) Find

\dfrac{dy}{dx}

for the ellipse in terms of

x

and

y

.},{ (ii) Obtain

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

in terms of

x

and

y

.},{ (iii) Evaluate

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

at the point

(2,\sqrt{2})

on the ellipse. (iv) Using the values of

\dfrac{dy}{dx}

and

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

(2,\sqrt{2})

, write the equation of the tangent line at that point and comment on the local concavity of the curve there.

Related theory

Higher-Order Derivatives
Advanced Differentiation
Implicit and Parametric Differentiation
Derivatives of implicit functions
Continuity and Differentiability

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