Mathematics

Problem

The ellipse

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

is drawn on a coordinate plane. a. (i) Find

\dfrac{dy}{dx}

for the ellipse in terms of

x

and

y

.},{ b. (ii) Obtain

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

in terms of

x

and

y

.},{ c. (iii) Evaluate

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

at the point

(2,\sqrt{2})

on the ellipse. d. (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.