A curve is given parametrically by and for all real

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

A curve is given parametrically by

x(t)=t^2+1

and

y(t)=t^3-3t

for all real

t

. At the instant

t=1

a particle moves along this curve. Answer the following parts to analyze the local shape of the trajectory near that instant. (a) Determine

\displaystyle \frac{dy}{dx}

t=1

. Show symbolic work before substitution. (b) Determine

\displaystyle \frac{d^{2}y}{dx^{2}}

t=1

. Express intermediate derivatives in terms of

t

before evaluating. (c) Using your result from part (b), describe whether the curve is bending upward or downward at

t=1

. Explain your reasoning in one or two complete sentences.