Mathematics

Problem

A particle moves so that its position at time

t

is given by

x(t)=t^{2}+1

and

y(t)=\dfrac{1}{t}

, where

t>0

. Determine the slope

\dfrac{dy}{dx}

of the particle’s path at

t=2

. a)

-\dfrac{1}{4}

\dfrac{1}{16}

-\dfrac{1}{16}

-\dfrac{1}{8}

Intuitive explanation

The key is recognizing that when position is given parametrically, you can't just divide $y$ by $x$ —you need to use the chain rule to relate how $y$ changes with respect to $x$ through their common dependence on $t$ . Find $\frac{dy}{dt}$ and $\frac{dx}{dt}$ separately, then divide them to get $\frac{dy}{dx}$ . This tells you the instantaneous slope of the particle's trajectory at that moment in time.

Intuitive explanation

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