Mathematics

Problem

For a parametric curve, at

t=2

it is given that

\dfrac{dx}{dt}<0

and

\dfrac{dy}{dt}>0

. Without computing

\dfrac{dy}{dx}

, which statement best describes the direction of motion of the point on the curve at that instant? a)

x

is increasing and

y

is increasing b)

x

is decreasing and

y

is increasing c) Direction cannot be determined without evaluating

\dfrac{dy}{dx}

x

is increasing and

y

is decreasing

Intuitive explanation

When a point moves along a parametric curve, the signs of $\frac{dx}{dt}$ and $\frac{dy}{dt}$ tell you directly which way it's moving in the $xy$ -plane—no need to find the slope $\frac{dy}{dx}$ . Since $x$ is decreasing (negative rate) and $y$ is increasing (positive rate), the point is moving left and upward simultaneously. Think of it as reading the velocity components: negative horizontal speed means leftward, positive vertical speed means upward.

Intuitive explanation

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