Suppose differentiable functions and satisfy the identity x(t)^2 + y(t)^2 = d^2

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

Suppose differentiable functions

x(t)

and

y(t)

satisfy the identity

x(t)^2 + y(t)^2 = d^2,

where

d

is a positive constant. For times when both

x(t)

and

y(t)

are positive, consider the signs of their time derivatives. (a) Show that

\dfrac{dy}{dt} = -\dfrac{x}{y}\,\dfrac{dx}{dt}

without substituting numerical values. (b) Assume

\dfrac{dx}{dt} < 0

for all such times. Using the relationship from part (a), explain why

\dfrac{dy}{dt} > 0

whenever

x>0

and

y>0

. Your explanation should refer explicitly to the structure of the equation obtained in part (a).