A parametrically defined curve has derivatives and whose signs on the parameter…

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

Problem

A parametrically defined curve CC has derivatives dxdt\dfrac{dx}{dt} and dydt\dfrac{dy}{dt} whose signs on the parameter intervals shown below are known, but explicit formulas for x(t)x(t) and y(t)y(t) are not provided. Parameter interval and sign information: • For −3<t<−1-3<t<-1: dxdt>0\dfrac{dx}{dt}>0 and dydt>0\dfrac{dy}{dt}>0. • At t=−1t=-1: dxdt=0\dfrac{dx}{dt}=0 while dydt>0\dfrac{dy}{dt}>0. • For −1<t<0-1<t<0: dxdt<0\dfrac{dx}{dt}<0 and dydt>0\dfrac{dy}{dt}>0. • For 0<t<20<t<2: dxdt<0\dfrac{dx}{dt}<0 and dydt<0\dfrac{dy}{dt}<0. • For 2<t<42<t<4: dxdt>0\dfrac{dx}{dt}>0 and dydt<0\dfrac{dy}{dt}<0. Throughout, assume dxdt\dfrac{dx}{dt} and dydt\dfrac{dy}{dt} are continuous on the interior of each open interval indicated above. (a) For each open parameter interval, determine whether yy is increasing or decreasing as a function of xx. Explicitly state the sign of dydx\dfrac{dy}{dx} on each interval and justify your conclusions. (b) Explain why the point t=−1t=-1 must form a boundary between separate xx–intervals for the purpose of your analysis, even though dydt≠0\dfrac{dy}{dt}\neq0 there.

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