A particle traces the path x(t)=e^ t +t

MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

A particle traces the path x(t)=et+t,y(t)=e−t,t∈R.x(t)=e^{t}+t,\qquad y(t)=e^{-t},\qquad t\in\mathbb{R}. Assume tt is measured in seconds and the coordinate units are meters. (a) Compute dydx\displaystyle \frac{dy}{dx} as a function of tt. (b) Compute d2ydx2\displaystyle \frac{d^{2}y}{dx^{2}} as a function of tt and simplify your answer. (c) Evaluate d2ydx2\displaystyle \frac{d^{2}y}{dx^{2}} at t=0t=0 and interpret the sign of this value in the context of the particle’s vertical acceleration relative to its horizontal motion. (d) Determine the limit of d2ydx2\displaystyle \frac{d^{2}y}{dx^{2}} as t→∞t\to\infty. What does this suggest about the eventual shape of the trajectory? Explain briefly.

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