Mathematics

Mathematics

Problem

A particle moves in the plane so that its position at time tt is given by x(t)=sin⁡2t,y(t)=t+cos⁡t,0≤t≤2π.x(t)=\sin 2t, \qquad y(t)=t+\cos t,\qquad 0\le t\le2\pi. The curve is traced exactly once on this interval. a. Calculate dxdt\dfrac{dx}{dt} and dydt\dfrac{dy}{dt}. b. Determine all values of tt in [0,2π][0,2\pi] for which the tangent line to the curve is vertical. Show all work leading to your answers. c. For each value of tt found in part (b), give the corresponding coordinate point (x,y)(x,y) on the curve.
Intuitive explanation

This problem asks you to understand parametric motion—how a point traces a path as time evolves. The key insight is that velocity v⃗(t)=(x′(t),y′(t))\vec{v}(t) = (x'(t), y'(t)) tells you both the direction and speed of motion at each instant. By analyzing when the velocity changes direction or magnitude, you can identify features like cusps, loops, or smooth arcs. Since the curve traces exactly once, you're looking at a simple path with no self-intersections—useful for understanding the overall shape and behavior of the trajectory.

Intuitive explanation

This problem asks you to understand parametric motion—how a point traces a path as time evolves. The key insight is that velocity v⃗(t)=(x′(t),y′(t))\vec{v}(t) = (x'(t), y'(t)) tells you both the direction and speed of motion at each instant. By analyzing when the velocity changes direction or magnitude, you can identify features like cusps, loops, or smooth arcs. Since the curve traces exactly once, you're looking at a simple path with no self-intersections—useful for understanding the overall shape and behavior of the trajectory.

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