Parametric Equations, Polar Coordinates, and Vector–Valued Functions (BC Only)

A particle is moving on the xy-plane. Its motion can be described by the parametric functions $y(t) = 3t^3 - 2t$ and $x(t) = \ln(t)$. What quadrant is the particle located in at time $t = 0.5$?

Which unconventional technique would allow determination of concavity changes in a parametrically defined curve without direct computation of its second derivative?

To find the slope of the tangent line to a curve at any point given by parametric equations, you must calculate which of these derivatives?

Given a set of parametric equations defining a curve, what would be an appropriate method for approximating arc length if you cannot solve analytically?

Which of the following best represents an equation for a normal line to a curve defined by parametric equations ($x(t)=t^4$, $y(t)=\sin{(2t)}$) at point where $t=\pi/4$?

What must be evaluated in order to determine whether there exists any vertical tangents or cusps on a curve described by differentiable functions $x=f(t)$ and $y=g(t)$?

Question #2: In using Euler's Method to approximate values along the solution curve of $\frac{dy}{dx} = y - x$, what step size would provide more accurate approximation between two consecutive points?

A particle is moving on the xy-plane. Its motion can be described by the parametric functions $y(t) = \sin(t)$ and $x(t) = \sqrt{t}$. Find the equation of the line tangent to particle at time $t = 16\pi$.

Given that a particle's position is defined by parametric equations with derivatives $\left( \dfrac{d^{n+1}x}{dt^{n+1}} \right) = (n+1)! \cdot e^t$ and $\left( \dfrac{d^n y}{dt^n} \right) = n! \cdot e^{-t}$, where $n$ is a positive in...

Greg is bowling with his friends and rolls the ball at time $t = 0$. Consider the center axis of the lane to correspond to line $x = 0$ and the pin deck to be at the line $y = 15$. The gutters correspond to the lines $x = -4$ and $ ...