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

The motion of a rolling ball on the coordinate plane is given by the set of parametric equations $x(t) = 12\sin(t)$ and $y(t) = 6t^2$. Which of the following derivatives is incorrect?

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 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 $y=g(t)$?

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

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 $x = 4$. If a ball falls into a gutter before hitting any pins, Greg’s score is ...

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?

Given a particle's position is described by parametric equations $x(t) = e^{2t}$ and $y(t) = \sin(3t)$, what is the second derivative $\frac{d^2y}{dx^2}$ at $t = 0$?

A rocket is launched from Earth. Its distance from Earth’s surface as a function of time is given by $D(t) = 12t^2 + 40t$. The amount of fuel in the tank as a function of distance is given by $F(D) = 1000 - 0.1D$. Convert distance and fuel into parametric functions with time, $t$, as a parameter and verify $\frac{dF}{dD}$.

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?