Calculus 2: Parametrized curves

Given a parametric function with a single input $T$ and a vector output defined as: $x = T \cdot \cos(T)$ and $y = T \cdot \sin(T)$, evaluate the function at different points and trace the curve it forms.

Show that a parametric function tracing a curve can be re-parameterized to alter the rate at which the curve is traced without changing its shape.

Using parameterization, find the coordinates of a point on the unit circle given an angle $\theta$.

Find the slope of the tangent line to a parameterized curve given functions $x(t)$ and $y(t)$.

Find the second derivative of a parameterized curve given functions $x(t)$ and $y(t)$.

Graph the parametric equations $x=2t$ and $y=t^2$ by picking some $t$ values and finding the corresponding $x$ and $y$ values. Plot the points and connect them to see the resulting parabola.

Find the area under the curve of a parametric function where x = t^2, y = 4t^2 - t^4, and t is bounded between 0 and 2.

Parametrize the unit circle in the XY-plane in XYZ space using trigonometry, with $R(T) = (\cos(T), \sin(T), 0)$ for T in $[0, 2\pi]$.

Parametrize the straight line segment from point P to point Q using $R(T) = (1-T) \cdot OP + T \cdot OQ$ for T in $[0,1]$.

Parametrize the parabola $Y = X^2$ that lies in the plane $Z = 1$ from the starting point where $X = -1$ to the ending point where $X = 2$ using $R(T) = (T, T^2, 1)$ for $T \in [-1, 2]$.

Parametrize the curve of intersection of surfaces $Z=X^3$ and $Y=X^2$ from the origin to the point $(2,4,8)$ using $X=T$, $Y=T^2$, $Z=T^3$ for $T\in [0,2]$.

Parametrize the intersection of the cylinder $X^2 + Y^2 = 4$ and the plane $2X + 3Y + 5Z = 10$ using polar coordinates for X and Y, and solving for Z. Use $R(T) = (2\cos(T), 2\sin(T), 2 - \frac{4}{5}\cos(T) - \frac{6}{5}\sin(T))$ for T in $[0, 2\pi]$.

Parametrize the curve $y = 2x^2 + 1$ by letting $x = t$, and then find the parameterization when $x = 2t$ and $x = t + 3$, and discuss the practical differences between these parameterizations.