Calculus 3: Vector Functions

The position of a particle in the xy plane at time t is $\vec{r}(t) = (t+1)\mathbf{i} \, + \, (t^2-1)\mathbf{j}$ Find an equation in x and y whose graph is the path of the particle. Then find the particle's velocity and acceleration vectors at t = 1.

dy/dt = $\frac{1}{25 + 9t^2}$, with initial condition $y\left(\frac{5}{3}\right) = \frac{\pi}{30}$.

Calculate the square of the magnitude of vector $\mathbf{a} = (2, 3)$.

Find the magnitude squared of vector $\mathbf{v} = (-4, 3)$.

Evaluate the function f at T = 0 and S = $\pi$ and determine the resulting point in three-dimensional space.

Find the derivative of the vector-valued function $\mathbf{R}(t) = (f(t), g(t), h(t))$ where $f$, $g$, and $h$ are scalar functions.

Describe a curve using a position vector-valued function.

Using vector valued functions, describe the path of a particle or object, taking into account time as a variable.

Parametrize the same curve using different rates and understand the derivative of a position vector valued function.

Describe the path of a particle in three-dimensional space using vector valued functions.

Given the vector-valued function $r(t) = \langle t, 3\cos(t), 5\sin(t) \rangle$, describe the curve in 3D space.

Given the vector-valued function $r(t) = \langle 2\cos(t), 2\sin(t), 6e^{-t/4} \rangle$, analyze how the curve behaves in 3D space and the effect of exponential decay in the $z$-component.

Give an example of a vector-valued function r(t) and determine its domain and range in $R^3$.

Visualize a vector-valued function and analyze its behavior as it does not intersect itself within a given domain subset from $t = 5$ to $t = 20$.