Calculus 3

Compute the dot product of vector $\mathbf{u} = (3, 12)$ with itself.

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

Calculate $(\mathbf{u} \cdot \mathbf{v}) \cdot \mathbf{v}$.

Find $w_1$, the projection of $\mathbf{u}$ onto $\mathbf{v}$, where $\mathbf{u} = (3, 5)$ and $\mathbf{v} = (2, 4)$.

Given a fixed point $P_0$ with coordinates $(x_0, y_0, z_0)$ and a direction vector $\vec{v}$ in three-dimensional space, find the vector equation of a line that passes through $P_0$ and is parallel to $\vec{v}$.

Find a vector equation and parametric equations for the line that passes through the point $(5, 1, 3)$ and is parallel to the vector $\mathbf{v} = (1, 4, -2)$. Then find two other points on the line.

Find the gradient of the function $f(x, y) = 3x^2 + y + 6$ at the point $(1, -1)$.

Calculate the partial derivative of a function Z with respect to X, holding Y constant.

Calculate the partial derivative of a function Z with respect to Y, holding X constant.

Find the gradient of a scalar function $z = f(x, y) = x^2 + y^2$, and evaluate it at the points (2, 1) and (-1, -1).

Imagine you have a function $z = x^2 + y^2$. How would you begin to plot this function in a 3-dimensional space?

Given a multivariable function $f(x,y) = x^2 + y^2$, find where the partial derivatives are equal to zero to identify candidates for maximums or minimums.

Given an ellipsoid represented by the equation $\frac{x^2}{1} + \frac{y^2}{4} + \frac{z^2}{9} = 1$, determine the lengths of the axes in the coordinate planes.

Sketch the quadric surface for the equation $x^2 + z^2 = 1$.

Sketch the graph for the equation $x \cdot y = 1$ and describe its properties.

Using the equation of a sphere, $x^2 + y^2 + z^2 = 1$, graph the sphere with the given axes.

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.

Given a function with two-dimensional input $(x, y)$ and a vector output, determine the vector at a specific input point such as $(1, 2)$ using the functions $(y^3 - 9y)$ for the x-component and $(x^3 - 9x)$ for the y-component.

Consider the vector field $F(x,y) = -yi + xj$. To get an idea of how this vector field looks, plug in a few coordinates and plot the resultant vectors.

Given the vector-valued function $r(t) = \langle 4\cos(t), 4\sin(t), 3 \rangle$, determine the curve it describes in 3D space.