Calculus 3

Using the chain rule, find $\frac{dz}{dt}$ for a function $z = f(x, y)$ where $x = x(t)$ and $y = y(t)$.

Find $\frac{dW}{dT}$ for $W = x \cdot \sin(y)$ where $x = e^t$ and $y = \pi - t$, and evaluate $\frac{dW}{dT}$ at $t = 0$.

Given a function $z = x^3 + y^3$ where $x = 2\sin(t)$ and $y = 3\cos(t)$, calculate $\frac{dz}{dt}$.

Calculate the dot product between vector $\mathbf{a} = (4, 5)$ and the sum of vectors $\mathbf{b} = (3, -6)$ and $\mathbf{c} = (-8, 2)$.

Find the cross product of the vectors $\mathbf{A} = \langle 1, 3, 4 \rangle$ and $\mathbf{B} = \langle 2, 7, -5 \rangle$.

Find the cross product of vectors $\mathbf{a}$ and $\mathbf{b}$, where $\mathbf{a} = 3\mathbf{i} + 5\mathbf{j} - 7\mathbf{k}$ and $\mathbf{b} = 2\mathbf{i} - 6\mathbf{j} + 4\mathbf{k}$.

Using the vector cross product, determine the vector perpendicular to two given initial vectors using the right-hand rule.

Calculate the dot product of $\mathbf{u}$ and $3\mathbf{v}$ by using a shortcut method for scalar multiplication.

Find $w_2$, the vector component of $\mathbf{u}$ orthogonal to $\mathbf{v}$, where $\mathbf{u} = (3, 5)$ and $\mathbf{v} = (2, 4)$.

For vectors $\mathbf{u} = 6\mathbf{i} - 3\mathbf{j} + 9\mathbf{k}$ and $\mathbf{v} = 4\mathbf{i} - \mathbf{j} + 8\mathbf{k}$, find the two components $w_1$ and $w_2$ of vector $\mathbf{u}$, where $w_1$ is the projection of $\mathbf{u}$ onto $\mathbf{v}$ and $w_2$ is the component orthogonal to $\mathbf{v}$.

Find the vector equation, parametric equations, and symmetric equations for the line that passes through the points $(1, 3, -2)$ and $(4, 1, 5)$.

Find the equation of a plane given the three points P(2, 1, 4), Q(4, -2, 7), and R(5, 3, -2).

Given a point $P_0 = (1, 2, 3)$ and a normal vector $\mathbf{n} = (4, 5, 6)$, find the equation of the plane in component form.

Find the parametric and symmetric equations of a line in space given two points.

Calculate the gradient vector for a given function $f(x, y)$ and describe its significance in the context of a 3D graph.

What direction should you travel to increase your height on a mountain as fast as possible?

What direction should you travel to keep your height constant (i.e. travel on a contour aka a level curve)?

Using a topographical map, analyze the contours to plan a route through the mountains with minimal elevation changes. Discuss the importance of this analysis in winter sports like skiing or snowshoeing.

Compute the gradient of the function $f(x, y) = x^2 \sin(y)$.

Compute the gradient of a multivariable function by finding its partial derivatives and forming a vector.