Calculus 3

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

Consider the planes x + y + z = 1 and x - 2y + 3z = 1

a. Find the angle between the two planes

b. Find symmetric equations for the line of intersection of the two planes

1. Find a vector and parametric equations for the line that passes through (4,2) and is parallel to v = <-1,5>. Then find 2 other points on that line.

2. Find parametric equations for the line segment joining points P(2, -4, -1) and Q(5, 0, 7). Where does this line intersect the xy-plane?

Plot the point $B(-2,-4,-3)$ in the 3D coordinate system.

Find the line through the points (5, -2, 3) and (7, 4, 1)

Plot the point $P(2, 4, 3)$ in the 3D coordinate system.

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.

Consider the function f(x,y) = xy^2 + x^3, and find the partial derivatives with respect to x and y.

Consider the function f(x,y,z) = $\frac{(x^5)(e^{2z})}{y}$, and find the gradient of the function.

What is the difference between a partial derivative and a total derivative of a function $f(x, y)$ when differentiated with respect to x?

Compute the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ for the function $f(x, y) = x^2 \cdot y + \sin(y)$.

If the temperature distribution over a flat slab of metal is described by a function of two variables, like $f(x, y) = 9 - x^2 - y^2$, what is the partial derivative of this function with respect to $x$ and $y$?

Find the partial derivative of $z$ with respect to $x$ and the partial derivative of $z$ with respect to $y$ using the implicit function theorem for the equation $x^2 + y^4 - z^3 + 3xy^2 - 8 = 0$.

Using the implicit function theorem, find the partial derivative of $z$ with respect to $x$ and $y$ for the equation $xy^3 + x^2z^2 - 6 = 0$.

Calculate the length of the vector valued function 3cos(2t), 3sin(2t), 2t over the interval for t from 0 to $2\pi$.

Calculate the length of the curve over the interval 1 to 4 for the vector-valued function \ln(t), 2t, t^2.

(x^2 + y^2 = 1). Parameterize the curve such that t is in the domain $[0, 2\pi]$.

Given a curve defined by a vector-valued function $R(T)$ where $T$ varies between $a$ and $b$, find the arclength of the curve.

Find the arc length of the curve given that $\vec{r}(t) = 3\cos(t)\hat{i} + 3\sin(t)\hat{j} + 6t\hat{k}$ where $t$ is from 0 to $\pi$.

Given the vector function $\mathbf{R}(t) = \langle 2t \mathbf{i} + e^t \mathbf{j} + e^{-t} \mathbf{k} \rangle$, find the arc length over the interval $[0, 1]$.