Calculus 3

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 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

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?

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.

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 $[0, 1]$.

Find the arc length of the vector-valued function $\mathbf{R}(t) = 3t\mathbf{i} - t\mathbf{j}$ over the interval \([0, 3]\).

Given a function $f(x, y) = x^2 \cdot y$, where $x(t) = 2t + 1$ and $y(t) = t^3$, find $\frac{dw}{dt}$ using the multi-variable chain rule.

What is the derivative of the function composition $F(x(T), y(T))$ given $F(x, y) = x^2 y$, $x(T) = \,\cos(T)$, and $y(T) = s(T)$?

Find the derivative of $s(6x)$.

Find the derivative of $\tan(x^3)$.

Find the derivative of $\sec(4x)$.

Find the derivative of $(\ln x)^7$.

Find the derivative of $(x^3 - 7)^{12}$.

Find the derivative of $\frac{1}{(x^2 + 8)^3}$.