Calculus 3

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

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

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?

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.

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

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 $y$?

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

Find the derivative of $(5x + 3)^4$.

Find the derivative of $(x^2 - 3x)^5$.

Find the derivative of $\cos(x^2)$.

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

Calculate the dot product of vectors $\mathbf{a} = (3, -4, 7)$ and $\mathbf{b} = (5, 2, -3)$.

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

Calculate the dot product of $a$ and $b$ times vector $a$, where $\mathbf{a} = (2, 3)$ and $\mathbf{b} = (5, -4)$.

Calculate the dot product between vector $b$ and $3a$, where $\mathbf{a} = (2, 3)$ and $\mathbf{b} = (5, -4)$.

Given the magnitudes of vectors $\mathbf{a}$ and $\mathbf{b}$ as 15 and 10 respectively, and the angle between them is 30 degrees, calculate the dot product of the two vectors.

Find the cross product of vectors $\mathbf{a}$ and $\mathbf{b}$, where $\mathbf{a} = 5 \mathbf{i} - 4 \mathbf{j} + 3 \mathbf{k}$ and $\mathbf{b} = -7 \mathbf{i} + 2 \mathbf{j} - 8 \mathbf{k}$.

Compute the dot product of vectors $\mathbf{u} = (3, 12)$ and $\mathbf{v} = (-4, 3)$.