Find r′(t), r(t_0), and r′(t_0) for the given value of t_0. Then sketch the plane curve represented by the vector-valued function, and sketch the vectors r(t_0) and r′(t_0). Position the vectors such that the initial point of r(t_0) is at the origin and the initial point of r′(t_0) is at the terminal point of r(t_0). What is the relationship between r′(t_0) and the curve? r(t) = ⟨e^-t, e^t⟩, t_0 = 0

Show that the line segment joining the points $\left(x_0, y_0\right)$ and $\left(x_1, y_1\right)$ can be represented parametrically as

$$\begin{aligned} & x=x_0+\left(x_1-x_0\right) \frac{t-t_0}{t_1-t_0}, \\ & y=y_0+\left(y_1-y_0\right) \frac{t-t_0}{t_1-t_0} \end{aligned} \quad\left(t_0 \leq t \leq t_1\right)$$

Evaluate $\int_{C}(y-x) d x+(2 x+5 y) d y$. C: line segments from (0, 0) to (2, −4) and (2, −4) to (4, −4)

Find the curvature K of the plane curve at the given value of the parameter. r(t) = 4ti - 2tj, t=1

Find an equation in rectangular coordinates for the surface represented by the cylindrical equation, and sketch its graph. $r=4 \sin \theta$

Find a set of parametric equations of the line. -The line passes through the point (2, 3, 4) and is perpendicular to the plane given by 3x + 2y -z = 6

Write an integral that represents the area of the shaded region of the figure. r = cos 2θ

Use cylindrical coordinates to find the volume of the solid. Solid inside x²+y²+z²=16 and outside z = √x²+y²

Find a set of parametric equations of the line. -The line passes through the point (-1, 4, -3) and is parallel to v = 5i - j.

Find the arc length of the graph of r(t). r(t)=3cos ti+3sin tj+tk; 0≤t≤2π

Determine the interval(s) on which the vector-valued function is continuous. r(t) = √t i + √(t-1) j

Find (a) d/dt [r(t) · u(t)] and (b) d/dt [r(t) × u(t)] in two different ways. (i) Find the product first, then differentiate. (ii) Apply the properties of Theorem 12.2. r(t) = cos ti + sin tj + tk, u(t) = j + tk

Find an equation of the plane. The plane passes through (0, 0, 0), (2, 0, 3), and (-3, -1, 5)

Find the unit tangent vector to the curve at the specified value of the parameter. r(t) = 6 cos ti + 2 sin tj, t = π/3