3.3 Arc length and curvature  (Page 7/18)

$\text{r}(t)=t^2\text{i}+14t\text{j},0\le t\le 7.$ This portion of the graph is shown here:

$\text{r}(t)=t^2\text{i}+(2t^2+1)\text{j},1\le t\le 3$

$\text{r}(t)=⟨2\text{sin}t,5t,2\text{cos}t⟩,0\le t\le \pi .$ This portion of the graph is shown here:

$\text{r}(t)=⟨t^2+1,4t^3+3⟩,-1\le t\le 0$

$\text{r}(t)=⟨e^{\text{−}t}\text{cos}t,e^{\text{−}t}\text{sin}t⟩$ over the interval $\left[0,\frac{\pi }{2}\right].$ Here is the portion of the graph on the indicated interval:

Find the length of one turn of the helix given by $\text{r}(t)=\frac{1}{2}\text{cos}t\text{i}+\frac{1}{2}\text{sin}t\text{j}+\sqrt{\frac{3}{4}}t\text{k}.$

Find the arc length of the vector-valued function $\text{r}(t)=-t\text{i}+4t\text{j}+3t\text{k}$ over $[0,1].$

A particle travels in a circle with the equation of motion $\text{r}(t)=3\text{cos}t\text{i}+3\text{sin}t\text{j}+0\text{k}.$ Find the distance traveled around the circle by the particle.

Set up an integral to find the circumference of the ellipse with the equation $\text{r}(t)=\text{cos}t\text{i}+2\text{sin}t\text{j}+0\text{k}.$

Find the length of the curve $\text{r}(t)=⟨\sqrt{2}t,e^t,e^{\text{−}t}⟩$ over the interval $0\le t\le 1.$ The graph is shown here:

Find the length of the curve $\text{r}(t)=⟨2\text{sin}t,5t,2\text{cos}t⟩$ for $t\in \left[\mathrm{-10},10\right].$

The position function for a particle is $\text{r}(t)=a\text{cos}(\omega t)\text{i}+b\text{sin}(\omega t)\text{j}.$ Find the unit tangent vector and the unit normal vector at $t=0.$

Given $\text{r}(t)=a\text{cos}(\omega t)\text{i}+b\text{sin}(\omega t)\text{j},$ find the binormal vector $\text{B}(0).$

Given $\text{r}(t)=⟨2e^t,e^t\text{cos}t,e^t\text{sin}t⟩,$ determine the unit tangent vector $\text{T}(t).$

Given $\text{r}(t)=⟨2e^t,e^t\text{cos}t,e^t\text{sin}t⟩,$ determine the unit tangent vector $\text{T}(t)$ evaluated at $t=0.$

Given $\text{r}(t)=⟨2e^t,e^t\text{cos}t,e^t\text{sin}t⟩,$ find the unit normal vector $\text{N}(t)$ evaluated at $t=0,$ $\text{N}(0).$

Given $\text{r}(t)=⟨2e^t,e^t\text{cos}t,e^t\text{sin}t⟩,$ find the unit normal vector evaluated at $t=0.$

Given $\text{r}(t)=t\text{i}+t^2\text{j}+t\text{k},$ find the unit tangent vector $\text{T}(t).$ The graph is shown here:

Find the unit tangent vector $\text{T}(t)$ and unit normal vector $\text{N}(t)$ at $t=0$ for the plane curve $\text{r}(t)=⟨t^3-4t,5t^2-2⟩.$ The graph is shown here:

Find the unit tangent vector $\text{T}(t)$ for $\text{r}(t)=3t\text{i}+5t^2\text{j}+2t\text{k}$

Find the principal normal vector to the curve $\text{r}(t)=⟨6\text{cos}t,6\text{sin}t⟩$ at the point determined by $t=\pi \text{/}3.$

Find $\text{T}(t)$ for the curve $\text{r}(t)=\left(t^3-4t\right)\text{i}+\left(5t^2-2\right)\text{j}.$

Find $\text{N}(t)$ for the curve $\text{r}(t)=\left(t^3-4t\right)\text{i}+\left(5t^2-2\right)\text{j}.$

Find the unit normal vector $\text{N}(t)$ for $\text{r}(t)=⟨2\text{sin}t,5t,2\text{cos}t⟩.$

Find the unit tangent vector $\text{T}(t)$ for $\text{r}(t)=⟨2\text{sin}t,5t,2\text{cos}t⟩.$

Find the arc-length function $s(t)$ for the line segment given by $\text{r}(t)=⟨3-3t,4t⟩.$ Write r as a parameter of s.

Parameterize the helix $\text{r}(t)=\text{cos}t\text{i}+\text{sin}t\text{j}+t\text{k}$ using the arc-length parameter s , from $t=0.$

Parameterize the curve using the arc-length parameter s , at the point at which $t=0$ for $\text{r}(t)=e^t\text{sin}t\text{i}+e^t\text{cos}t\text{j}.$

Find the curvature of the curve $\text{r}(t)=5\text{cos}t\text{i}+4\text{sin}t\text{j}$ at $t=\pi \text{/}3.$ ( Note: The graph is an ellipse.)

Find the x -coordinate at which the curvature of the curve $y=1\text{/}x$ is a maximum value.

Find the curvature of the curve $\text{r}(t)=5\text{cos}t\text{i}+5\text{sin}t\text{j}.$ Does the curvature depend upon the parameter t ?

Find the curvature $\kappa$ for the curve $y=x-\frac{1}{4}{x}^2$ at the point $x=2.$

Find the curvature $\kappa$ for the curve $y=\frac{1}{3}{x}^3$ at the point $x=1.$

Find the curvature $\kappa$ of the curve $\text{r}(t)=t\text{i}+6t^2\text{j}+4t\text{k}.$ The graph is shown here:

Find the curvature of $\text{r}(t)=⟨2\text{sin}t,5t,2\text{cos}t⟩.$

Find the curvature of $\text{r}(t)=\sqrt{2}t\text{i}+e^t\text{j}+e^{\text{−}t}\text{k}$ at point $P\left(0,1,1\right).$

At what point does the curve $y=e^x$ have maximum curvature?

What happens to the curvature as $x\to \infty$ for the curve $y=e^x?$

Find the point of maximum curvature on the curve $y=\text{ln}x.$

Find the equations of the normal plane and the osculating plane of the curve $\text{r}(t)=⟨2\text{sin}(3t),t,2\text{cos}(3t)⟩$ at point $\left(0,\pi ,\mathrm{-2}\right).$

Find equations of the osculating circles of the ellipse $4y^2+9x^2=36$ at the points $(2,0)$ and $(0,3).$

Find the equation for the osculating plane at point $t=\pi \text{/}4$ on the curve $\text{r}(t)=\text{cos}(2t)\text{i}+\text{sin}(2t)\text{j}+t.$

Find the radius of curvature of $6y=x^3$ at the point $\left(2,\frac{4}{3}\right).$

Find the curvature at each point $\left(x,y\right)$ on the hyperbola $\text{r}(t)=⟨a\text{cosh}(t),b\text{sinh}(t)⟩.$

Calculate the curvature of the circular helix $\text{r}(t)=r\text{sin}(t)\text{i}+r\text{cos}(t)\text{j}+t\text{k}.$

Find the radius of curvature of $y=\text{ln}(x+1)$ at point $\left(2,\text{ln}3\right).$

Find the radius of curvature of the hyperbola $xy=1$ at point $(1,1).$

Find the length of the curve over the interval $\left[0,2\right].$

Find the curvature of the plane curve at $t=0,1,2.$

Describe the curvature as t increases from $t=0$ to $t=2.$

[T] Use technology to graph the surface.

Find the curvature $\kappa$ of the generating curve as a function of x.

[T] Use technology to graph the curvature function.