Chapter 3

Given \(\mathbf{r}(t)=\left\langle 2 e^{t}, e^{t} \cos t, e^{t} \sin t\right\rangle,\) determine the tangent vector \(\mathbf{T}(t)\)

Given \(\mathbf{r}(t)=\left\langle 2 e^{t}, e^{t} \cos t, e^{t} \sin t\right\rangle,\) determine the unit tangent vector \(\mathbf{T}(t)\) evaluated at \(t=0\).

. Given \(\mathbf{r}(t)=\left\langle 2 e^{t}, e^{t} \cos t, e^{t} \sin t\right\rangle, \quad\) find the unit normal vector \(\mathbf{N}(t)\) evaluated at \(t=0, \quad \mathbf{N}(0)\)

Given \(\mathbf{r}(t)=\left\langle 2 e^{t}, e^{t} \cos t, e^{t} \sin t\right\rangle, \quad\) find the unit normal vector evaluated at \(t=0\).

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

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

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

Find the principal normal vector to the curve \(\mathbf{r}(t)=\langle 6 \cos t, 6 \sin t\rangle\) at the point determined by \(t=\pi / 3 .\)

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

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

Find the unit normal vector \(\mathbf{N}(t)\) for \(\mathbf{r}(t)=\langle 2 \sin t, 5 t, 2 \cos t\rangle\)

Find the unit tangent vector \(\mathbf{T}(t)\) for \(\mathbf{r}(t)=\langle 2 \sin t, 5 t, 2 \cos t\rangle\)

Find the arc-length function \(s(t)\) for the line segment given by \(\mathbf{r}(t)=\langle 3-3 t, 4 t\rangle\). Write \(r\) as a parameter of S.

Parameterize the helix \(\mathbf{r}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}+t \mathbf{k}\) 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 \(\mathbf{r}(t)=e^{t} \sin t \mathbf{i}+e^{t} \cos t \mathbf{j}\)

Find the curvature of the curve \(\mathbf{r}(t)=5 \cos t \mathbf{i}+4 \sin t \mathbf{j}\) at \(t=\pi / 3\).

Find the \(x\) -coordinate at which the curvature of the curve \(y=1 / x\) is a maximum value.

Find the curvature of the curve \(\mathbf{r}(t)=5 \cos t \mathbf{i}+5 \sin t \mathbf{j}\). 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 \(\mathbf{r}(t)=t \mathbf{i}+6 t^{2} \mathbf{j}+4 t \mathbf{k}\). The graph is shown here:

Find the curvature of \(\mathbf{r}(t)=\langle 2 \sin t, 5 t, 2 \cos t\rangle\).

Find the curvature of \(\mathbf{r}(t)=\sqrt{2}\) ti \(+e^{t} \mathbf{j}+e^{-t} \mathbf{k}\) at point \(P(0,1,1)\).

Find the point of maximum curvature on the curve \(y=\ln x\).

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

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

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

Find the curvature at each point \((x, y)\) on the hyperbola \(\mathbf{r}(t)=\langle a \cosh (t), b \sinh (t)\rangle\).

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

Find the radius of curvature of \(y=\ln (x+1)\) at point \((2, \ln 3)\).

Find the radius of curvature of the hyperbola \(x y=1\) at point (1,1) .

A particle moves along the plane curve \(\mathrm{C}\) described by \(\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j} .\) Solve the following problems. Find the length of the curve over the interval [0,2] .

A particle moves along the plane curve \(\mathrm{C}\) described by \(\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j} .\) Solve the following problems. Find the curvature of the plane curve at \(t=0,1,2\).

The surface of a large cup is formed by revolving the graph of the function \(y=0.25 x^{1.6}\) from \(x=0\) to \(x=5\) about the \(y\) -axis (measured in centimeters). Use technology to graph the surface.

The surface of a large cup is formed by revolving the graph of the function \(y=0.25 x^{1.6}\) from \(x=0\) to \(x=5\) about the \(y\) -axis (measured in centimeters). Find the curvature \(\kappa\) of the generating curve as a function of \(x\).

The surface of a large cup is formed by revolving the graph of the function \(y=0.25 x^{1.6}\) from \(x=0\) to \(x=5\) about the \(y\) -axis (measured in centimeters). Use technology to graph the curvature function.

Given \(\mathbf{r}(t)=\left(3 t^{2}-2\right) \mathbf{i}+(2 t-\sin (t)) \mathbf{j}\), find the velocity of a particle moving along this curve.

Given \(\mathbf{r}(t)=\left(3 t^{2}-2\right) \mathbf{i}+(2 t-\sin (t)) \mathbf{j},\) find the acceleration vector of a particle moving along the curve in the preceding exercise.

Given the following position functions, find the velocity, acceleration, and speed in terms of the parameter \(t\). \(\mathbf{r}(t)=\left\langle 3 \cos t, 3 \sin t, t^{2}\right\rangle\)

Given the following position functions, find the velocity, acceleration, and speed in terms of the parameter \(t\). \(\mathbf{r}(t)=e^{-t} \mathbf{i}+t^{2} \mathbf{j}+\tan t \mathbf{k}\)

Given the following position functions, find the velocity, acceleration, and speed in terms of the parameter \(t\). \(\mathbf{r}(t)=2 \cos t \mathbf{j}+3 \sin t \mathbf{k}\). The graph is shown here:

Find the velocity, acceleration, and speed of a particle with the given position function. \(\mathbf{r}(t)=\left\langle t^{2}-1, t\right\rangle\)

Find the velocity, acceleration, and speed of a particle with the given position function. \(\mathbf{r}(t)=\left\langle e^{t}, e^{-t}\right\rangle\)

Find the velocity, acceleration, and speed of a particle with the given position function. \(\mathbf{r}(t)=\langle\sin t, t, \cos t\rangle\). The graph is shown here:

The position function of an object is given by \(\mathbf{r}(t)=\left\langle t^{2}, 5 t, t^{2}-16 t\right\rangle\). At what time is the speed a minimum?

Let \(\mathbf{r}(t)=r \cosh (\omega t) \mathbf{i}+r \sinh (\omega t) \mathbf{j} .\) Find the velocity and acceleration vectors and show that the acceleration is proportional to \(\mathbf{r}(t)\)

Consider the motion of a point on the circumference of a rolling circle. As the circle rolls, it generates the cycloid \(\mathbf{r}(t)=(\omega t-\sin (\omega t) \mid \mathbf{i}+(1-\cos (\omega t)) \mathbf{j}\), where \(\omega\) is the angular velocity of the circle and \(b\) is the radius of the circle: Find the equations for the velocity, acceleration, and speed of the particle at any time.

A person on a hang glider is spiraling upward as a result of the rapidly rising air on a path having position vector \(\mathbf{r}(t)=(3 \cos t) \mathbf{i}+(3 \sin t) \mathbf{j}+t^{2} \mathbf{k}\). The path is similar to that of a helix, although it is not a helix. The graph is shown here: Find the following quantities: The velocity and acceleration vectors

A person on a hang glider is spiraling upward as a result of the rapidly rising air on a path having position vector \(\mathbf{r}(t)=(3 \cos t) \mathbf{i}+(3 \sin t) \mathbf{j}+t^{2} \mathbf{k}\). The path is similar to that of a helix, although it is not a helix. The graph is shown here: Find the following quantities: The glider's speed at any time

Given that \(\mathbf{r}(t)=\left\langle e^{-5 t} \sin t, e^{-5 t} \cos t, 4 e^{-5 t}\right\rangle\) is the position vector of a moving particle, find the following quantities: The velocity of the particle