Chapter 3

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

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

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

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

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

Given \(\mathbf{r}(t)=\left\langle 2 e^{t}, e^{t} \cos t, e^{t} \sin t\right\rangle,\) 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 \(\quad \mathbf{N}(t) \quad\) at \(\quad t=0 \quad\) 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 \(\mathbf{r}(t)=\langle 6 \cos t, 6 \sin t\rangle \quad\) at the point determined by \(t=\pi / 3\)

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

Find the unit tangent vector \(\mathbf{T}(t) \quad\) 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 \(s, \quad\) 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 \(t=\pi / 3 .\) (Note: The graph is an ellipse.)

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

What happens to the curvature as \(x \rightarrow \infty\) for the curve \(y=e^{x} ?\)

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

Find the equations of the normal plane and the osculating plane of the curve \(\mathbf{r}(t)=\langle 2 \sin (3 t), t, 2 \cos (3 t)\rangle\) at point \((0, \pi,-2)\) .

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 \(t=0,1,2\)

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. Describe the curvature as \(t\) increases from \(t=0\) to \(t=2\)

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). [T] 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). [T] Use technology to graph the curvature function.

Given \(\mathbf{r}(t)=\left(3 t^{2}-2\right) \mathbf{i}+(2 t-\sin (t)) \mathbf{j}, \quad\) 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?

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:

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

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 speed of the particle