?True or False? Justify your answer with a proof or a counterexample.The curvature of a

True or False? Justify your answer with a proof or a counterexample. A parametric equation that passes through points \(\mathbf{P}\) and \(\mathbf{Q}\) can be given by \(\mathbf{r}(t)=\left\langle t^{2}, 3 t+1, t-2\right\rangle\), where \(P\)(1,4,-1) and \(Q\)(16,11,2). Text Transcription: \mathbf{P} \mathbf{Q} \mathbf{r}(t)=\left\langle t^{2}, 3 t+1, t-2\right\rangle \(P\)(1,4,-1) \(Q\)(16,11,2)

True or False? Justify your answer with a proof or a counterexample. \(\frac{d}{d t}[\mathbf{u}(t) \times \mathbf{u}(t)]=2 \mathbf{u}^{\prime}(t) \times \mathbf{u}(t)\) Text Transcription: \frac{d}{d t}[\mathbf{u}(t) \times \mathbf{u}(t)]=2 \mathbf{u}^{\prime}(t) \times \mathbf{u}(t)

True or False? Justify your answer with a proof or a counterexample. The curvature of a circle of radius \(r\) is constant everywhere. Furthermore, the curvature is equal to 1 / \(r\). Text Transcription: \(r\) 1 / \(r\)

True or False? Justify your answer with a proof or a counterexample. The speed of a particle with a position function \(\mathbf{r}(t)\) is \(\left(\mathbf{r}^{\prime}(t)\right) /\left(\left|\mathbf{r}^{\prime}(t)\right|\right)\). Text Transcription: \(\mathbf{r}(t)\) \left(\mathbf{r}^{\prime}(t)\right)\left(\mid\left(\mathbf{r}^{\prime}(t) \mid\right)\right

Find the domains of the vector-valued functions. \(\mathbf{r}(t)=\langle\sin (t), \ln (t), \sqrt{t}\rangle\) Text Transcription: \mathbf{r}(t)=\langle\sin (t), \ln (t), \sqrt{t}\rangle

Find the domains of the vector-valued functions. \(\mathbf{r}(t)=\left\langle e^{t}, \frac{1}{\sqrt{4-t}}, \sec (t)\right\rangle\) Text Transcription: \mathbf{r}(t)=\left\langle e^{t}, \frac{1}{\sqrt{4-t}}, \sec (t)\right\rangle

Sketch the curves for the following vector equations. Use a calculator if needed. \(\text { [T] } \mathbf{r}(t)=\left\langle t^{2}, t^{3}\right\rangle\) Text Transcription: \text { [T] } \mathbf{r}(t)=\left\langle t^{2}, t^{3}\right\rangle

Sketch the curves for the following vector equations. Use a calculator if needed. \(\text { [T] } \mathbf{r}(t)=\left\langle\sin (20 t) e^{-t}, \cos (20 t) e^{-t}, e^{-t}\right\rangle\) Text Transcription: \text { [T] } \mathbf{r}(t)=\left\langle\sin (20 t) e^{-t}, \cos (20 t) e^{-t}, e^{-t}\right\rangle

Find a vector function that describes the following curves. Intersection of the cylinder \(x^{2}+y^{2}=4\) with the plane \(x\) + \(z\) = \(6\) Text Transcription: x^{2}+y^{2}=4 \(x\) + \(z\) = \(6\)

Find a vector function that describes the following curves. Intersection of the cone \(z=\sqrt{x^{2}+y^{2}}\) and plane \(z\) = \(y\) - \(\mathbf{4}\) Text Transcription: z=\sqrt{x^{2}+y^{2}} \(z\) = \(y\) - \(\mathbf{4}\)

Find the derivatives of \mathbf{u}(t), \mathbf{u}^{\prime}(t), \mathbf{u}^{\prime}(t) \times \mathbf{u}(t), \mathbf{u}(t) \times \mathbf{u}^{\prime}(t), and \mathbf{u}(t) \cdot \mathbf{u}^{\prime}(t). Find the unit tangent vector. \(\mathbf{u}(t)=\left\langle e^{t}, e^{-t}\right\rangle\) Text Transcription: \mathbf{u}(t)=\left\langle e^{t}, e^{-t}\right\rangle

Find the derivatives of \mathbf{u}(t), \mathbf{u}^{\prime}(t), \mathbf{u}^{\prime}(t) \times \mathbf{u}(t), \mathbf{u}(t) \times \mathbf{u}^{\prime}(t), and \mathbf{u}(t) \cdot \mathbf{u}^{\prime}(t). Find the unit tangent vector. \(\mathbf{u}(t)=\left\langle t^{2}, 2 t+6,4 t^{5}-12\right\rangle\) Text Transcription: \mathbf{u}(t)=\left\langle t^{2}, 2 t+6,4 t^{5}-12\right\rangle

Evaluate the following integrals. \(\int\left(\tan (t) \sec (t) \mathbf{i}-t e^{3 t} \mathbf{j}\right) \(d\) \(t\)\) Text Transcription: \int\left(\tan (t) \sec (t) \mathbf{i}-t e^{3 t} \mathbf{j}\right) \(d\) \(t\)

Evaluate the following integrals. \(\int_{1}^{4} \mathbf{u}(t) d t, \quad \text { with } \mathbf{u}(t)=\left\langle\frac{\ln (t)}{t}, \frac{1}{\sqrt{t}}, \sin \left(\frac{t \pi}{4}\right)\right\rangle\) Text Transcription: \int_{1}^{4} \mathbf{u}(t) d t, \quad \text { with } \mathbf{u}(t)=\left\langle\frac{\ln (t)}{t}, \frac{1}{\sqrt{t}}, \sin \left(\frac{t \pi}{4}\right)\right\rangle

Find the length for the following curves. \(\mathbf{r}(t)=\langle 3(t), 4 \cos (t), 4 \sin (t)\rangle for 1 \leq t \leq 4\) Text Transcription: \mathbf{r}(t)=\langle 3(t), 4 \cos (t), 4 \sin (t)\rangle for 1 \leq t \leq 4

Find the length for the following curves. \(\mathbf{r}(t)=2 \mathbf{i}+\mathrm{t} \mathbf{j}+3 t^{2} \mathbf{k} \text { for } 0 \leq t \leq 1\) Text Transcription: mathbf.r(t)=2.mathbf.i+mathrm.t.mathbf.j+3 t^2.mathbf.k.text for 0 leq t leq 1

Reparameterize the following functions with respect to their arc length measured from t = 0 in direction of increasing t. r(t) = 2t i + (4t ? 5)j + (1 ? 3t)k

Reparameterize the following functions with respect to their arc length measured from t = 0 in direction of increasing t. r(t) = cos(2t)i + 8t j ? sin(2t)k

Find the curvature for the following vector functions. r(t) = (2sint)i ? 4t j + (2cost)k

Find the curvature for the following vector functions. \(\mathbf{r}(t)=\sqrt{2} e^{t} \mathbf{i}+\sqrt{2} e^{-t} \mathbf{j}+2 t \mathbf{k}\) Text Transcription: mathbf.r(t)=sqrt2 e^t mathbf.i+sqrt2 e^-t mathbf.j+2 t mathbf.k

Find the curvature for the following vector functions. Find the unit tangent vector, the unit normal vector, and the binormal vector for r(t) = 2cost i + 3t j + 2sintk.

Find the curvature for the following vector functions. Find the tangential and normal acceleration components with the position vector \(\mathbf{r}(t)=\left\langle\cos t, \sin t, e^{t}\right\rangle/) Text Transcription: mathbf.r(t)=left.langle.cos t, sin t, e^t.right.rangle

Find the curvature for the following vector functions. A Ferris wheel car is moving at a constant speed v and has a constant radius r. Find the tangential and normal acceleration of the Ferris wheel car.

Find the curvature for the following vector functions. The position of a particle is given by \(\mathbf{r}(t)=\left\langle t^{2}, \ln (t), \sin (\pi t)\right\rangle\) where t is measured in seconds and r is measured in meters. Find the velocity, acceleration, and speed functions. What are the position, velocity, speed, and acceleration of the particle at 1 sec? The following problems consider launching a cannonball out of a cannon. The cannonball is shot out of the cannon with an angle ? and initial velocity \(\(\mathbf{v}_{0}\). The only force acting on the cannonball is gravity, so we begin with a constant acceleration a(t) = ?g j. Text Transcription: mathbf.r(t)=left.langle t^2, ln (t), sin (pi t)right.rangle mathbf.v_0

Find the curvature for the following vector functions. Find the velocity vector function v(t).

Find the curvature for the following vector functions. Find the position vector r(t) and the parametric representation for the position.

The following problems consider launching a cannonball out of a cannon. The cannonball is shot out of the cannon with an angle \(\theta\) and initial velocity \(\mathbf{v}_{0}\) . The only force acting on the cannonball is gravity, so we begin with a constant acceleration a(t) = ?g j. At what angle do you need to fire the cannonball for the horizontal distance to be greatest? What is the total distance it would travel? Text Transcription: theta v_0