5E: Excercises

Exercise \(\PageIndex{1}\)

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

1. A parametric equation that passes through points P and Q can be given by \(\vec{r(t)}=⟨t^2, 3t+1, t−2⟩⟩\), where \(P(1, 4, −1)\) and \(Q(16, 11, 2)\).

2. \( \dfrac{d}{dt}[\vec{u(t)}×\vec{u(t)}]=2 \vec{u′(t)} × \vec{u(t)}\)

Answer

False, \( \dfrac{d}{dt}[\vec{u(t)}×\vec{u(t)}]=0 \)

3. The curvature of a circle of radius \(r\) is constant everywhere. Furthermore, the curvature is equal to \(\dfrac{1}{r}.\)

4. The speed of a particle with a position function\(\vec{r(t)}\) is \( \dfrac{\vec{r′(t)}}{|\vec{r′(t)}|}.\)

Answer

False, it is \(|\vec{r′(t)}|\).

Exercise \(\PageIndex{2}\)

Find the domains of the vector-valued functions.

1. \(\vec{r(t)}=⟨sin(t),ln(t),t⟩\)

2. \(\vec{r(t)}=⟨e^t,14−t,sec(t)⟩\)

Answer

\(t<4, t≠nπ^2\)

Exercise \(\PageIndex{3}\)

Sketch the curves for the following vector equations. Use a calculator if needed.

[T] 1. \(\vec {r(t)}=⟨t^2,t^3⟩\)

[T] 2. \(\vec{ r(t)}=⟨sin(20t)e^{−t},cos(20t)e^{−t},e^{−t}⟩\)

Answer

Exercise \(\PageIndex{4}\)

Find a vector function that describes the following curves.

1. The intersection of the cylinder \(x^2+y^2=4\) with the plane \(x+z=6\).

2. The intersection of the cone \(z=x^2+y^2\) and plane \(z=y−4. \)

Answer

\(\vec{r(t)}=⟨t,2−t,−2−t⟩\)

Exercise \(\PageIndex{5}\)

Find the derivatives of \(\vec{u(t)}, \vec{ u′(t)},\vec{ u′(t)} × \vec{u(t)}, \vec{u(t)}× \vec{u′(t)}\), and \(\vec{u(t)}⋅ \vec{u′(t)}\). Also find the unit tangent vector.

1. \(\vec{u(t)}=⟨e^t,e−t⟩\)

2. \(\vec{u(t)}=⟨t^2,2t+6,4t^5−12⟩\)

Answer

\(\vec{u′(t)}=⟨2t,2,20t^4⟩, \vec{{u′′(t)}=⟨2,0,80t^3⟩, \dfrac{d}{dt}[\vec{u′(t)}× \vec{u(t)}]=⟨−480t^3−160t^4,24+75t^2,12+4t⟩, \dfrac{d} {dt}[\vec{u(t)}×\vec{u′(t)}]=⟨480t^3+160t^4,−24−75t^2,−12−4t⟩, \dfrac{d}{dt}[\vec{u(t)}⋅\vec{u′(t)}]=720t^8−9600t^3+6t^2+4,\) unit tangent vector: \(\vec{T(t)}=<2t400t^8+4t^2+4, 2400t^8+4t^2+4,20t4400t^8+4t^2+4>\)

Exercise \(\PageIndex{6}\)

Evaluate the following integrals.

1) \(\int(tan(t)sec(t) \hat{\mathbf{i}}−te^{3t} \hat{\mathbf{j}})dt\)

2) \(\int14 \vec{u(t)} dt\), where \(\vec{u(t)}=⟨ln(t)t,t,sin(πt)⟩\)

Answer

TBA

Exercise \(\PageIndex{7}\)

Find the length for the following curves.

1) \(\vec{r(t)}=⟨3(t),4cos(t),4sin(t)⟩\) for \(1≤t≤4\)

2) \(\vec{r(t)}=2\hat{\mathbf{i}}+t\hat{\mathbf{j}}+3t^2\hat{\mathbf{k}}\) for \(0≤t≤1\)

Answer

TBA

Exercise \(\PageIndex{8}\)

Reparameterize the following functions with respect to their arc length measured from \(t=0\) in direction of increasing \(t.\)

1) \(\vec{r(t)}=2t\hat{\mathbf{i}}+(4t-5) \hat{\mathbf{j}}+(1-3t)\hat{\mathbf{k}}\)

2) \(\vec{r(t)}= cos(2t) \hat{\mathbf{i}}+8t \hat{\mathbf{j}}+sin(2t)\hat{\mathbf{k}}\)

3) \(\vec{r((s))}= cos(2s) \hat{\mathbf{i}}+tan(2s) \hat{\mathbf{j}}+sin(3s) \hat{\mathbf{k}}\)

Answer

Add texts here. Do not delete this text first.

Exercise \(\PageIndex{9}\)

Find the curvature for the following vector functions.

1) \(\vec{r(t)}= 2sin(t) \hat{\mathbf{i}}-4t \hat{\mathbf{j}}+2cos(t)\hat{\mathbf{k}}\)

2) \(\vec{r(t)}= 2e^t \hat{\mathbf{i}}+2e^{-t} \hat{\mathbf{j}}+2t\hat{\mathbf{k}}\)

Answer

Add texts here. Do not delete this text first.

Exercise \(\PageIndex{10}\)

1) Find the unit tangent vector, the unit normal vector, and the binormal vector for \(\vec{r(t)}= 2cos(t) \hat{\mathbf{i}}-+3t \hat{\mathbf{j}}+2sin(t)\hat{\mathbf{k}}\).

2) Find the tangential and normal acceleration components with the position vector \(\vec{r(t)}=⟨cost,sint,e^t⟩.\)

3) 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.

4) The position of a particle is given by \(\vec{r(t)}=⟨t^2,ln(t),sin(πt)⟩\), 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?

Answer

\(\vec{v(t)}=⟨2t,\frac{1}{t},π cos(πt)⟩\), \(\vec{a(t)}=⟨2,-\frac{1}{t^2},-π^2 sin(πt)⟩\), speed=\(\sqrt{4t^2+ \frac{1}{t^2}+π^2 cos^(πt)}, and at \(t=1\), \(\vec{r(1)}=⟨1,0,0⟩\), \(\vec{v(t)}=⟨2,1,-π ⟩\), \(\vec{a(t)}=⟨2,-1,0⟩\), speed=\(\sqrt{5+π^2}\).

Exercise \(\PageIndex{11}\)

The following problems consider launching a cannonball out of a cannon. The cannonball is shot out of the cannon with an angle \(θ\) and initial velocity \(v_0.\) The only force acting on the cannonball is gravity, so we begin with a constant acceleration \(\vec{a(t)}=−g\hat{\mathbf{j}}\)

a) Find the velocity vector function \(\vec{v(t)}\)

b) Find the position vector \(\vec{r(t)}\) and the parametric representation for the position.

c) 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?