13.R: Chapter 13 Review Exercises

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 \(\vecs r(t)=⟨t^2,\, 3t+1,\, t−2⟩,\) where \(P(1,4,−1)\) and \(Q(16,11,2).\)

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

Answer

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

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

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

Answer

False, it is \(\|\vecs r′(t)\|\)

Find the domains of the vector-valued functions.

5. \(\vecs r(t)=⟨\sin(t),\, \ln(t),\, \sqrt{t}⟩\)

6. \(\vecs r(t)=\left\langle e^t,\,\dfrac{1}{\sqrt{4−t}},\,\sec t\right\rangle\)

Answer

\(t<4, \; t≠\dfrac{nπ}{2}\)

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

7. [T] \(\vecs r(t)=⟨t^2,\, t^3⟩\)

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

Answer

Find a vector function that describes the following curves.

9. Intersection of the cylinder \(x^2+y^2=4\) with the plane \(x+z=6\)

10. Intersection of the cone \(z=\sqrt{x^2+y^2}\) and plane \(z=y−4\)

Answer

\(\vecs r(t)=\left\langle t, \, 2-\frac{t^2}{8},\, -2 - \frac{t^2}{8}\right\rangle\)

Find the derivatives of \(\vecs u(t), \, \vecs u′(t), \, \vecs u′(t)×\vecs u(t), \, \vecs u(t)×\vecs u′(t),\) and \(\vecs u(t)·\vecs u′(t)\). Find the unit tangent vector.

11. \(\vecs u(t)=⟨e^t, \, e^{−t}⟩\)

12. \(\vecs u(t)=⟨t^2,\, 2t+6, \, 4t^5−12⟩\)

Answer

\(\vecs u′(t)=⟨2t, \, 2, \, 20t^4⟩,\)
\(\vecs u″(t)=⟨2, \, 0, \, 80t^3⟩,\)
\(\dfrac{d}{dt}\Big[\vecs u′(t)×\vecs u(t)\Big]=⟨−480t^3−160t^4, \, 24+75t^2, \, 12+4t⟩,\)
\(\dfrac{d}{dt}\Big[\vecs u(t)×\vecs u′(t)\Big]=⟨480t^3+160t^4, \, -24-75t^2, \, -12-4t⟩,\)
\(\dfrac{d}{dt}\Big[\vecs u(t)⋅\vecs u′(t)\Big]=720t^8−9600t^3+6t^2+4,\)
unit tangent vector: \(\vecs T(t)=\dfrac{2t}{\sqrt{400t^8+4t^2+4}}\,\mathbf{\hat i}+\dfrac{2}{\sqrt{400t^8+4t^2+4}}\,\mathbf{\hat j}+\dfrac{20t^4}{\sqrt{400t^8+4t^2+4}}\,\mathbf{\hat k}\)

Evaluate the following integrals.

13. \(\displaystyle ∫\left(\tan(t)\sec(t)\,\mathbf{\hat i}−te^{3t}\,\mathbf{\hat j}\right)\, dt\)

14. \(\displaystyle ∫_1^4 \vecs u(t) \, dt,\) with \(\vecs u(t)=\left\langle\dfrac{\ln t}{t}, \, \dfrac{1}{\sqrt{t}}, \, \sin\left(\frac{tπ}{4}\right)\right\rangle\)

Answer

\(\dfrac{\ln(4^2)}{2}\,\mathbf{\hat i}+2\,\mathbf{\hat j}+\dfrac{2(2+\sqrt{2})}{\pi}\,\mathbf{\hat k}\)

Find the length for the following curves.

15. \(\vecs r(t)=⟨3t,\, 4\cos t, \, 4\sin t ⟩\) for \(1≤t≤4\)

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

Answer

\(\dfrac{\sqrt{37}}{2}+\frac{1}{12}\sinh^{−1} 6\)

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

17. \(\vecs r(t)=2t\,\mathbf{\hat i}+(4t−5)\,\mathbf{\hat j}+(1−3t)\,\mathbf{\hat k}\)

18. \(\vecs r(t)=\cos(2t)\,\mathbf{\hat i}+8t\,\mathbf{\hat j}−\sin(2t)\,\mathbf{\hat k}\)

Answer

\(\vecs r(t(s))=\cos\left(\frac{2s}{\sqrt{65}}\right)\,\mathbf{\hat i}+\frac{8s}{\sqrt{65}}\,\mathbf{\hat j}−\sin\left(\frac{2s}{\sqrt{65}}\right)\,\mathbf{\hat k}\)

Find the curvature for the following vector functions.

19. \(\vecs r(t)=(2\sin t)\,\mathbf{\hat i}−4t\,\mathbf{\hat j}+(2\cos t)\,\mathbf{\hat k}\)

20. \(\vecs r(t)=\sqrt{2}e^t\,\mathbf{\hat i}+\sqrt{2}e^{−t}\,\mathbf{\hat j}+2t\,\mathbf{\hat k}\)

Answer

\(\dfrac{e^{2t}}{\left(e^{2t}+1\right)^2}\)

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

22. Find the tangential and normal acceleration components with the position vector \(\vecs r(t)=⟨\cos t,\, \sin t, \, e^t⟩.\)

Answer

\(a_T=\dfrac{e^{2t}}{1+e^{2t}},\)

\(a_N=\dfrac{\sqrt{2e^{2t}+4e^{2t}\sin t\cos t+1}}{1+e^{2t}}\)

23. 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.

24. The position of a particle is given by \(\vecs r(t)=⟨t^2, \, \ln t, \, \sin(πt)⟩,\) 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

\(\vecs v(t)=\left\langle 2t,\, \frac{1}{t}, \, \pi\cos(πt)\right\rangle\text{ m/sec},\)
\(\vecs a(t)=\left\langle 2, \, −\frac{1}{t^2}, \, −\pi^2\sin(πt) \right\rangle\text{ m/sec}^2,\)
\(\text{speed}(t)=\sqrt{4t^2+\frac{1}{t^2}+\pi^2\cos^2(πt)}\text{ m/sec}\);
At \(t=1,\; \vecs r(1)=⟨1,0,0⟩\) m, \(\vecs v(1)=⟨2,−1,\pi⟩\) m/sec, \(\vecs a(1)=⟨2,−1,0⟩\) m/sec², and \(\text{speed}(1) =\sqrt{5+\pi^2}\) m/sec

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 \(\vecs v_0\). The only force acting on the cannonball is gravity, so we begin with a constant acceleration \(\vecs a(t)=−g\,\mathbf{\hat j}\).

25. Find the velocity vector function \(\vecs v(t).\)

26. Find the position vector \(\vecs r(t)\) and the parametric representation for the position.

Answer

\(\vecs r(t)=\vecs v_0t−\dfrac{gt^2}{2}\,\mathbf{\hat j},\)
\(\vecs r(t)=⟨v_0(\cos θ)t,\,v_0(\sin θ)t,−\dfrac{gt^2}{2}⟩\) where \(v_0 = \|\vecs v_0\|.\)

27. 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?

Contributors

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.