# Chapter 12 Review Exercises

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

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 $$\dfrac{\vecs r′(t)}{\|\vecs r′(t)\|}.$$

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

$$t<4, \; t≠\dfrac{nπ}{2}$$

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

7. [T] $$\vecs r(t)=⟨t^2,\, t^3⟩$$

8. [T] $$\vecs r(t)=⟨\sin(20t)e^{−t}, \, \cos(20t)e^{−t}, \, e^{−t}⟩$$ 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 $$z=y−4$$

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

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

$$\vecs u′(t)=⟨2t, \, 2, \, 20t^4⟩,$$
$$\vecs u″(t)=⟨2, \, 0, \, 80t^3⟩,$$
$$\dfrac{d}{dt}\Big[\vecs u′(t)×\vecs u(t)\Big]=⟨−480t^3−160t^4, \, 24+75t^2, \, 12+4t⟩,$$
$$\dfrac{d}{dt}\Big[\vecs u(t)×\vecs u′(t)\Big]=⟨480t^3+160t^4, \, -24-75t^2, \, -12-4t⟩,$$
$$\dfrac{d}{dt}\Big[\vecs u(t)⋅\vecs u′(t)\Big]=720t^8−9600t^3+6t^2+4,$$
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 $$\vecs u(t)=\left\langle\dfrac{\ln t}{t}, \, \dfrac{1}{\sqrt{t}}, \, \sin\left(\frac{tπ}{4}\right)\right\rangle$$

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

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 $$0≤t≤1$$

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

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

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

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

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

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 $$\vecs r(t)=⟨\cos t,\, \sin t, \, e^t⟩.$$

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

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

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?

$$\vecs v(t)=\left\langle 2t,\, \frac{1}{t}, \, \pi\cos(πt)\right\rangle\text{ m/sec},$$
$$\vecs a(t)=\left\langle 2, \, −\frac{1}{t^2}, \, −\pi^2\sin(πt) \right\rangle\text{ m/sec}^2,$$
$$\text{speed}(t)=\sqrt{4t^2+\frac{1}{t^2}+\pi^2\cos^2(πt)}\text{ m/sec}$$;
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.

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