1. Multiple Choice If a smooth curve $C$ is traced out by the vector function $\mathbf{r}=\mathbf{r}(t)$, $a\leq t\leq b,$ then the parameter $t$ is the arc length if and only if [(a) $\left\Vert \mathbf{r}(t)\right\Vert\;=1$, (b) $\left\Vert \mathbf{r}^{\prime}(t)\right\Vert\;=0$, (c) $\left\Vert \mathbf{r}^{\prime\prime}(t)\right\Vert\;=1$, (d) $\left\Vert \mathbf{r}^{\prime}(t)\right\Vert\;=1]$ for all $t.$

2. True or False The curvature $\kappa$ of a straight line equals the slope of the line.

3. True or False The curvature $\kappa$ of a circle equals the radius of the circle.

4. True or False The curvature $\kappa$ of a twice differentiable smooth curve $C$ equals the magnitude of the rate of change of the unit tangent vector $\mathbf{T}$ with respect to arc length.

5. True or False The curvature $\kappa$ of a twice differentiable smooth curve $C$: $\mathbf{r}=\mathbf{r}(t)$ is $\kappa\;=\;\dfrac{\left\Vert \mathbf{r}^{\prime} (t) \times \mathbf{r} ^{\prime \prime} (t) \right\Vert }{\left\Vert \mathbf{r}(t) \right\Vert ^{3}}.$

6. The curvature of a twice differentiable function $y=f(x)$ is $\kappa$ = _______.

7. The curvature of the circle $x^{2}+\left( y-2\right) ^{2}=9$ is $\kappa$ = _______.

8. True or False The radius $\rho$ of the osculating circle at a point $P$ of a smooth curve $C$ equals $\kappa,$ provided $\kappa \neq 0.$

9. $\mathbf{r}(t)=4\cos t\mathbf{i}-4\sin t\mathbf{j},\quad 0\leq t\leq 2\pi$

10. $\mathbf{r}(t)=\sin (3t)\mathbf{i}+\cos (3t)\mathbf{j,} \quad 0\leq t\leq 2\pi$

11. $\mathbf{r}(t)=t^{2}\mathbf{i}+t\mathbf{j},\quad 0\leq t\leq 4$

12. $\mathbf{r}(t)=t\mathbf{i}+t^{3}\mathbf{j,}\quad 0\leq t\leq 2$

13. $\mathbf{r}(t)=(2t+1)\mathbf{i}+(3t-2)\mathbf{j},\quad 0\leq t\leq 5$

14. $\mathbf{r}(t)=\mathbf{i}+t\mathbf{j},\quad 0\leq t\leq 1$

15. $\mathbf{r}(t)=\left( \dfrac{2}{\sqrt{13}}t+1\right) \mathbf{i}+\left( \dfrac{3}{\sqrt{13}}t-2\right) \mathbf{j}$, $0\leq t\leq 5\sqrt{13}$

16. $\mathbf{r}(t)=\mathbf{i}+t^{2}\mathbf{j},\quad 0\leq t\leq 1$

17. $\mathbf{r}(t)=\sin t\mathbf{i}+\cos t\mathbf{j}+t\mathbf{k}$

18. $\mathbf{r}(t)=a\sin t\mathbf{i}+a\cos t\mathbf{j}+\sqrt{1-a^{2}}\kern1pt t\mathbf{k}$

21. $\mathbf{r}(t)=t^{2}\mathbf{i}+\dfrac{2}{t}\mathbf{j}$

22. $\mathbf{r}(t)=2t\mathbf{i}+t^{3}\mathbf{j}$

23. $\mathbf{r}(t)=2\sin t \mathbf{i}+2\cos t\mathbf{j}$

24. $\mathbf{r}(t)=\cos t\mathbf{i}+2\sin t\mathbf{j}$

25. $\mathbf{r}(t)=(3t-t^{3})\mathbf{i}+3t^{2} \mathbf{j}$

26. $\mathbf{r}(t)=(3t-t^{3})\mathbf{i}+(3t+t^{3})\mathbf{j}$

27. $\mathbf{r}(t)=t\mathbf{i}+2t\mathbf{j}+t\mathbf{k}$

28. $\mathbf{r}(t)=2t\mathbf{i}+t\mathbf{j}+3t\mathbf{k}$

29. $\mathbf{r}(t)=\sin ( 2t) \mathbf{i}+\cos ( 2t) \mathbf{j}+t\mathbf{k}$

30. $\mathbf{r}(t)=\sin t\mathbf{i}\,{+}\,\cos t\mathbf{j}\,{+}\,bt\mathbf{k}$

31. $\mathbf{r}(t)=e^{t}\mathbf{i}+e^{-t}\mathbf{j}+\sqrt{2}t\mathbf{k}$

32. $\mathbf{r}(t)=e^{t}\mathbf{i}+e^{2t}\mathbf{j}+e^{-t}\mathbf{k}$

33. $\mathbf{r}(t)=\cos ^{3}t\mathbf{i}+\sin ^{3}t\mathbf{j}+\mathbf{k}$

34. $\mathbf{r}(t)=4\cos ^{3}t\mathbf{i}+ 3\mathbf{j} + 4\sin^{3}t\mathbf{k}$

35. $y=x^{2}$ at $(1,1)$

36. $y=2x-x^{2}$ at $(1,1)$

37. $y=x^{2}-x^{3}$ at $(1,0)$

38. $y=x^{-3/2}$ at $(1,1)$

39. $y=\sqrt{x}$ at $\big(2,\sqrt{2}\big)$

40. $y=\dfrac{1}{\sqrt{x}}$ at $(1,1)$

41. $4x^{2}+9y^{2}=36$ at $(0,2)$

42. $y=\sec x-1$ at $\left(\! \dfrac{\pi}{4}, \sqrt{2}-1\!\right)$

43. $y=e^{x}$ at $(0,1)$

44. $y=\ln (x+1)$ at $( 2,\ln 3)$

45. $y=x^{3}-6x$ at $(1,-5)$

46. $y=\dfrac{1}{x^{2}}$ at $(-1,1)$

47. $y=\sin x$ at $\left( \dfrac{\pi }{2},1\right)$

48. $y=e^{-x}$ at $(0,1)$

49. $x^{2}+xy+y^{2}=3$ at $(1,1)$

50. $y^{2}-y+x=0$ at $(0,0)$

51. $y=\ln ( \sec x)$ at $\left(\dfrac{\pi }{4}, \ln \sqrt{2}\right)$

52. $y=\cosh x$ at $(0,1)$

53. $\mathbf{r}(t)=3t^{2}\mathbf{i}+(3t-t^{3})\mathbf{j,}\quad t=1$

54. $\mathbf{r}(t)=t\mathbf{i}+t^{2}\mathbf{j}+t^{3}\mathbf{k},\quad t=1$

55. $\mathbf{r}(t)=\sin t\mathbf{i}+\cos ( 2t) \mathbf{j},\quad t=\dfrac{\pi }{4}$

56. $\mathbf{r}(t)=\sin t\mathbf{i}+\cos t\mathbf{j}+bt\mathbf{k,}\quad b>0,\quad t=\dfrac{\pi }{4}$

57. $\mathbf{r}(t)=\sin ( 2t) \mathbf{i}+\cos ( 2t) \mathbf{j}+t\mathbf{k,}\quad t=\dfrac{\pi }{4}$

58. $\mathbf{r}(t)=\cos ^{3}t\mathbf{i}+\sin ^{3}t\mathbf{j}+\mathbf{k,}\quad t=\dfrac{\pi }{3}$

59. $\mathbf{r}(t)=e^{t}\mathbf{i}+e^{-t}\mathbf{j}+\sqrt{2}\kern1ptt\mathbf{k},\quad t=0$

60. $\mathbf{r}(t)=a(3t-t^{3})\mathbf{i}+3at^{2}\mathbf{j}+a(3t+t^{3})\mathbf{k,}\quad a>0, \quad t=1$

61. $\mathbf{r}(t)=4a\cos ^{3}t\mathbf{i}+4a\sin ^{3}t\mathbf{j}+3a\cos ( 2t) \mathbf{k,}\quad a>0,\quad t=\dfrac{\pi }{4}$

62. $\mathbf{r}(t)=t\mathbf{i}+2t\mathbf{j}+\sqrt{1-5t^{2}}\kern1pt\mathbf{k},\quad -\dfrac{\sqrt{5}}{5} < t< \dfrac{\sqrt{5}}{5},\quad t=0$

63. Radius of Curvature Show that the radius of curvature of the parabola $y=ax^{2}+bx+c$ is a minimum at its vertex.

64. Radius of Curvature Show that the radii of curvature at the ends of the axes of the ellipse $b^{2}x^{2}+a^{2}y^{2}=a^{2}b^{2}$ are $\dfrac{b^{2}}{a}$ and $\dfrac{a^{2}}{b},\quad a>0, b>0.$

65. Maximum Curvature Find the point on the curve $y=\ln x$ at which the curvature is maximum.

66. Maximum Curvature Find the point on the curve $y=e^{x}$ at which the curvature is maximum.

67. Maximum Curvature Find the point(s) on the curve $y=\dfrac{1}{3}x^{3}$ at which the curvature is maximum.

68. Maximum Curvature Find the point(s) on the curve $y=\sin x$ at which the curvature is maximum.

69. Maximum Curvature Find $\alpha >0$ so that $\mathbf{r} (t)= \alpha \cos t\mathbf{i} + \alpha \sin t\mathbf{j}+t\mathbf{k}$ has maximum curvature.

70. Curvature of a Plane Curve What is the curvature at a point of inflection of a plane curve?

71. Curvature of a Catenary Show that the curvature of the catenary $y=a\cosh \dfrac{x}{a}$, $a>0$, at any point $(x,y)$ is $\dfrac{a}{y^{2}}$.

72. Curvature of a Cissoid Find the curvature of the cissoid $y^{2}(2-x)=x^{3}$ at the point $(1,1)$. See the figure.

    

73. Curvature of a Cycloid Find the curvature of the cycloid $x( \theta)\;=\;\theta -\sin \theta$ and $y( \theta )\;=\;1-\cos \theta$ at the highest point of an arch. See the figure.

    

74. 1. (a) Find the curvature of the curve $\mathbf{r} (t)=(1-t^{3})\mathbf{i}+t^{2}\mathbf{j}$.
    2. (b) Graph $\mathbf{r}=\mathbf{r}(t)$. Where is the curvature undefined? Do you see any geometric reason for this?

75. Curvature of a Spiral Find the curvature of the spiral $\mathbf{r}(t)=e^{-t}\cos t\mathbf{i}+e^{-t}\sin t\mathbf{j}$ shown in the figure. How does the curvature behave when $t\rightarrow \infty$? Do you see any geometric reason for this?

    

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76. Curvature Find the curvature $\kappa$ of the curve $\mathbf{r}(t)=2a\cos t\mathbf{i}+2a\sin t\mathbf{j}+bt^{2}\mathbf{k},\qquad a>0, b>0.$

77. Curvature Show that the curvature of an ellipse $\dfrac{x^{2}}{ a^{2}}+\dfrac{y^{2}}{b^{2}}=1,$ $a> b,$ is a maximum at the points $\left( \pm a,0\right)$ and is a minimum at the points $\left( 0,\pm b\right).$

    Source: Contributed by the students at the University of Missouri.

78. Curvature Show that the circular helix $\mathbf{r}( t)\;=\;a\cos t\mathbf{i}+a\sin t\mathbf{j}+t\mathbf{k}$ has constant curvature.

    Source: Contributed by the students at the University of Missouri.

79. Write a vector equation for the curve $C$ traced out by $\mathbf{r}(t)=2t\mathbf{i}+(2t-1)\mathbf{j}+t\mathbf{k}$, $0\leq t\leq 2$, using arc length $s$ as the parameter. [Hint: For each $t$, $0\leq t\leq 2$, calculate the length $s(t)$ of the curve from $0$ to $t$.]

80. Compare the solutions of Problems 13 and 15. Then show how to change the parameter $t$ of the line $\mathbf{r}(t)=(at+b)\mathbf{i}+(ct+d) \mathbf{j}$, where either $a\neq 0$ or $c\neq 0$, to one that is arc length as measured along the line.

81. Suppose a smooth curve $C$ is traced out by a twice differentiable vector function $\mathbf{r}=\mathbf{r}(t)$, $a\leq t\leq b$. If the curvature $\kappa \neq 0$ at a point $P$ on $C$, show that the position vector $\mathbf{C}$ of the center of the osculating circle at $P$ is given by $$\begin{equation*} \mathbf{C}(t)=\mathbf{r}(t)+\rho \mathbf{N}(t) \end{equation*}$$ where $\mathbf{N}$ is the principal unit normal vector to $C$ at $P$, and $\rho\;=\;\dfrac{1}{\kappa }$.

82. Use the result of Problem 81 to find the center and radius of the osculating circle for the helix $\mathbf{r}(t)=a\sin t\mathbf{i}+a\cos t\mathbf{j}+a^{2}t\mathbf{k}$, $a>0;$ $$\begin{equation*} {\bf (a)}\enspace at\enspace t=\dfrac{\pi }{2}.\qquad {\bf (b)}\enspace at\enspace t=\pi. \end{equation*}$$

83. Show that the three vectors $\mathbf{T}, \mathbf{N}$, and $\mathbf{B}$ form a collection of mutually orthogonal unit vectors at each point on $C$.

84. Show that $\dfrac{d\mathbf{T}}{ds}=\kappa (s)\mathbf{N}(s)$.

85. Show that $\dfrac{d\mathbf{B}}{ds}$ is orthogonal to both $\mathbf{B}(s)$ and $\mathbf{T}(s).$

86. If the torsion $\tau (s)$ of $C$ is defined by the equation $\dfrac{d\mathbf{B}}{ds}=-\tau \mathbf{N}$, show that $\dfrac{d\mathbf{N}}{ds}=\tau \mathbf{B}-\kappa \mathbf{T}$.

87. Find $\kappa, \mathbf{T}, \mathbf{N}$, and $\mathbf{B}$ for $\mathbf{r}(s)=\dfrac{1}{\sqrt{2}}\left[ \sin s\mathbf{i}+\cos s\mathbf{j}+s\mathbf{k}\right].$

88. Curvature of a Polar Curve Show that the formula for the curvature of a polar curve $r=f(\theta )$ is $$\begin{eqnarray*} \\[-24pt] \kappa\;=\;\frac{\left\vert r^{2}+2\!\left( \dfrac{dr}{d\theta }\right) ^{2}-r\left( \dfrac{d^{2}r}{d\theta ^{2}}\right) \right\vert }{\left[ r^{2}+\left( \dfrac{dr}{d\theta }\right) ^{2}\right] ^{3/2}} \end{eqnarray*}$$

89. $r=2\cos ( 2\theta )$ at $\theta\;=\;\dfrac{\pi }{12}$

90. $r=e^{a\theta }$ at $\theta\;=\;\dfrac{\pi }{2}, a>0$

91. $r=a\theta$ at $\theta\;=\;1$ and $a>0$

92. $r=1-\cos \theta$ at $\theta\;=\;0$

93. $r=3-2\sin \theta$ at $\theta\;=\;\dfrac{\pi }{6}$

94. $r=2+3\cos \theta$ at $\theta\;=\;\dfrac{\pi }{3}$

95. Use the figure below to show that the coordinates $(h,k)$ of the center of curvature of $y=f(x)$ are $$\begin{equation*} h=x-\rho \sin \phi\qquad k=y+\rho \cos \phi \end{equation*}$$ where $\rho$ is the radius of curvature. Show that $$\begin{equation*} \sin \phi\;=\;\frac{y^{\prime} }{\sqrt{1+( y^{\prime} ) ^{2}}}\qquad \hbox{and}\qquad \cos \phi\;=\;\frac{1}{\sqrt{1+( y^{\prime} ) ^{2}}} \end{equation*}$$ so $$\begin{equation*} h=x-\frac{y^{\prime} \left[ 1+( y^{\prime} ) ^{2}\right] }{y^{\prime \prime}}\qquad \hbox{and}\qquad k=y+\frac{1+( y^{\prime} ) ^{2}}{y^{\prime \prime} } \end{equation*}$$

96. $y=x^{2}$ at $x=1$

97. $y=\sin x$ at $x=\dfrac{\pi }{2}$

98. $y=\dfrac{x}{x+1}$ at $(0, 0)$

99. $x^{3}+y^{3}=4xy$ at $(2, 2)$

100. $xy=4$ at $x=2$

101. As a point $P$ moves along a curve $C$, the center of curvature corresponding to $P$ traces out a curve $C_{1}$ called the evolute of $C$; conversely, $C$ is the involute of $C_{1}$. Show that parametric equations of the evolute of $y=\dfrac{1}{2}x^{2}$ are $h=-x^{3}$, $k=\dfrac{3}{2}x^{2}+1$. Then eliminate the parameter $x$ to obtain $$\begin{equation*} h^{2}=\frac{8}{27}\left( k-1\right) ^{3} \end{equation*}$$

102. Refer to Problem 101. Find parametric equations and a rectangular equation for the evolute of $\mathbf{r}( t)\;=\;a\cos t\mathbf{i}+b\sin t\mathbf{j}$, $a>0, b>0, 0\leq t\leq 2\pi$.