Solution: Finding radii of curvature Find the radius of

15-22. Curvature Find the unit tangent vector T and the curvature k for the following parameterized curves. \(\mathbf{r}(t)=\langle t, \ln (\cos t)\rangle\)

50-53. Curvature for plane curves Use the result of Exercise 49 to find the curvature function of the following curves. \(\mathbf{r}(t)=\langle a \sin t, b \cos t\rangle\) (ellipse)

9-14. Arc length parameterization Determine whether the following curves use arc length as a parameter If not, find a description that uses arc length as a parameter. \(\mathbf{r}(t)=\langle t, 2 t\rangle, \text { for } 0 \leq t \leq 3\)

Finding radii of curvature Find the radius of curvature (see Exercise 64) of the following curves at the given point. Then write the equation of the circle of curvature at the point. \(\mathbf{r}(t)=\left\langle t, t^{2}\right\rangle \quad \text { (parabola) } \quad \text { at } t=0\)

When appropriate, consider using the special formulas derived in Exercises 44 and 49 in the remaining exercises. Same paths, different velocity The position functions of objects A and B describe different motion along the same path for \(t \geq 0\). a. Sketch the path followed by both A and B. b. Find the velocity and acceleration of A and B and discuss the differences c. Express the acceleration of A and B in terms of the tangential and normal components and discuss the differences. \(A: \mathbf{r}(t)=\langle\cos t, \sin t\rangle, B: \mathbf{r}(t)=\langle\cos 3 t, \sin 3 t\rangle\)

29-36. Principal unit normal vector Find the unit tangent vector T and the principal unit normal vector N for the following parameterized curves. In each case, verify that \(|\mathbf{T}|=|\mathbf{N}|=1 \text { and } \mathbf{T} \cdot \mathbf{N}=0\). \(\mathbf{r}(t)=\left\langle\cos ^{3} t, \sin ^{3} t\right\rangle\)

Give a practical formula for computing the curvature.

23-28. Alternative curvature formula Use the alternative curvature formula \(\kappa=|\mathbf{a} \times \mathbf{v}| /|\mathbf{v}|^{3}\) to find the curvature of the following parameterized curves. \(\mathbf{r}(t)=\left\langle e^{t} \cos t, e^{t} \sin t, t\right\rangle\)

15-22. Curvature Find the unit tangent vector T and the curvature k for the following parameterized curves. \(\mathbf{r}(t)=\left\langle\cos t^{2}, \sin t^{2}\right\rangle\)

Practical formula for N Show that the definition of the principal unit normal vector \(\mathbf{N}=\frac{d \mathbf{T} / d s}{|d \mathbf{T} / d s|}\) implies the practical formula \(\mathbf{N}=\frac{d \mathbf{T} / d t}{|d \mathbf{T} / d t|}\). Use the Chain Rule and recall that |v|=ds/dt>0.

23-28. Alternative curvature formula Use the alternative curvature formula \(\kappa=|\mathbf{a} \times \mathbf{v}| /|\mathbf{v}|^{3}\) to find the curvature of the following parameterized curves. \(\mathbf{r}(t)=\left\langle 4+t^{2}, t, 0\right\rangle\)

Is the curve \(\mathbf{r}(t)=\langle t, t, t\rangle\) parameterized by its arc length? Explain.

Explain what it means for a curve to be parameterized by its arc length.

When appropriate, consider using the special formulas derived in Exercises 44 and 49 in the remaining exercises. 54-57. Same paths, different velocity The position functions of objects A and B describe different motion along the same path for \(t \geq 0\). a. Sketch the path followed by both A and B. b. Find the velocity and acceleration of A and B and discuss the differences c. Express the acceleration of A and B in terms of the tangential and normal components and discuss the differences. \(A: \mathbf{r}(t)=\langle t, 2 t, 3 t\rangle, B: \mathbf{r}(t)=\left\langle t^{2}, 2 t^{2}, 3 t^{2}\right\rangle\)

When appropriate, consider using the special formulas derived in Exercises 44 and 49 in the remaining exercises. 54-57. Same paths, different velocity The position functions of objects A and B describe different motion along the same path for \(t \geq 0\). a. Sketch the path followed by both A and B. b. Find the velocity and acceleration of A and B and discuss the differences c. Express the acceleration of A and B in terms of the tangential and normal components and discuss the differences. \(A: \mathbf{r}(t)=\langle 1+2 t, 2-3 t, 4 t\rangle, B: \mathbf{r}(t)=\langle 1+6 t, 2-9 t, 12 t\rangle\)

Explain in words the meaning of the curvature of a curve. Is it a scalar function or a vector function?

Graphs of the curvature Consider the following curves. a. Graph the curve. b. Compute the curvature (using either of the two formulas). c. Graph the curvature as a function of the parameter. d. identify the points (if any) at which the curve has a minimum and maximum curvature. e. Verify that the graph of the curvature is consistent with the graph of the curve. \(\mathbf{r}(t)=\left\langle t^{2} / 2, t^{3} / 3\right\rangle, \quad\) for \(t>0\)

Give a practical formula for computing the principal unit normal vector.

15-22. Curvature Find the unit tangent vector T and the curvature k for the following parameterized curves. \(\mathbf{r}(t)=\langle\sqrt{3} \sin t, \sin t, 2 \cos t\rangle\)

29-36. Principal unit normal vector Find the unit tangent vector T and the principal unit normal vector N for the following parameterized curves. In each case, verify that \(|\mathbf{T}|=|\mathbf{N}|=1 \text { and } \mathbf{T} \cdot \mathbf{N}=0\). \(\mathbf{r}(t)=\langle 2 \sin t, 2 \cos t\rangle\)

Arc length parameterization Prove that the line \(\mathbf{r}(t)=\left\langle x_{0}+a t, y_{0}+b t\), \(z_{0}+c t\right\rangle\) is parameterized by arc length provided \(a^{2}+b^{2}+c^{2}=1\).

15-22. Curvature Find the unit tangent vector T and the curvature k for the following parameterized curves. \(\mathbf{r}(t)=\langle 2 \cos t,-2 \sin t\rangle\)

Graphs of the curvature Consider the following curves. a. Graph the curve. b. Compute the curvature (using either of the two formulas). c. Graph the curvature as a function of the parameter. d. identify the points (if any) at which the curve has a minimum and maximum curvature. e. Verify that the graph of the curvature is consistent with the graph of the curve. \(\mathbf{r}(t)=\left\langle t, t^{2}\right\rangle, \quad\) for \(-2 \leq t \leq 2\) (parabola)

37-42. Components of the acceleration Consider the following trajectories of moving objects. Find the tangential and normal components of the acceleration. \(\mathbf{r}(t)=\left\langle t^{3}, t^{2}\right\rangle\)

23-28. Alternative curvature formula Use the alternative curvature formula \(\kappa=|\mathbf{a} \times \mathbf{v}| /|\mathbf{v}|^{3}\) to find the curvature of the following parameterized curves. \(\mathbf{r}(t)=\langle-3 \cos t, 3 \sin t, 0\rangle\)

Explain how to decompose the acceleration vector of a moving object into its tangential and normal components.

Special formula: Curvature for y =f(x) Assume that f is twice differentiable and prove that the curve y = f(x) has curvature \(\kappa(x)=\frac{\left|f^{\prime \prime}(x)\right|}{\left(1+f^{\prime}(x)^{2}\right)^{3 / 2}}\). (Hint: Use the parametric description x = t, y = f(t).)

29-36. Principal unit normal vector Find the unit tangent vector T and the principal unit normal vector N for the following parameterized curves. In each case, verify that \(|\mathbf{T}|=|\mathbf{N}|=1 \text { and } \mathbf{T} \cdot \mathbf{N}=0\). \(\mathbf{r}(t)=\left\langle\cos t^{2}, \sin t^{2}\right\rangle\)

9-14. Arc length parameterization Determine whether the following curves use arc length as a parameter If not, find a description that uses arc length as a parameter. \(\mathbf{r}(t)=\langle 2 \cos t, 2 \sin t\rangle, \text { for } 0 \leq t \leq 2 \pi\)

37-42. Components of the acceleration Consider the following trajectories of moving objects. Find the tangential and normal components of the acceleration. \(\mathbf{r}(t)=\left\langle t, t^{2}+1\right\rangle\)

Descartes' four-circle solution Consider the four mutually tangent circles shown in the figure that have radii a, b, c, and d, and curvatures A=1/a, B=1/b, C=1/c, and D=1/d. Prove Descartes' result (1643) that \((A+B+C+D)^{2}=2\left(A^{2}+B^{2}+C^{2}+D^{2}\right)\).

15-22. Curvature Find the unit tangent vector T and the curvature k for the following parameterized curves. \(\mathbf{r}(t)=\left\langle\cos ^{3} t, \sin ^{3} t\right\rangle\)

Finding radii of curvature Find the radius of curvature (see Exercise 64) of the following curves at the given point. Then write the equation of the circle of curvature at the point. y=sin x at \(x=\pi / 2\)

23-28. Alternative curvature formula Use the alternative curvature formula \(\kappa=|\mathbf{a} \times \mathbf{v}| /|\mathbf{v}|^{3}\) to find the curvature of the following parameterized curves. \(\mathbf{r}(t)=\langle\sqrt{3} \sin t, \sin t, 2 \cos t\rangle\)

45-48. Curvature for y = f(x) Use the result of Exercise 44 to find the curvature function of the following curves. \(f(x)=\ln x\)

37-42. Components of the acceleration Consider the following trajectories of moving objects. Find the tangential and normal components of the acceleration. \(\mathbf{r}(t)=\langle t, 1+4 t, 2-6 t\rangle\)

15-22. Curvature Find the unit tangent vector T and the curvature k for the following parameterized curves. \(\mathbf{r}(t)=\langle 2 t, 4 \sin t, 4 \cos t\rangle\)

Interpret the principal unit normal vector of a curve. Is it a scalar function or a vector function?

Graphs of the curvature Consider the following curves. a. Graph the curve. b. Compute the curvature (using either of the two formulas). c. Graph the curvature as a function of the parameter. d. identify the points (if any) at which the curve has a minimum and maximum curvature. e. Verify that the graph of the curvature is consistent with the graph of the curve. \(\mathbf{r}(t)=\langle t-\sin t, 1-\cos t\rangle, \quad\) for \(0 \leq t \leq 2 \pi\) (cycloid)

15-22. Curvature Find the unit tangent vector T and the curvature k for the following parameterized curves. \(\mathbf{r}(t)=\left\langle t, 2 t^{2}\right\rangle\)

37-42. Components of the acceleration Consider the following trajectories of moving objects. Find the tangential and normal components of the acceleration. \(\mathbf{r}(t)=\langle\cos t, 6 \sin t, \sqrt{5} \cos t\rangle\)

Arc length parameterization Prove that the curve \(\mathbf{r}(t)=\langle a \cos t, b \sin t, c \sin t\rangle\) is parameterized by arc length provided \(a^{2}=b^{2}+c^{2}=1\).

When appropriate, consider using the special formulas derived in Exercises 44 and 49 in the remaining exercises. Same paths, different velocity The position functions of objects A and B describe different motion along the same path for \(t \geq 0\). a. Sketch the path followed by both A and B. b. Find the velocity and acceleration of A and B and discuss the differences c. Express the acceleration of A and B in terms of the tangential and normal components and discuss the differences. \(A: \mathbf{r}(t)=\langle\cos t, \sin t\rangle, B: \mathbf{r}(t)=\left\langle\cos t^{2}, \sin t^{2}\right\rangle\)

29-36. Principal unit normal vector Find the unit tangent vector T and the principal unit normal vector N for the following parameterized curves. In each case, verify that \(|\mathbf{T}|=|\mathbf{N}|=1 \text { and } \mathbf{T} \cdot \mathbf{N}=0\). \(\mathbf{r}(t)=\left\langle t^{2}, t\right\rangle\)

Zero curvature Prove that the curve \(\mathbf{r}(t)=\left\langle a+b t^{p}, c+d t^{p}, e+f t^{p}\right\rangle\), where a, b, c, d, e, f are real numbers and p is a positive integer, has zero curvature. Give an explanation.

50-53. Curvature for plane curves Use the result of Exercise 49 to find the curvature function of the following curves. \(\mathbf{r}(t)=\left\langle t, a t^{2}\right\rangle\) (parabola)

Finding radii of curvature Find the radius of curvature (see Exercise 64) of the following curves at the given point. Then write the equation of the circle of curvature at the point. y=ln x at x=1

37-42. Components of the acceleration Consider the following trajectories of moving objects. Find the tangential and normal components of the acceleration. \(\mathbf{r}(t)=\langle 10 \cos t,-10 \sin t\rangle\)

29-36. Principal unit normal vector Find the unit tangent vector T and the principal unit normal vector N for the following parameterized curves. In each case, verify that \(|\mathbf{T}|=|\mathbf{N}|=1 \text { and } \mathbf{T} \cdot \mathbf{N}=0\). \(\mathbf{r}(t)=\langle t, \ln (\cos t)\rangle\)

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The position, unit tangent, and principal unit normal vectors (r, T, and N) at a point lie in the same plane. b. The vectors T and N at a point depend on the orientation of a Curve. c. The curvature at a point depends on the orientation of a curve. d. An object with unit speed (|v| = 1) on a circle of radius R has an acceleration of a = N/R.

Curvature of the sine curve The function f(x)=sin nx, where n is a positive real number, has a local maximum at \(x=\pi /(2 n)\). Compute the curvature \(\kappa\) of f at this point. How does \(\kappa\) vary (if at all) as n varies?

45-48. Curvature for y = f(x) Use the result of Exercise 44 to find the curvature function of the following curves. \(f(x)=x^{2}\)

15-22. Curvature Find the unit tangent vector T and the curvature k for the following parameterized curves. \(\mathbf{r}(t)=\langle 2 t+1,4 t-5,6 t+12\rangle\)

37-42. Components of the acceleration Consider the following trajectories of moving objects. Find the tangential and normal components of the acceleration. \(\mathbf{r}(t)=\langle 20 \cos t, 20 \sin t, 30 t\rangle\)

23-28. Alternative curvature formula Use the alternative curvature formula \(\kappa=|\mathbf{a} \times \mathbf{v}| /|\mathbf{v}|^{3}\) to find the curvature of the following parameterized curves. \(\mathbf{r}(t)=\langle 7 \cos t, \sqrt{3} \sin t, 2 \cos t\rangle\)

23-28. Alternative curvature formula Use the alternative curvature formula \(\kappa=|\mathbf{a} \times \mathbf{v}| /|\mathbf{v}|^{3}\) to find the curvature of the following parameterized curves. \(\mathbf{r}(t)=\langle t, 8 \sin t, 8 \cos t\rangle\)

Is the curve \(\mathbf{r}(t)=\langle\cos t, \sin t\rangle\) parameterized by its arc length? Explain.

45-48. Curvature for y = f(x) Use the result of Exercise 44 to find the curvature function of the following curves. \(f(x)=\sqrt{a^{2}-x^{2}}\)

50-53. Curvature for plane curves Use the result of Exercise 49 to find the curvature function of the following curves. \(\mathbf{r}(t)=\langle a \sin t, a \cos t\rangle\) (circle)

Parabolic trajectory In Example 8 it was shown that for the parabolic trajectory \(\mathbf{r}(t)=\left\langle t, t^{2}\right\rangle\), \(\mathbf{a}=\langle 0,2\rangle\) and \(\mathbf{a}=\frac{2}{\sqrt{1+4 t^{2}}}(\mathbf{N}+2 t \mathbf{T})\). Show that the second equation for a reduces to the first equation.

Graphs of the curvature Consider the following curves. a. Graph the curve. b. Compute the curvature (using either of the two formulas). c. Graph the curvature as a function of the parameter. d. identify the points (if any) at which the curve has a minimum and maximum curvature. e. Verify that the graph of the curvature is consistent with the graph of the curve. \(\mathbf{r}(t)=\langle t, \sin t\rangle, \quad\) for \(0 \leq t \leq \pi\) (sine curve)

Curvature of In x Find the curvature of f(x) = In x, for x > 0, and find the point at which it is a maximum. What is the value of the maximum curvature?

9-14. Arc length parameterization Determine whether the following curves use arc length as a parameter If not, find a description that uses arc length as a parameter. \(\mathbf{r}(t)=\left\langle t^{2}, 2 t^{2}, 4 t^{2}\right\rangle, \text { for } 1 \leq t \leq 4)

Curvature of \(e^{x}\) Find the curvature of \(f(x)=e^{x}\) and find the point at which it is a maximum. What is the value of the maximum curvature?

29-36. Principal unit normal vector Find the unit tangent vector T and the principal unit normal vector N for the following parameterized curves. In each case, verify that \(|\mathbf{T}|=|\mathbf{N}|=1 \text { and } \mathbf{T} \cdot \mathbf{N}=0\). \(\mathbf{r}(t)=\left\langle t^{2} / 2,4-3 t, 1\right\rangle\)

29-36. Principal unit normal vector Find the unit tangent vector T and the principal unit normal vector N for the following parameterized curves. In each case, verify that \(|\mathbf{T}|=|\mathbf{N}|=1 \text { and } \mathbf{T} \cdot \mathbf{N}=0\). \(\mathbf{r}(t)=\langle 4 \sin t, 4 \cos t, 10 t\rangle\)

45-48. Curvature for y = f(x) Use the result of Exercise 44 to find the curvature function of the following curves. \(f(x)=\ln (\cos x)\)

9-14. Arc length parameterization Determine whether the following curves use arc length as a parameter If not, find a description that uses arc length as a parameter. \(\mathbf{r}(t)=\left\langle\cos t^{2}, \sin t^{2}\right\rangle, \text { for } 0 \leq t \leq \sqrt{\pi}\)

9-14. Arc length parameterization Determine whether the following curves use arc length as a parameter If not, find a description that uses arc length as a parameter. \(\mathbf{r}(t)=\langle 5 \cos t, 3 \sin t, 4 \sin t\rangle, \text { for } 0 \leq t \leq \pi\)

50-53. Curvature for plane curves Use the result of Exercise 49 to find the curvature function of the following curves. \(\mathbf{r}(t)=\left\langle a \cos ^{3} t, a \sin ^{3} t\right\rangle\) (astroid)

Special formula: Curvature for plane curves Show that the curve \(\mathbf{r}(t)=\langle f(t), g(t)\rangle\), where f and g are twice differentiable, has curvature \(\kappa(t)=\frac{\left|f^{\prime} g^{\prime \prime}-f^{\prime \prime} g^{\prime}\right|}{\left(\left(f^{\prime}\right)^{2}+\left(g^{\prime}\right)^{2}\right)^{3 / 2}}\), where all derivatives are taken with respect to t.

9-14. Arc length parameterization Determine whether the following curves use arc length as a parameter If not, find a description that uses arc length as a parameter. \(\mathbf{r}(t)=\langle t+1,2 t-3,6 t\rangle, \text { for } 0 \leq t \leq 10\)

29-36. Principal unit normal vector Find the unit tangent vector T and the principal unit normal vector N for the following parameterized curves. In each case, verify that \(|\mathbf{T}|=|\mathbf{N}|=1 \text { and } \mathbf{T} \cdot \mathbf{N}=0\). \(\mathbf{r}(t)=\left\langle t^{2} / 2, t^{3} / 3\right\rangle\)

Circle and radius of curvature Choose a point P on a smooth curve C in the plane. The circle of curvature (or osculating circle) at the point P is the circle that (a) is tangent to C at P, (b) has the same curvature as C at P, and (c) lies on the same side of C as the principal unit normal N (see figure). The radius of curvature is the radius of the circle of curvature. Show that the radius of curvature is \(1 / \kappa\), where \(\kappa\) is the curvature of C at P.

Finding radii of curvature Find the radius of curvature (see Exercise 64) of the following curves at the given point. Then write the equation of the circle of curvature at the point. \(\mathbf{r}(t)=\langle t-\sin t, 1-\cos t\rangle\) (cycloid) at \(t=\pi\)

Parabolic trajectory Consider the parabolic trajectory \(x=\left(V_{0} \cos \alpha\right) t\), \(y=\left(V_{0} \sin \alpha\right) t-\frac{1}{2} g t^{2}\), where \(V_{0}\) is the initial speed, \(\alpha\) is the angle of launch, and g is the acceleration due to gravity. Consider all times [0, T] for which \(y \geq 0\). (a) Find and graph the speed for \(0 \leq t \leq T\). (b) Find and graph the curvature for \(0 \leq t \leq T\). (c) At what times (if any) do the speed and curvature have minimum and maximum values?

Relationship between T, N, and a Show that if an object accelerates in the sense that \(d^{2} s / d t^{2}>0\) and \(\kappa \neq 0\), then the acceleration vector lies between T and N in the plane of T and N. If an object decelerates in the sense that \(d^{2} s / d t^{2}<0\), then the acceleration vector lies in the plane of T and N, but not between T and N.

Maximum curvature Consider the "superparabolas" \(f_{n}(x)=x^{2 n}\), where n is a positive integer. (a) Find the curvature functions of \(f_{n}\), for n=1, 2, 3. (b) Plot \(f_{n}\) and their curvature functions, for n=1, 2, 3, and check for consistency. (c) At what points does the maximum curvature occur for n=1, 2, 3? (d) Let the maximum curvature for \(f_{n}\) occur at \(x=\pm z_{n}\). Using either analytical methods or a calculator determine \(\lim _{n \rightarrow \infty} z_{n}\). Interpret your result.