Solution: In Exercises 2124, find the curvature of the curve, where s is the arc length

In Exercises 1–6, sketch the plane curve and find its length over the given interval. \(\mathbf{r}(t)=3 t \mathbf{i}-t \mathbf{j}\), [0,3] Text Transcription: r(t)=3 t i-t j

In Exercises 1–6, sketch the plane curve and find its length over the given interval. \(\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}\), [0,4] Text Transcription: r(t)=t i+t^2 j

In Exercises 1–6, sketch the plane curve and find its length over the given interval. \(\mathbf{r}(t)=t^{3} \mathbf{i}+t^{2} \mathbf{j}\) [0,1] Text Transcription: r(t)=t^3 i+t^2 j

In Exercises 1–6, sketch the plane curve and find its length over the given interval. \(\mathbf{r}(t)=(t+1) \mathbf{i}+t^{2} \mathbf{j}, \quad[0,6]\) Text Transcription: r(t)=(t+1) i+t^2 j [0,6]

In Exercises 1–6, sketch the plane curve and find its length over the given interval. \(\mathbf{r}(t)=a \cos ^{3} t \mathbf{i}+a \sin ^{3} t \mathbf{j}, \quad[0,2 \pi]\) Text Transcription: r(t)=a cos ^3 t i+a sin ^3 t j, [0,2 pi]

In Exercises 1–6, sketch the plane curve and find its length over the given interval. \(\mathbf{r}(t)=a \cos t \mathbf{i}+a \sin t \mathbf{j}, \quad[0,2 \pi]\) Text Transcription: r(t)=a cos t i+a sin t j [0,2 pi]

Projectile Motion A baseball is hit 3 feet above the ground at 100 feet per second and at an angle of \(45^{\circ}\) with respect to the ground. (a) Find the vector-valued function for the path of the baseball. (b) Find the maximum height. (c) Find the range. (d) Find the arc length of the trajectory Text Transcription: 45^circ

Projectile Motion An object is launched from ground level. Determine the angle of the launch to obtain (a) the maximum height, (b) the maximum range, and (c) the maximum length of the trajectory. For part (c), let \(v_{0}\)=96 feet per second. Text Transcription: v_0

In Exercises 9–14, sketch the space curve and find its length over the given interval. Function \(\mathbf{r}(t)=-t \mathbf{i}+4 t \mathbf{j}+3 t \mathbf{k}\) Interval [0,1] Text Transcription: r(t)=-t i+4 t j+3 t k

In Exercises 9–14, sketch the space curve and find its length over the given interval. Function \(\mathbf{r}(t)=\mathbf{i}+t^{2} \mathbf{j}+t^{3} \mathbf{k}\) Interval [0,2] Text Transcription: r(t)=i+t^2 j+t^3 k

In Exercises 9–14, sketch the space curve and find its length over the given interval. Function \(\mathbf{r}(t)=\langle 4 t,-\cos t, \sin t\rangle\) Interval \(\left[0, \frac{3 \pi}{2}\right]\) Text Transcription: r(t)= langle 4 t,- cos t, sin t rangle [0, frac3 pi/2 ]

In Exercises 9–14, sketch the space curve and find its length over the given interval. Function \(\mathbf{r}(t)=\langle 2 \sin t, 5 t, 2 \cos t\rangle\) Interval [0, \(\pi\)] Text Transcription: r(t)= langle 2 sin t, 5 t, 2 cos t rangle [0, pi]

In Exercises 9–14, sketch the space curve and find its length over the given interval. Function \(\mathbf{r}(t)=a \cos t \mathbf{i}+a \sin t \mathbf{j}+b t \mathbf{k}\) Interval [0, 2 \(\pi\)] Text Transcription: r(t)=a cos t i+a sin t j+b t k [0, 2 pi]

In Exercises 9–14, sketch the space curve and find its length over the given interval. Function \(\mathbf{r}(t)=\left\langle\cos t+t \sin t, \sin t-t \cos t, t^{2}\right\rangle \) Interval \([0, \frac{\pi}{2}]\) Text Transcription: r(t)= langle cos t+t sin t, sin t-t cos t, t^2 rangle [0, frac pi/2 ]

In Exercises 15 and 16, use the integration capabilities of a graphing utility to approximate the length of the space curve over the given interval. Function \(\mathbf{r}(t)=t^{2} \mathbf{i}+t \mathbf{j}+\ln t \mathbf{k}\) Interval \(1 \leq t \leq 3\) Text Transcription: r(t)=t^2 i+t j+ln t k 1 leq t leq 3

In Exercises 15 and 16, use the integration capabilities of a graphing utility to approximate the length of the space curve over the given interval. Function \(\mathbf{r}(t)=\sin \pi t \mathbf{i}+\cos \pi t \mathbf{j}+t^{3} \mathbf{k}\) Interval \(0 \leq t \leq 2\) Text Transcription: r(t)=sin pi t i+cos pi t j+t^3 k 0 leq t leq 2

Investigation Consider the graph of the vector-valued function \(\mathbf{r}(t)=t \mathbf{i}+\left(4-t^{2}\right) \mathbf{j}+t^{3} \mathbf{k}\) on the interval [0,2] (a) Approximate the length of the curve by finding the length of the line segment connecting its endpoints. (b) Approximate the length of the curve by summing the lengths of the line segments connecting the terminal points of the vectors r(0), r(0.5), r(1), r(1.5) and r(2) (c) Describe how you could obtain a more accurate approximation by continuing the processes in parts (a) and (b). (d) Use the integration capabilities of a graphing utility to approximate the length of the curve. Compare this result with the answers in parts (a) and (b). Text Transcription: r(t)=t i+ (4-t^2 ) j+t^3 k

Investigation Repeat Exercise 17 for the vector-valued function \(\mathbf{r}(t)=6 \cos (\pi t / 4) \mathbf{i}+2 \sin (\pi t / 4) \mathbf{j}+t \mathbf{k}\) Text Transcription: r(t)=6 cos (pi t / 4) i+2 sin (pi t / 4) j+t k

Investigation Consider the helix represented by the vector valued function \(\mathbf{r}(t)=\langle 2 \cos t, 2 \sin t, t\rangle\) (a) Write the length of the arc s on the helix as a function of t by evaluating the integral \(s=\int_{0}^{t}\sqrt{\left[x^{\prime}(u)\right]^{2}+\left[y^{\prime}(u)\right]^{2}+\left[z^{\prime}(u)\right]^{2}} d u\) (b) Solve for in the relationship derived in part (a), and substitute the result into the original set of parametric equations. This yields a parametrization of the curve in terms of the arc length parameter s (c) Find the coordinates of the point on the helix for arc lengths \(s=\sqrt{5}\) and s=4 (d) Verify that ||r’(s)||=1 Text Transcription: r(t)= langle 2 cos t, 2 sin t, t rangle s= int_0^t sqrt [x^'(u) ]^2+ [y^'(u) ]^2+ [z^'(u) ]^2 d u s=sqrt5

Investigation Repeat Exercise 19 for the curve represented by the vector-valued function \(\mathbf{r}(t)=\left\langle 4(\sin t-t \cos t), 4(\cos t+t \sin t), \frac{3}{2} t^{2}\right\rangle\) Text Transcription: r(t)= langle 4( sin t-t cos t), 4( cos t+t sin t), 3/2 t^2 rangle

In Exercises 21–24, find the curvature K of the curve, where s is the arc length parameter. \(\mathbf{r}(s)=\left(1+\frac{\sqrt{2}}{2} s\right) \mathbf{i}+\left(1-\frac{\sqrt{2}}{2} s\right) \mathbf{j}\) Text Transcription: r(s)=(1+ sqrt2/2 s) i+(1- sqrt2/2 s) j

In Exercises 21–24, find the curvature K of the curve, where s is the arc length parameter. \(\mathbf{r}(s)=(3+s) \mathbf{i}+\mathbf{j}\) Text Transcription: r(s)=(3+s) i+j

In Exercises 21–24, find the curvature K of the curve, where s is the arc length parameter. Helix in Exercise 19: \(\mathbf{r}(t)=\langle 2 \cos t, 2 \sin t, t\rangle\) Text Transcription: r(t)= langle 2 cos t, 2 sin t, t rangle

In Exercises 21–24, find the curvature K of the curve, where s is the arc length parameter. Curve in Exercise 20: \(\mathbf{r}(t)=\left\langle 4(\sin t-t \cos t), 4(\cos t+t \sin t), \frac{3}{2} t^{2}\right\rangle\) Text Transcription: r(t)= langle 4( sin t-t cos t), 4( cos t+t sin t), 3/2 t^2 rangle

In Exercises 25–30, find the curvature K of the plane curve at the given value of the parameter. \(\mathbf{r}(t)=4 t \mathbf{i}-2 t \mathbf{j}\), t=1 Text Transcription: r(t)=4 t i-2 t j

In Exercises 25–30, find the curvature K of the plane curve at the given value of the parameter. \(\mathbf{r}(t)=t^{2} \mathbf{i}+\mathbf{j}\) t=2 Text Transcription: r(t)=t^2 i+j

In Exercises 25–30, find the curvature K of the plane curve at the given value of the parameter. \(\mathbf{r}(t)=t \mathbf{i}+\frac{1}{t} \mathbf{j}\), t=1 Text Transcription: r(t)=t i+1/t j

In Exercises 25–30, find the curvature K of the plane curve at the given value of the parameter. \(\mathbf{r}(t)=t \mathbf{i}+\frac{1}{9} t^{3} \mathbf{j}\), t=2 Text Transcription: r(t)=t i+1/9 t^3 j

In Exercises 25–30, find the curvature K of the plane curve at the given value of the parameter. \(\mathbf{r}(t)=\langle t, \sin t\rangle\) \(t=\frac{\pi}{2}\) Text Transcription: r(t)= langle t, sin t rangle t= pi/2

In Exercises 25–30, find the curvature K of the plane curve at the given value of the parameter. \(\mathbf{r}(t)=\langle 5 \cos t, 4 \sin t\rangle\) \(t=\frac{\pi}{3}\) Text Transcription: r(t)= langle 5 cos t, 4 sin t rangle t= pi/3

In Exercises 31–40, find the curvature K of the curve. \(\mathbf{r}(t)=4 \cos 2 \pi t \mathbf{i}+4 \sin 2 \pi t \mathbf{j}\) Text Transcription: r(t)=4 cos 2 pi t i+4 sin 2 pi t j

In Exercises 31–40, find the curvature K of the curve. \(\mathbf{r}(t)=2 \cos \pi t \mathbf{i}+\sin \pi t \mathbf{j}\) Text Transcription: r(t)=2 cos pi t i+ sin pi t j

In Exercises 31–40, find the curvature K of the curve. \(\mathbf{r}(t)=a \cos \omega t \mathbf{i}+a \sin \omega t \mathbf{j}\) Text Transcription: r(t)=a cos omega t i+a sin omega t j

In Exercises 31–40, find the curvature K of the curve. \(\mathbf{r}(t)=a \cos \omega t \mathbf{i}+b \sin \omega t \mathbf{j}\) Text Transcription: r(t)=a cos omega t i+b sin omega t j

In Exercises 31–40, find the curvature K of the curve. \(\mathbf{r}(t)=\langle a(\omega t-\sin \omega t), a(1-\cos \omega t)\rangle\) Text Transcription: r(t)= langle a( omega t- sin omega t), a(1- cos omega t) rangle

In Exercises 31–40, find the curvature K of the curve. \(\mathbf{r}(t)=\langle\cos \omega t+\omega t \sin \omega t, \sin \omega t-\omega t \cos \omega t\rangle\) Text Transcription: r(t)= langle cos omega t+ omega t sin omega t, sin omega t- omega t cos omega t rangle

In Exercises 31–40, find the curvature K of the curve. \(\mathbf{r}(t)=t \mathbf{i}+\left|t^{2} \mathbf{j}\right + \frac{t^{2}}{2} \mathbf{k}\) Text Transcription: r(t)=t i + |t^2 j + t^2/2 k

In Exercises 31–40, find the curvature K of the curve. \(\mathbf{r}(t)=2 t^{2} \mathbf{i}+t \mathbf{j} +\frac{1}{2} t^{2} \mathbf{k}\) Text Transcription: r(t)=2 t^2 i + t j + 1/2 t^2 k

In Exercises 31–40, find the curvature K of the curve. \(\mathbf{r}(t)=4 t \mathbf{i}+3 \cos t \mathbf{j}+3 \sin t \mathbf{k}\) Text Transcription: r(t)=4 t i+3 cos t j+3 sin t k

In Exercises 31–40, find the curvature K of the curve. \(\mathbf{r}(t)=e^{2 t} \mathbf{i}+e^{2 t} \cos t \mathbf{j}+e^{2 t} \sin t \mathbf{k}\) Text Transcription: r(t)=e^2 t i+e^2 t cos t j+e^2 t sin t k

In Exercises 41– 44, find the curvature K of the curve at the point P \(\mathbf{r}(t)=3 t \mathbf{i}+2 t^{2} \mathbf{j}\) P(-3,2) Text Transcription: r (t)=3 t i +2 t^ 2 j

In Exercises 41– 44, find the curvature K of the curve at the point P \(\mathbf{r}(t)=e^{t} \mathbf{i}+4 t \mathbf{j}\), P(1,0) Text Transcription: r (t)=e^ t i +4 t j

In Exercises 41– 44, find the curvature K of the curve at the point P \(\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+\frac{t^{3}}{4} \mathbf{k}\), P(2,4,2) Text Transcription: r (t)=t i +t^ 2 j + t^ 3 /4 k

In Exercises 41– 44, find the curvature K of the curve at the point P \(\mathbf{r}(t)=e^{t} \cos t \mathbf{i}+c^{t} \sin t \mathbf{j}+c^{t} \mathbf{k}\)P(1,0,1) Text Transcription: r (t)=e^ t cos t i +e^ t sin t j +e^ t

In Exercises 45–54, find the curvature and radius of curvature of the plane curve at the given value of x y= 3x-2 , x=a

In Exercises 45–54, find the curvature and radius of curvature of the plane curve at the given value of x y= mx+b, x=a

In Exercises 45–54, find the curvature and radius of curvature of the plane curve at the given value of x y=2 \(x^{2}\)+3, x=-1 Text Transcription: x^2

In Exercises 45–54, find the curvature and radius of curvature of the plane curve at the given value of x \(y=2 x+\frac{4}{x}\), x=1 Text Transcription: y=2 x+4/x

In Exercises 45–54, find the curvature and radius of curvature of the plane curve at the given value of x. \(y=\cos 2 x, \quad x=2 \pi\) Text Transcription: y=cos 2 x, x=2 pi

In Exercises 45–54, find the curvature and radius of curvature of the plane curve at the given value of x. \(y=e^{3 x}\), x=0 Text Transcription: y=e^3 x

In Exercises 45–54, find the curvature and radius of curvature of the plane curve at the given value of x. \(y=\sqrt{a^{2}-x^{2}}\), x =0 Text Transcription: y=sqrta^2-x^2

In Exercises 45–54, find the curvature and radius of curvature of the plane curve at the given value of x. \(y=\frac{3}{4} \sqrt{16-x^{2}}\), x=0 Text Transcription: y=3/4 sqrt16-x^2

In Exercises 45–54, find the curvature and radius of curvature of the plane curve at the given value of x. \(y= x^{3}\), x=2 Text Transcription: y=x^3

In Exercises 45–54, find the curvature and radius of curvature of the plane curve at the given value of x. \(y=x^{n}\), x=1, \(n \geq 2\) Text Transcription: y=x^n n geq 2

Writing In Exercises 55 and 56, two circles of curvature to the graph of the function are given. (a) Find the equation of the smaller circle, and (b) write a short paragraph explaining why the circles have different radii. f(x) = sin x

Writing In Exercises 55 and 56, two circles of curvature to the graph of the function are given. (a) Find the equation of the smaller circle, and (b) write a short paragraph explaining why the circles have different radii. \(f(x)=4 x^{2} /\left(x^{2}+3\right)\) Text Transcription: f(x)=4 x^2 /(x^2+3)

In Exercises 57–60, use a graphing utility to graph the function. In the same viewing window, graph the circle of curvature to the graph at the given value of x. \(y=x+\frac{1}{x}\), x =1 Text Transcription: y=x+1/x

In Exercises 57–60, use a graphing utility to graph the function. In the same viewing window, graph the circle of curvature to the graph at the given value of x. y = ln x, x = 1

In Exercises 57–60, use a graphing utility to graph the function. In the same viewing window, graph the circle of curvature to the graph at the given value of x. \(y=e^{x}\), x=0 Text Transcription: y=e^x

In Exercises 57–60, use a graphing utility to graph the function. In the same viewing window, graph the circle of curvature to the graph at the given value of x. \(y=\frac{1}{3} x^{3}\), x=1 Text Transcription: y=1/3 x^3

Evolute An evolute is the curve formed by the set of centers of curvature of a curve. In Exercises 61 and 62, a curve and its evolute are given. Use a compass to sketch the circles of curvature with centers at points A and B. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. Cycloid: x = t - sin t y = 1 - cos t Evolute: x = sin t + t y = cos t - 1

Evolute An evolute is the curve formed by the set of centers of curvature of a curve. In Exercises 61 and 62, a curve and its evolute are given. Use a compass to sketch the circles of curvature with centers at points A and B. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. Ellipse: x = 3 cos t y = 2 sin t Evolute: \(x=\frac{5}{3} \cos ^{3} t\) \(y=\frac{5}{2} \sin ^{3} t\) Text Transcription: x=5/3 cos^3 t y=5/2 sin^3 t

In Exercises 63–70, (a) find the point on the curve at which the curvature K is a maximum and (b) find the limit of K as \(x \rightarrow \infty\) \(y=(x-1)^{2}+3\) Text Transcription: y=(x-1)^2 + 3 x rightarrow infty

In Exercises 63–70, (a) find the point on the curve at which the curvature K is a maximum and (b) find the limit of K as \(x \rightarrow \infty\) \(y=x^{3}\) Text Transcription: x rightarrow infty y=x^3

In Exercises 63–70, (a) find the point on the curve at which the curvature K is a maximum and (b) find the limit of K as \(x \rightarrow \infty\) \(y=x^{2 / 3}\) Text Transcription: x rightarrow infty y=x^2/3

In Exercises 63–70, (a) find the point on the curve at which the curvature K is a maximum and (b) find the limit of K as \(x \rightarrow \infty\) \(y=\frac{1}{x}\) Text Transcription: x rightarrow infty y=1/x

In Exercises 63–70, (a) find the point on the curve at which the curvature K is a maximum and (b) find the limit of K as \(x \rightarrow \infty\) y=ln x Text Transcription: x rightarrow infty

In Exercises 63–70, (a) find the point on the curve at which the curvature K is a maximum and (b) find the limit of K as \(x \rightarrow \infty\) \(y=e^{x}\) Text Transcription: x rightarrow infty y=e^x

In Exercises 63–70, (a) find the point on the curve at which the curvature K is a maximum and (b) find the limit of K as \(x \rightarrow \infty\) y=sinh x Text Transcription: x rightarrow infty

In Exercises 63–70, (a) find the point on the curve at which the curvature K is a maximum and (b) find the limit of K as \(x \rightarrow \infty\) y=cosh x Text Transcription: x rightarrow infty

In Exercises 71–74, find all points on the graph of the function such that the curvature is zero. \(y=1-x^{3}\) Text Transcription: y=1-x^3

In Exercises 71–74, find all points on the graph of the function such that the curvature is zero. \(y=(x-1)^{3}+3\) Text Transcription: y=(x-1)^3 + 3

In Exercises 71–74, find all points on the graph of the function such that the curvature is zero. y=cos x

In Exercises 71–74, find all points on the graph of the function such that the curvature is zero. y=sin x

(a) Give the formula for the arc length of a smooth curve in space. (b) Give the formulas for curvature in the plane and in space.

Describe the graph of a vector-valued function for which the curvature is 0 for all values of t in its domain.

Given a twice-differentiable function y = f(x), determine its curvature at a relative extremum. Can the curvature ever be greater than it is at a relative extremum? Why or why not?

A particle moves along the plane curve C described by \(\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}\). (a) Find the length of C on the interval \(0 \leq t \leq 2\). (b) Find the curvature K of the plane curve at t = 0, t = 1 and t = 2. (c) Describe the curvature of C as t changes from t = 0 and t = 2. Text Transcription: mathbf r(t) = t mathbf i + t^2 mathbf j 0 leq t leq 2

Show that the curvature is greatest at the endpoints of the major axis, and is least at the endpoints of the minor axis, for the ellipse given by \(x^{2}+4 y^{2}=4\). Text Transcription: x^2 + 4y^2 = 4

Investigation Find all and such that the two curves given by \(y_{1}=a x(b-x)\) and \(y_{2}=\frac{x}{x+2}\) intersect at only one point and have a common tangent line and equal curvature at that point. Sketch a graph for each set of values of a and b. Text Transcription: y_1 = ax(b-x) y_2 = x / x+2

Investigation Consider the function \(f(x)=x^{4}-x^{2}\). (a) Use a computer algebra system to find the curvature K of the curve as a function of x. (b) Use the result of part (a) to find the circles of curvature to the graph of f when x = 0 and x = 1. Use a computer algebra system to graph the function and the two circles of curvature. (c) Graph the function K(x) and compare it with the graph of f(x). For example, do the extrema of f and K occur at the same critical numbers? Explain your reasoning. Text Transcription: f(x)=x^4 - x^2

Investigation The surface of a goblet is formed by revolving the graph of the function \(y=\frac{1}{4} x^{8 / 5}\), \(0 \leq x \leq 5\) about the axis. The measurements are given in centimeters. (a) Use a computer algebra system to graph the surface. (b) Find the volume of the goblet. (c) Find the curvature K of the generating curve as a function of x. Use a graphing utility to graph K. (d) If a spherical object is dropped into the goblet, is it possible for it to touch the bottom? Explain. Text Transcription: y=1/4 x^8/5 0 leq x leq 5

A sphere of radius 4 is dropped into the paraboloid given by \(z=x^{2}+y^{2}\). a) How close will the sphere come to the vertex of the paraboloid? (b) What is the radius of the largest sphere that will touch the vertex? Text Transcription: z=x^2 + y^2

Speed The smaller the curvature of a bend in a road, the faster a car can travel. Assume that the maximum speed around a turn is inversely proportional to the square root of the curvature. A car moving on the path \(y=\frac{1}{3} x^{3}\) (x and y are measured in miles) can safely go 30 miles per hour at \(\left(1, \frac{1}{3}\right)\). How fast can it go at \(\left(\frac{3}{2}, \frac{9}{8}\right)\)? Text Transcription: y=1/3 x^3 (1, 1/3) (3/2, 9/8)

Let C be a curve given by y = f(x). Let K be the curvature \((K \neq 0)\) at the point \(P\left(x_{0}, y_{0}\right)\) and let \(z=\frac{1+f^{\prime}\left(x_{0}\right)^{2}}{f^{\prime \prime}\left(x_{0}\right)}\). Show that the coordinates \((\alpha, \beta)\) of the center of curvature at P are \((\alpha, \beta)=\left(x_{0}-f^{\prime}\left(x_{0}\right) z, y_{0}+z\right)\). Text Transcription: (K neq 0) P(x_0, y_0) z=1+f^prime (x_0)^2 / f^prime prime (x_0) (alpha, beta)

Use the result of Exercise 85 to find the center of curvature for the curve at the given point. (a) \(y=e^{x}\), (0,1) (b) \(y=\frac{x^{2}}{2}\), \(\left(1, \frac{1}{2}\right)\) (c) \(y=x^{2}\), (0,0) Text Transcription: y=e^x y=x^2 / 2 (1, 1/2) y=x^2

A curve C is given by the polar equation \(r=f(\theta)\). Show that the curvature K at the point \((r, \theta)\) is \(K=\frac{\left|2\left(r^{\prime}\right)^{2}-r r^{\prime \prime}+r^{2}\right|}{\left[\left(r^{\prime}\right)^{2}+r^{2}\right]^{3 / 2}}\) [Hint: Represent the curve by \(\mathbf{r}(\theta)=r \cos \theta \mathbf{i}+r \sin \theta \mathbf{j}\).] Text Transcription: r=f(theta) (r, theta) K=|2(r^prime)^2 - rr^prime prime + r^2| / [(r^prime)^2 + r^2]^3/2 mathbf r(theta) = r cos theta mathbf i + r sin theta mathbf j

Use the result of Exercise 87 to find the curvature of each polar curve. (a) \(r=1+\sin \theta\) (b) \(r=\theta\) (c) \(r=a \sin \theta\) (d) \(r=e^{\theta}\) Text Transcription: r=1 + sin theta r=theta r=a sin theta r=e^theta

Given the polar curve \(r=e^{a \theta}\), a>0, find the curvature K and determine the limit of K as (a) \(\theta \rightarrow \infty\) and (b) \(a \rightarrow \infty\) Text Transcription: r=e^a theta theta rightarrow infty a rightarrow infty

Show that the formula for the curvature of a polar curve \(r=f(\theta)\) given in Exercise 87 reduces to \(K=2 /\left|r^{\prime}\right|\) for the curvature at the pole. Text Transcription: r=f(theta) K=2/|r^prime|

In Exercises 91 and 92, use the result of Exercise 90 to find the curvature of the rose curve at the pole. \(r=4 \sin 2 \theta\) Text Transcription: r=4 sin 2 theta

In Exercises 91 and 92, use the result of Exercise 90 to find the curvature of the rose curve at the pole. \(r=6 \cos 3 \theta\) Text Transcription: r=6 cos 3 theta

For a smooth curve given by the parametric equations x = f(t) and y = g(t) prove that the curvature is given by \(K=\frac{\left|f^{\prime}(t) g^{\prime \prime}(t)-g^{\prime}(t) f^{\prime \prime}(t)\right|}{\left\{\left[f^{\prime}(t)\right]^{2}+\left[g^{\prime}(t)\right]^{2}\right\}^{3 / 2}}\) Text Transcription: K=|f^prime (t) g^prime prime (t) - g^prime (t) f^prime prime (t)| / {[f^prime (t)]^2 + [g^prime (t)]^2}^3/2

Use the result of Exercise 93 to find the curvature K of the curve represented by the parametric equations \(x(t)=t^{3}\) and \(y(t)=\frac{1}{2} t^{2}\). Use a graphing utility to graph K and determine any horizontal asymptotes. Interpret the asymptotes in the context of the problem. Text Transcription: x(t)=t^3 y(t)=1/2 t^2

Use the result of Exercise 93 to find the curvature K of the cycloid represented by the parametric equations \(x(\theta)=a(\theta-\sin \theta)\) and \(y(\theta)=a(1-\cos \theta)\). What are the minimum and maximum values of K? Text Transcription: x(theta)=a(theta - sin theta) y(theta)=a(1 - cos theta)

Use Theorem 12.10 to find \(a_{\mathbf{T}}\) and \(a_{\mathbf{N}}\) for each curve given by the vector-valued function. (a) \(\mathbf{r}(t)=3 t^{2} \mathbf{i}+\left(3 t-t^{3}\right) \mathbf{j}\) (b) \(\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+\frac{1}{2} t^{2} \mathbf{k}\) Text Transcription: a_mathbf T a_mathbf N mathbf r(t) = 3t^2 mathbf i + (3t-t^3) mathbf j mathbf r(t) = t mathbf i + t^2 mathbf j + 1/2 t^2 mathbf k

Frictional Force A 5500-pound vehicle is driven at a speed of 30 miles per hour on a circular interchange of radius 100 feet. To keep the vehicle from skidding off course, what frictional force must the road surface exert on the tires?

Frictional Force A 6400-pound vehicle is driven at a speed of 35 miles per hour on a circular interchange of radius 250 feet. To keep the vehicle from skidding off course, what frictional force must the road surface exert on the tires?

Verify that the curvature at any point (x, y) on the graph of y = cosh x is \(1 / y^{2}\). Text Transcription: 1/y^2

Use the definition of curvature in space, \(K=\left\|\mathbf{T}^{\prime}(s)\right\|=\left\|\mathbf{r}^{\prime \prime}(s)\right\|\), to verify each formula. (a) \(K=\frac{\left\|\mathbf{T}^{\prime}(t)\right\|}{\left\|\mathbf{r}^{\prime}(t)\right\|}\) (b) \(K=\frac{\left\|\mathbf{r}^{\prime}(t) \times \mathbf{r}^{\prime \prime}(t)\right\|}{\left\|\mathbf{r}^{\prime}(t)\right\|^{3}}\) (c) \(K=\frac{\mathbf{a}(t) \cdot \mathbf{N}(t)}{\|\mathbf{v}(t)\|^{2}}\) Text Transcription: K=|mathbf T^prime(s)| = |mathbf r^prime prime(s)| K=|mathbf T^prime(t)| / |mathbf r^prime(t)| K=|mathbf r^prime(t) times mathbf r^prime prime(t)| / |mathbf r^prime(t)|^3 K=mathbf a(t) cdot mathbf N(t) / |mathbf v(t)|^2

True or False? In Exercises 101–104, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The arc length of a space curve depends on the parametrization.

True or False? In Exercises 101–104, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The curvature of a circle is the same as its radius.

True or False? In Exercises 101–104, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The curvature of a line is 0.

True or False? In Exercises 101–104, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The normal component of acceleration is a function of both speed and curvature.

Kepler’s Laws In Exercises 105–112, you are asked to verify Kepler’s Laws of Planetary Motion. For these exercises, assume that each planet moves in an orbit given by the vector-valued function r. Let \(r=\|\mathbf{r}\|\), let G represent the universal gravitational constant, let M represent the mass of the sun, and let m represent the mass of the planet. Prove that \(\mathbf{r} \cdot \mathbf{r}^{\prime}=r \frac{d r}{d t}) Text Transcription: r=|mathbf r| mathbf r cdot mathbf r^prime = r dr/dt

Kepler’s Laws In Exercises 105–112, you are asked to verify Kepler’s Laws of Planetary Motion. For these exercises, assume that each planet moves in an orbit given by the vector-valued function r. Let \(r=\|\mathbf{r}\|\), let G represent the universal gravitational constant, let M represent the mass of the sun, and let m represent the mass of the planet. Using Newton’s Second Law of Motion, F = ma and Newton’s Second Law of Gravitation, \(\mathbf{F}=-\left(G m M / r^{3}\right) \mathbf{r}\) show that a and r are parallel, and that \(\mathbf{r}(t) \times \mathbf{r}^{\prime}(t)=\mathbf{L}\) is a constant vector. So, r(t) moves in a fixed plane, orthogonal to L. Text Transcription: r=|mathbf r| mathb F = -(GmM/r^3) mathbf r mathbf r(t) times mathbf r^prime(t) = mathbf L

Kepler’s Laws In Exercises 105–112, you are asked to verify Kepler’s Laws of Planetary Motion. For these exercises, assume that each planet moves in an orbit given by the vector-valued function r. Let \(r=\|\mathbf{r}\|\), let G represent the universal gravitational constant, let M represent the mass of the sun, and let m represent the mass of the planet. Prove that \(\frac{d}{d t}\left[\frac{\mathbf{r}}{r}\right]=\frac{1}{r^{3}}\left\{\left[\mathbf{r} \times \mathbf{r}^{\prime}\right] \times \mathbf{r}\right\}\) Text Transcription: r=|mathbf r| d/dt [r/r] = 1/r^3 {[mathbf r times mathbf r^prime] times mathbf r}

Kepler’s Laws In Exercises 105–112, you are asked to verify Kepler’s Laws of Planetary Motion. For these exercises, assume that each planet moves in an orbit given by the vector-valued function r. Let \(r=\|\mathbf{r}\|\), let G represent the universal gravitational constant, let M represent the mass of the sun, and let m represent the mass of the planet. Show that \(\frac{\mathbf{r}^{\prime}}{G M} \times \mathbf{L}-\frac{\mathbf{r}}{r}=\mathbf{e}\) is a constant vector. Text Transcription: r=|mathbf r| mathbf r^prime / GM times mathbf L - mathbf r/r = mathbf e

Kepler’s Laws In Exercises 105–112, you are asked to verify Kepler’s Laws of Planetary Motion. For these exercises, assume that each planet moves in an orbit given by the vector-valued function r. Let \(r=\|\mathbf{r}\|\), let G represent the universal gravitational constant, let M represent the mass of the sun, and let m represent the mass of the planet. Prove Kepler’s First Law: Each planet moves in an elliptical orbit with the sun as a focus. Text Transcription: r=|mathbf r|

Kepler’s Laws In Exercises 105–112, you are asked to verify Kepler’s Laws of Planetary Motion. For these exercises, assume that each planet moves in an orbit given by the vector-valued function r. Let \(r=\|\mathbf{r}\|\), let G represent the universal gravitational constant, let M represent the mass of the sun, and let m represent the mass of the planet. Assume that the elliptical orbit \(r=e d /(1+e \cos \theta)\) is in the xy-plane, with L along the z-axis. Prove that \(\|\mathbf{L}\|=r^{2} \ d \theta / d t\) Text Transcription: r=|mathbf r| r=ed /(1+e cos theta) |mathbf L| = r^2 d theta/dt

Kepler’s Laws In Exercises 105–112, you are asked to verify Kepler’s Laws of Planetary Motion. For these exercises, assume that each planet moves in an orbit given by the vector-valued function r. Let \(r=\|\mathbf{r}\|\), let G represent the universal gravitational constant, let M represent the mass of the sun, and let m represent the mass of the planet. Prove Kepler’s Second Law: Each ray from the sun to a planet sweeps out equal areas of the ellipse in equal times.

Kepler’s Laws In Exercises 105–112, you are asked to verify Kepler’s Laws of Planetary Motion. For these exercises, assume that each planet moves in an orbit given by the vector-valued function r. Let \(r=\|\mathbf{r}\|\), let G represent the universal gravitational constant, let M represent the mass of the sun, and let m represent the mass of the planet. Prove Kepler’s Third Law: The square of the period of a planet’s orbit is proportional to the cube of the mean distance between the planet and the sun.