?(a) Find the slope of the tangent to the astroid \(x=a\ \cos^3 \theta,\ y=a\ \sin^3

Find dx/dt, dy/dt, and dy/dx.

Find dx/dt, dy/dt, and dy/dx.

Find dx/dt, dy/dt, and dy/dx.

Find \(d x / d t, \ d y / d t\), and \(d y / d x\). \(x=t+\sin \left(t^{2}+2\right), \ \ y=\tan \left(t^{2}+2\right)\)

Find the slope of the tangent to the parametric curve at the indicated point.

Find the slope of the tangent to the parametric curve at the indicated point.

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. \(x=t^{3}+1, \ \ y=t^{4}+t; \ \ t=-1\)

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter.

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter.

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter.

Find an equation of the tangent to the curve at the given point by two methods: (a) without eliminating the parameter and (b) by first eliminating the parameter.

Find an equation of the tangent to the curve at the given point by two methods: (a) without eliminating the parameter and (b) by first eliminating the parameter.

Find an equation of the tangent to the curve at the given point. Then graph the curve and the tangent.

Find an equation of the tangent to the curve at the given point. Then graph the curve and the tangent.

Find dy/dx and d2y/dx2. For which values of t is the curve concave upward?

Find dy/dx and d2y/dx2. For which values of t is the curve concave upward?

Find \(dy/dx\) and \(d^{2} y / d x^{2}\). For which values of \(t\) is the curve concave upward? \(x=e^t,\quad\ \ \ y=te^{-t}\) Equation Transcription: Text Transcription: dy/dx d^2 y/dx^2 t x = e^t, y = te^-t

Find \(dy/dx\) and \(d^{2} y / d x^{2}\). For which values of \(t\) is the curve concave upward? \(x=t^2+1,\quad\ \ \ y=e^t-1\) Equation Transcription: Text Transcription: dy/dx d^2 y/dx^2 t x = t^2 + 1, y = e^t - 1

Find \(dy/dx\) and \(d^{2} y / d x^{2}\). For which values of \(t\) is the curve concave upward? \(x=t-\ln t,\quad\ \ \ y=t+\ln t\) Equation Transcription: Text Transcription: dy/dx d^2 y/dx^2 t x = t - ln t, y = t + ln t

Find \(dy/dx\) and \(d^{2} y / d x^{2}\). For which values of \(t\) is the curve concave upward? \(x=\cos\ t,\quad\ \ \ y=\sin\ 2t,\quad\ \ \ 0<t<\pi\) Equation Transcription: Text Transcription: dy/dx d^2 y/dx^2 t x = cos t, y = sin 2t, 0 < t < pi

Find the points on the curve where the tangent is horizontal or vertical. You may want to use a graph from a calculator or computer to check your work. \(x=t^3-3t,\quad\ \ \ y=t^2-3\) Equation Transcription: Text Transcription: x = t^3 - 3t, y = t^2 - 3

Find the points on the curve where the tangent is horizontal or vertical. You may want to use a graph from a calculator or computer to check your work. \(x=t^3-3t,\quad\ \ \ y=t^3-3t^2\) Equation Transcription: Text Transcription: x = t^3 - 3t, y = t^3 - 3t^2

Find the points on the curve where the tangent is horizontal or vertical. You may want to use a graph from a calculator or computer to check your work. \(x=\cos\ \theta,\quad\ \ \ y=\cos\ 3\theta\) Equation Transcription: Text Transcription: x = cos theta , y = cos 3 theta

Find the points on the curve where the tangent is horizontal or vertical. You may want to use a graph from a calculator or computer to check your work. \(x=e^{\sin\ \theta},\quad y=e^{\cos\ \theta}\) Equation Transcription: Text Transcription: x = e^sin theta , y = e^cos theta

Use a graph to estimate the coordinates of the rightmost point on the curve \(x=t-t^6,\ y=e^t\). Then use calculus to find the exact coordinates. Equation Transcription: Text Transcription: x = t - t^6, y = e^t

Use a graph to estimate the coordinates of the lowest point and the leftmost point on the curve \(x=t^4-2t,\ y=t+t^4\). Then find the exact coordinates. Equation Transcription: Text Transcription: x = t^4 - 2t, y = t + t^4

Graph the curve in a viewing rectangle that displays all the important aspects of the curve. \(x=t^4-2t^3-2t^2,\quad\ \ \ y=t^3-t\) Equation Transcription: Text Transcription: x = t^4 - 2t^3 - 2t^2, y = t^3 - t

Graph the curve in a viewing rectangle that displays all the important aspects of the curve. \(x=t^4+4t^3-8t^2,\quad\ \ \ y=2t^2-t\) Equation Transcription: Text Transcription: x = t^4 + 4t^3 - 8t^2, y = 2t^2 - t

Show that the curve \(x=\cos\ t,\ y=\sin\ t\ \cos\ t\) has two tangents at \((0, \ 0)\) and find their equations. Graph the curve. Equation Transcription: Text Transcription: x = cos t, y = sin t cos t (0, 0)

Graph the curve \(x=-2\ \cos\ t, \ y=\sin\ t+\sin\ 2t\) to discover where it crosses itself. Then find equations of both tangents at that point.

(a) Find the slope of the tangent line to the trochoid \(x=r\theta-d\ \sin\ \theta,\ y=r-d\ \cos\ \theta\) in terms of \(\theta\). (See Exercise 10.1.49.) (b) Show that if \(d<r\), then the trochoid does not have a vertical tangent. Equation Transcription: Text Transcription: x = r theta - d sin , y = r - d cos theta theta d<r

(a) Find the slope of the tangent to the astroid \(x=a\ \cos^3 \theta,\ y=a\ \sin^3 \theta\) in terms of \(\theta\). (Astroids are explored in the Discovery Project following Section 10.1.) (b) At what points is the tangent horizontal or vertical? (c) At what points does the tangent have slope \(1\) or \(-1\)? Equation Transcription: Text Transcription: x = a cos^3 theta, y = a sin^3 theta theta 1 -1

At what point(s) on the curve \(x=3t^2+1, \ y=t^3-1\) does the tangent line have slope \(\frac{1}{2}\)?

Find equations of the tangents to the curve \(x=3t^2+1,\ y=2 t^{3}+1\) that pass through the point \((4, \ 3)\). Equation Transcription: Text Transcription: x = 3t^2 + 1, y = 2t^3 + 1 (4, 3)

Find the area enclosed by the given parametric curve and the \(x\)-axis. \(x=t^{3}+1, \quad y=2 t-t^{2}\) Equation Transcription: Text Transcription: x x = t^3 + 1, y = 2t - t^2

Find the area enclosed by the given parametric curve and the x-axis. \(x=\sin \ t, \quad y=\sin \ t \cos \ t, \quad 0 \leqslant t \leqslant \pi / 2\) Equation Transcription: ? ? Text Transcription: x = sin t, y = sin t cos t, 0 leqslant t leqslant pi/2

Find the area enclosed by the given parametric curve and the \(y\)-axis. \(x=\sin^2t\) \(y=\cos t\) Equation Transcription: Text Transcription: y x = sin^2 t y = cos t

Find the area enclosed by the given parametric curve and the y-axis. \(x=t^{2}-2 t\), \(y=\sqrt{t}\)

Use the parametric equations of an ellipse, \(x=a\ \cos\ \theta,\ \ y=b \sin \theta, \ \ 0 \leqslant \theta \leqslant 2 \pi\), to find the area that it encloses.

Find the area of the region enclosed by the loop of the curve \(x=1-t^2,\quad\ \ \ y=t-t^3\) Equation Transcription: Text Transcription: x = 1 - t^2, y = t - t^3

Find the area under one arch of the trochoid of Exercise 10.1.49 for the case \(d<r\). Equation Transcription: Text Transcription: d < r

Let \(\mathscr{R}\) be the region enclosed by the loop of the curve in Example 1. (a) Find the area of \(\mathscr{R}\). (b) If \(\mathscr{R}\) is rotated about the \(x\)-axis, find the volume of the resulting solid. (c) Find the centroid of \(\mathscr{R}\). Equation Transcription: ? Text Transcription: ? x Image text transcription: Let 5 be the region enclosed by the loop of the curve in Example 1. (a) Find the area of 5. (b) If 5 is rotated about the x-axis, find the volume of the resulting solid. (c) Find the centroid of 5.

Set up an integral that represents the length of the part of the parametric curve shown in the graph. Then use a calculator (or computer) to find the length correct to four decimal places. \(x=3t^2-t^3,\quad\ \ \ y=t^2-2t\) Equation Transcription: Text Transcription: x = 3t^2 - t^3, y = t^2 - 2t

Set up an integral that represents the length of the part of the parametric curve shown in the graph. Then use a calculator (or computer) to find the length correct to four decimal places. \(x=t+e^{-t},\quad\ \ \ y=t^2+t\) Equation Transcription: Text Transcription: x = t + e^-t, y = t^2 + t

Set up an integral that represents the length of the part of the parametric curve shown in the graph. Then use a calculator (or computer) to find the length correct to four decimal places. \(x=t-2 \ \sin \ t, \quad y=1-2 \ \cos \ t, \quad 0 \leqslant t \leqslant 4 \pi\) Equation Transcription: ? ? Text Transcription: x = t - 2 sin t, y = 1 - 2 cos t, 0 leqslant t leqslant 4

Set up an integral that represents the length of the part of the parametric curve shown in the graph. Then use a calculator (or computer) to find the length correct to four decimal places. \(x=t\ \cos\ t,\quad\ \ \ y=t-5\ \sin\ t\) Equation Transcription: Text Transcription: x = t cos t, y = t - 5 sin t

Find the exact length of the curve. \(x=\frac{2}{3} t^{3}, \ \ y=t^{2}-2, \ \ 0 \leqslant t \leqslant 3\)

Find the exact length of the curve. \(x=e^{t}-t, \quad y=4 e^{t / 2}, \quad 0 \leqslant t \leqslant 2\) Equation Transcription: ? ? Text Transcription: x = e^t - t, y = 4e^t/2, 0 leqslant t leqslant 2

Find the exact length of the curve. \(x=t \ \sin \ t, \quad y=t \ \cos \ t, \quad 0 \leqslant t \leqslant 1\) Equation Transcription: ? ? Text Transcription: x = t sin t, y = t cos t, 0 leqslant t leqslant 1

Find the exact length of the curve. \(x=3 \ \cos \ t-\cos \ 3 t, \quad y=3 \ \sin \ t-\sin \ 3 t, \quad 0 \leqslant t \leqslant \pi\) Equation Transcription: ? ? Text Transcription: x = 3 cos t - cos 3t, y = 3 sin t - sin 3t, 0 leqslant t leqslant

Graph the curve and find its exact length. \(x=e^{t} \ \cos \ t, \quad y=e^{t} \ \sin \ t, \quad 0 \leqslant t \leqslant \pi\) Equation Transcription: ? ? Text Transcription: x = e^t cos t, y = e^t sin t, 0 leqslant t leqslant pi

Graph the curve and find its exact length. \(x=\cos \ t+\ln \left(\tan \frac{1}{2} t\right), \quad y=\sin \ t, \quad \pi / 4 \leqslant t \leqslant 3 \pi / 4\) Equation Transcription: ? ? Text Transcription: x = cos t + ln(tan 1/2 t), y = sin t, pi/4 leqslant t leqslant 3pi/4

Graph the curve \(x=\sin\ t+\sin\ 1.5t,\ y=\cos\ t\) and find its length correct to four decimal places. Equation Transcription: Text Transcription: x = sin t + sin 1.5t, y = cos t

Find the length of the loop of the curve \(x=3t-t^3,\quad\ y=3t^2\). Equation Transcription: Text Transcription: x = 3t - t^3, y = 3t^2

Find the distance traveled by a particle with position \((x, \ y)\) as \(t\) varies in the given time interval. Compare with the length of the curve. \(x=\sin ^{2} t, \quad y=\cos ^{2} t, \quad 0 \leqslant t \leqslant 3 \pi\) Equation Transcription: ? ? Text Transcription: (x, y) t x = sin^2 t, y = cos^2 t, 0 leqslant t leqslant 3pi

Find the distance traveled by a particle with position \((x, \ y)\) as \(t\) varies in the given time interval. Compare with the length of the curve. \(x=\cos ^{2} t, \quad y=\cos \ t, \quad 0 \leqslant t \leqslant 4 \pi\) Equation Transcription: ? ? Text Transcription: (x, y) t x = cos^2 t, y = cos t, 0 leqslant t leqslant 4pi

The parametric equations give the position (in meters) of a moving particle at time \(t\) (in seconds). Find the speed of the particle at the indicated time or point. \(x=2t-3,\quad\ \ \ y=2t^2-3t+6;\quad\ \ \ t=5\) Equation Transcription: Text Transcription: t x = 2t - 3, y = 2t^2 - 3t + 6; t = 5

The parametric equations give the position (in meters) of a moving particle at time \(t\) (in seconds). Find the speed of the particle at the indicated time or point. \(x=2+5\cos\left(\frac{\pi}{3}t\right),\quad\ \ \ y=-2+7\sin\left(\frac{\pi}{3}t\right);\quad\ \ \ t=3\) Equation Transcription: Text Transcription: t x = 2 + 5 cos (pi/3 t), y = -2 + 7 sin (pi/3 t); t = 3

The parametric equations give the position (in meters) of a moving particle at time \(t\) (in seconds). Find the speed of the particle at the indicated time or point. \(x=e^t,\quad\ \ \ y=te^t;\quad\ \ \ (e,\ e)\) Equation Transcription: Text Transcription: t x = e^t, y = te^t; (e, e)

The parametric equations give the position (in meters) of a moving particle at time \(t\) (in seconds). Find the speed of the particle at the indicated time or point. \(x=t^2+1,\quad\ \ \ y=t^4+2t^2+1;\quad\ \ \ (2,\ 4)\) Equation Transcription: Text Transcription: t x = t^2 + 1, y = t^4 + 2t^2 + 1; (2, 4)

A projectile is fired from the point \((0, \ 0)\) with an initial velocity of \(v_0\mathrm{\ m}/\mathrm{s}\) at an angle \(\alpha\) above the horizontal. (See Exercise 10.1.58.) If we assume that air resistance is negligible, then the position (in meters) of the projectile after \(t\) seconds is given by the parametric equations \(x=\left(v_0\ \cos\ \alpha\right)t\) \(y=\left(v_0\ \sin\ \alpha\right)t-\frac{1}{2}gt^2\) where \(g=9.8\mathrm{\ m}/\mathrm{s}^2\) is the acceleration due to gravity. (a) Find the speed of the projectile when it hits the ground. (b) Find the speed of the projectile at its highest point. Equation Transcription: Text Transcription: (0, 0) v_0 m/s alpha t x = (v_0 cos alpha)t y = (v_0 sin alpha)t - 1/2 gt^2 g = 9.8 m/s^2

Show that the total length of the ellipse \(x=a\ \sin\ \theta,\ y=b\ \cos\ \theta,\ a>b>0\), is \(L=4a\ \int_0^{\pi/2}\ \sqrt{1-e^2\sin^2\theta}\ d\theta\) where \(e\) is the eccentricity of the ellipse (\(e=c/a\), where \(c=\sqrt{a^{2}-b^{2}}\)). Equation Transcription: Text Transcription: x = a sin theta, y = b cos theta, a > b > 0 L = 4a int_0 ^pi/2 sqrt 1 - e^2 sin^2 theta d theta e e = c/a c = sqrt a^2 - b^2

(a) Graph the epitrochoid with equations \(x=11 \ cos \ t?4 \ cos(11t/2)\) \(y=11 \ sin \ t?4 \ sin(11t/2)\) What parameter interval gives the complete curve? (b) Use a calculator or computer to find the approximate length of this curve. Equation Transcription: Text Transcription: x = 11 cos t - 4 cos(11t/2) y = 11 sin t - 4 sin(11t/2)

A curve called Cornu’s spiral is defined by the parametric equations \(x=C(t)=\int_0^t\ \cos\left(\pi\ u^2/2\right)\ du\) \(y=S(t)=\int_0^t\ \sin\left(\pi\ u^2/2\right)\ du\) where \(C\) and \(S\) are the Fresnel functions that were introduced in Chapter 5. (a) Graph this curve. What happens as \(t \rightarrow \infty\) and as \(t \rightarrow-\infty\)? (b) Find the length of Cornu’s spiral from the origin to the point with parameter value \(t\). Equation Transcription: Text Transcription: x = C(t) = int_0 ^t cos(pi u^2 /2) du y = S(t) = int_0 ^t sin(pi u^2 /2) du C S t rightarrow infty t rightarrow - infty t

The curve shown in the figure is the astroid \(x=a \cos ^3 \theta, y=a \sin ^3 \theta\). (Astroids are explored in the Discovery Project following Section 10.1.) Find the area of the region enclosed by the astroid.

The curve shown in the figure is the astroid \(x=a\ \cos^3\ \theta\), \(y=a\ \sin^3\ \theta\). (Astroids are explored in the Discovery Project following Section 10.1.) Find the perimeter of the astroid. Equation Transcription: Text Transcription: x = a cos^3 theta y = a sin^3 theta

Set up an integral that represents the area of the surface obtained by rotating the given curve about the \(x\)-axis. Then use a calculator or computer to find the surface area correct to four decimal places. \(x=t \ \sin \ t, \quad y=t \ \cos \ t, \quad 0 \leqslant t \leqslant \pi / 2\) Equation Transcription: ? ? Text Transcription: x x = t sin t, y = t cos t, 0 leqslant t leqslant pi/2

Set up an integral that represents the area of the surface obtained by rotating the given curve about the \(x\)-axis. Then use a calculator or computer to find the surface area correct to four decimal places. \(x=\sin \ t, \quad y=\sin \ 2 t, \quad 0 \leqslant t \leqslant \pi / 2\) Equation Transcription: ? ? Text Transcription: x x = sin t, y = sin 2t, 0 leqslant t leqslant pi/2

Set up an integral that represents the area of the surface obtained by rotating the given curve about the \(x\)-axis. Then use a calculator or computer to find the surface area correct to four decimal places. \(x=t+e^{t}, \quad y=e^{-t}, \quad 0 \leqslant t \leqslant 1\) Equation Transcription: ? ? Text Transcription: x x = t + e^t, y = e^-t, 0 leqslant t leqslant 1

Set up an integral that represents the area of the surface obtained by rotating the given curve about the \(x\)-axis. Then use a calculator or computer to find the surface area correct to four decimal places. \(x=t^{2}-t^{3}, \quad y=t+t^{4}, \quad 0 \leqslant t \leqslant 1\) Equation Transcription: ? ? Text Transcription: x x = t^2 - t^3, y = t + t^4, 0 leqslant t leqslant 1

Find the exact area of the surface obtained by rotating the given curve about the \(x\)-axis. \(x=t^{3}, \quad y=t^{2}, \quad 0 \leqslant t \leqslant 1\) Equation Transcription: ? ? Text Transcription: x x = t^3, y = t^2, 0 leqslant t leqslant 1

Find the exact area of the surface obtained by rotating the given curve about the x-axis. \(x=2 t^{2}+1 / t, \ \ y=8 \sqrt{t}, \ \ 1 \leqslant t \leqslant 3\)

Find the exact area of the surface obtained by rotating the given curve about the \(x\)-axis. \(x=a \cos ^{3} \theta, \quad y=a \sin ^{3} \theta, \quad 0 \leqslant \theta \leqslant \pi / 2\)

Graph the curve \(x=2\ \cos\ \theta-\cos\ 2\theta\) \(y=2\ \sin\ \theta-\sin\ 2\theta\) If this curve is rotated about the \(x\)-axis, find the exact area of the resulting surface. (Use your graph to help find the correct parameter interval.) Equation Transcription: Text Transcription: x = 2 cos theta - cos 2 theta y=2 sin theta - sin 2 theta x

Find the surface area generated by rotating the given curve about the y-axis. \(x=3 t^2, \quad y=2 t^3, \quad 0 \leqslant t \leqslant 5\)

Find the surface area generated by rotating the given curve about the \(y\)-axis. \(x=e^{t}-t, \quad y=4 e^{t / 2}, \quad 0 \leqslant t \leqslant 1\) Equation Transcription: ? ? Text Transcription: y x = e^t - t, y = 4e^t/2, 0 leqslant t leqslant 1

If \(f^{\prime}\) is continuous and \(f^{\prime}(t) \neq 0\) for \(a \leqslant t \leqslant b\), show that the parametric curve \(x=f(t), \ y=g(t), \ a \leqslant t \leqslant b\), can be put in the form \(y=F(x)\). [Hint: Show that \(f^{-1}\) exists.] Equation Transcription: ? ? ? ? Text Transcription: f’ f'(t) neq 0 a leqslant t leqslant b x = f(t), y = g(t), a leqslant t leqslant b y = F(x) f^-1

Use Formula 1 to derive Formula 9 from Formula 8.2.5 for the case in which the curve can be represented in the form \(y=F(x), \ a \leqslant x \leqslant b\). Equation Transcription: ? ? Text Transcription: y = F(x), a leqslant x leqslant b

The curvature at a point \(P\) of a curve is defined as \(\boldsymbol{\kappa}=\left|\frac{d \phi}{d s}\right|\) where \(\phi\) is the angle of inclination of the tangent line at \(P\), as shown in the figure. Thus the curvature is the absolute value of the rate of change of \(\phi\) with respect to arc length. It can be regarded as a measure of the rate of change of direction of the curve at \(P\) and will be studied in greater detail in Chapter 13. For a parametric curve \(x=x(t),\ y=y(t)\), derive the formula where the dots indicate derivatives with respect to \(t\), so \(\dot{x}=d x / d t\). [Hint: Use \(\phi=\tan ^{-1}(d y / d x)\) and Formula 2 to find \(d \phi / d t\). Then use the Chain Rule to find \(d \phi / d s\).] Equation Transcription: Text Transcription: P Kappa = d phi / ds phi P phi x = x(t), y = y(t) t x = dx/dt phi = tan^-1(dy/dx) d phi / dt d phi / ds

The curvature at a point \(P\) of a curve is defined as \(\boldsymbol{\kappa}=\left|\frac{d \phi}{d s}\right|\) where \(\phi\) is the angle of inclination of the tangent line at \(P\), as shown in the figure. Thus the curvature is the absolute value of the rate of change of \(\phi\) with respect to arc length. It can be regarded as a measure of the rate of change of direction of the curve at \(P\) and will be studied in greater detail in Chapter 13. By regarding a curve \(y=f(x)\) as the parametric curve \(x=x, \ y=f(x)\) with parameter \(x\), show that the formula in Exercise 79 becomes \(\kappa=\frac{\left|d^{2} y / d x^{2}\right|}{\left[1+(d y / d x)^{2}\right]^{3 / 2}}\) Equation Transcription: Text Transcription: P Kappa = d phi / ds phi P phi P y = f(x) x = x, f = f(x) x Kappa = |d^2 y / dx^2| / [1 + (dy/dx)^2 ]^3/2

The curvature at a point \(P\) of a curve is defined as \(\boldsymbol{\kappa}=\left|\frac{d \phi}{d s}\right|\) where \(\phi\) is the angle of inclination of the tangent line at \(P\), as shown in the figure. Thus the curvature is the absolute value of the rate of change of \(\phi\) with respect to arc length. It can be regarded as a measure of the rate of change of direction of the curve at \(P\) and will be studied in greater detail in Chapter 13. Use the formula in Exercise 79 to find the curvature of the cycloid \(x=\theta-\sin\ \theta,\ \ y=1-\cos\ \theta\) at the top of one of its arches. Equation Transcription: Text Transcription: P Kappa = d phi / ds phi P phi P x = theta - sin theta, y = 1 - cos theta

The curvature at a point \(P\) of a curve is defined as \(\boldsymbol{\kappa}=\left|\frac{d \phi}{d s}\right|\) where \(\phi\) is the angle of inclination of the tangent line at \(P\), as shown in the figure. Thus the curvature is the absolute value of the rate of change of \(\phi\) with respect to arc length. It can be regarded as a measure of the rate of change of direction of the curve at \(P\) and will be studied in greater detail in Chapter 13. (a) Use the formula in Exercise 80 to find the curvature of the parabola \(y=x^{2}\) at the point \((1, \ 1)\). (b) At what point does this parabola have maximum curvature? Equation Transcription: Text Transcription: P Kappa = d phi / ds phi P phi P y = x^2 (1, 1)

The curvature at a point \(P\) of a curve is defined as \(\boldsymbol{\kappa}=\left|\frac{d \phi}{d s}\right|\) where \(\phi\) is the angle of inclination of the tangent line at \(P\), as shown in the figure. Thus the curvature is the absolute value of the rate of change of \(\phi\) with respect to arc length. It can be regarded as a measure of the rate of change of direction of the curve at \(P\) and will be studied in greater detail in Chapter 13. (a) Show that the curvature at each point of a straight line is \(\kappa=0\). (b) Show that the curvature at each point of a circle of radius r is \(\kappa=1/r\). Equation Transcription: Text Transcription: P kappa = d phi / ds phi P phi P kappa = 0 kappa = 1/r

A cow is tied to a silo with radius \(r\) by a rope just long enough to reach the opposite side of the silo. Find the grazing area available for the cow. Equation Transcription: Text Transcription: r

A string is wound around a circle and then unwound while being held taut. The curve traced by the point \(P\) at the end of the string is called the involute of the circle. If the circle has radius \(r\) and center \(O\) and the initial position of \(P\) is \((r, \ 0)\), and if the parameter \(\theta\) is chosen as in the figure, show that parametric equations of the involute are \(x=r(\cos\ \theta+\theta\ \sin\ \theta)\) \(y=r(\sin\ \theta-\theta\ \cos\ \theta)\) Equation Transcription: Text Transcription: P r O P (r, 0) theta x = r(cos theta + theta sin theta) y = r(sin theta + theta cos theta)