?For the following exercises, write the equation of the tangent line in Cartesian

For the following exercises, each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line. \(x=3+t, \quad y=1-t\) Text Transcription: x=3+t, \quad y=1-t

For the following exercises, each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line. \(x=8+2 t, \quad y=1\) Text Transcription: x=8+2 t, \quad y=1

For the following exercises, each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line. \(x=4-3 t, \quad y=-2+6 t\) Text Transcription: x=4-3 t, \quad y=-2+6 t

For the following exercises, each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line. \(x=-5 t+7, \quad y=3 t-1\) Text Transcription x=-5 t+7, \quad y=3 t-1

For the following exercises, determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter. \(x=3 \sin t, \quad y=3 \cos t, \quad t=\frac{\pi}{4}\) Text Transcription: x=3 \sin t, \quad y=3 \cos t, \quad t=pi/4

For the following exercises, determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter. \(x=\cos t, \quad y=8 \sin t, t=\frac{\pi}{2}\) Text Transcription: x=\cos t, \quad y=8 \sin t, t=pi/2

For the following exercises, determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter. \(x=2 t, \quad y=t^{3}, \quad t=-1\) Text Transcription: x=2 t, \quad y=t^3, \quad t=-1

For the following exercises, determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter. \(x=t+\frac{1}{t}, \quad y=t-\frac{1}{t}, \quad t=1\) Text Transcription: x=t+1/t, \quad y=t-1/t, \quad t=1

For the following exercises, determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter. \(x=\sqrt{t}, \quad y=2 t, \quad t=4\) Text Transcription: x=\sqrt t, \quad y=2 t, \quad t=4

For the following exercises, find all points on the curve that have the given slope. \(x=4 \cos t, \quad y=4 \sin t, \quad \text { slope }=0.5\) Text Transcription: x=4 \cos t, \quad y=4 \sin t, \quad \text slope =0.5

For the following exercises, find all points on the curve that have the given slope. \(x=2 \cos t, \quad y=8 \sin t, \text { slope }=-1\) Text Transcription: x=2 \cos t, \quad y=8 \sin t, \text slope =-1

For the following exercises, find all points on the curve that have the given slope. \(x=t+\frac{1}{t}, \quad y=t-\frac{1}{t}, \text { slope }=1\) Text Transcription: x=t+1/t, \quad y=t-1/t, \text slope =1

For the following exercises, find all points on the curve that have the given slope. \(x=2+\sqrt{t}, \quad y=2-4 t, \text { slope }=0\) Text Transcription: x=2+\sqrt t, \quad y=2-4 t, \text slope =0

For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t. \(x=e^{\sqrt{t}}, \quad y=1-\ln t^{2}, \quad t=1\) Text Transcription: x=e^\sqrt t, \quad y=1-\ln t^2, \quad t=1

For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t. \(x=t \ln t, \quad y=\sin ^{2} t, t=\frac{\pi}{4}\) Text Transcription: x=t \ln t, \quad y=\sin ^2 t, t=pi/4

For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t. \(x=e^{t}, \quad y=(t-1)^{2}, \quad \operatorname{at}(1,1)\) Text Transcription: x=e^t, \quad y=(t-1)^2, \quad \operatorname at (1,1)

For \(x=\sin (2 t), y=2 \sin t\) where \(0 \leq t<2 \pi\). Find all values of t at which a horizontal tangent line exists. Text Transcription: x=\sin (2 t), y=2 \sin t 0 \leq t<2 \pi

For \(x=\sin (2 t), y=2 \sin t\) where \(0 \leq t<2 \pi\). Find all values of t at which a vertical tangent line exists. Text Transcription: x=\sin (2 t), y=2 \sin t 0 \leq t<2 \pi

Find all points on the curve \(x=4 \cos (t), y=4 \sin (t)\) that have the slope of \(\frac{1}{2}\). Text Transcription: x=4 \cos (t), y=4 \sin (t) 1/2

For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t. Find \(\frac{dy}{dx}\) for x = sin(t), y = cos(t). Text Transcription: dy/dx

For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t. Find the equation of the tangent line to x = sin(t), y = cos(t) at \(t=\frac{\pi}{4}\). Text Transcription: t = pi/4

For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t. For the curve x = 4t, y = 3t ? 2, find the slope and concavity of the curve at t = 3.

For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t. For the parametric curve whose equation is \(x=4 \cos \theta, y=4 \sin \theta\), find the slope and concavity of the curve at \(\theta=\frac{\pi}{4}\) . Text Transcription: x=4 cos theta, y=4 sin theta theta=pi/4

For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t. Find the slope and concavity for the curve whose equation is \(x=2+\sec \theta, y=1+2 \tan \theta\) at \(\theta=\frac{\pi}{6}\) . Text Transcription: x=2+sec theta, y=1+2 tan theta theta=pi/6

For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t. Find all points on the curve x = t + 4, \(y=t^{3}-3 t\) at which there are vertical and horizontal tangents. Text Transcription: y=t^3 - 3t

For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t. Find all points on the curve \(x=\sec \theta, y=\tan \theta\) at which horizontal and vertical tangents exist. Text Transcription: x=sec theta, y=tan theta

For the following exercises, find \(d^{2} y / d x^{2}\) \(x=t^{4}-1,\ \ \ \ \ y=t-t^{2}\) Text Transcription: d^2 y / dx^2 x=t^4-1, y=t-t^2

For the following exercises, find \(d^{2} y / d x^{2}\) \(x=\sin(\pi t),\ \ \ \ \ y=\cos(\pi t)\) Text Transcription: d^2 y / dx^2 x=sin(pi t), y=cos(pi t)

For the following exercises, find \(d^{2} y / d x^{2}\) \(x=e^{-t},\ \ \ \ \ y=te^{2t}\) Text Transcription: d^2 y / dx^2 x=e^-t, y=te^2t

For the following exercises, find points on the curve at which tangent line is horizontal or vertical. \(x=t\left(t^{2}-3\right), \quad y=3\left(t^{2}-3\right)\) Text Transcription: x=tt(t^2-3), y=3(t^2-3)

For the following exercises, find points on the curve at which tangent line is horizontal or vertical. \(x=\frac{3 t}{1+t^{3}}, \quad y=\frac{3 t^{2}}{1+t^{3}}\) Text Transcription: x= 3t / 1+t^3, y=3t^2 / 1+t^3

For the following exercises, find dy/dx at the value of the parameter. x = cos t, y = sin t, \(t=\frac{3 \pi}{4}\) Text Transcription: t=3pi/4

For the following exercises, find dy/dx at the value of the parameter. \(x=\sqrt{t}\) y = 2t + 4, t = 9 Text Transcription: x=\sqrt{t}

For the following exercises, find dy/dx at the value of the parameter. x = 4 cos(2?s), y = 3 sin(2?s), \(s=-\frac{1}{4}\) Text Transcription: s=-1/4

For the following exercises, find \(d^{2} y / d x^{2}\) at the given point without eliminating the parameter. \(x=\frac{1}{2} t^{2}, \quad y=\frac{1}{3} t^{3}\), t = 2 Text Transcription: d^2 y / dx^2 x=1/2 t^2, y=1/3 t^3

For the following exercises, find \(d^{2} y / d x^{2}\) at the given point without eliminating the parameter. \(x=\sqrt{t}\), y = 2t + 4, t = 1 Text Transcription: d^2 y / dx^2 x=sqrt t

For the following exercises, find \(d^{2} y / d x^{2}\) at the given point without eliminating the parameter. Find t intervals on which the curve \(x=3 t^{2}, y=t^{3}-t\) is concave up as well as concave down. Text Transcription: d^2 y / dx^2 x=3t^2, y=t^3 - t

For the following exercises, find \(d^{2} y / d x^{2}\) at the given point without eliminating the parameter. Determine the concavity of the curve x = 2t + ln t, y = 2t ? ln t.

For the following exercises, find \(d^{2} y / d x^{2}\) at the given point without eliminating the parameter. Sketch and find the area under one arch of the cycloid \(x=r(\theta-\sin \theta), y=r(1-\cos \theta)\). Text Transcription: d^2 y / dx^2 x=r(theta - sin theta), y=r(1- cos theta)

For the following exercises, find \(d^{2} y / d x^{2}\) at the given point without eliminating the parameter. Find the area bounded by the curve x = cos t, \(y=e^{t}\), \(0 \leq t \leq \frac{\pi}{2}\) and the lines y = 1 and x = 0. Text Transcription: d^2 y / dx^2 y=e^t 0 leq t leq pi/2

For the following exercises, find \(d^{2} y / d x^{2}\) at the given point without eliminating the parameter. Find the area enclosed by the ellipse \(x=a \cos \theta, y=b \sin \theta, 0 \leq \theta<2 \pi\). Text Transcription: d^2 y / dx^2 x=a cos theta, y=b sin theta, 0 leq theta<2pi

For the following exercises, find \(d^{2} y / d x^{2}\) at the given point without eliminating the parameter. Find the area of the region bounded by \(x=2 \sin ^{2} \theta, y=2 \sin ^{2} \theta \tan \theta\), for \(0 \leq \theta \leq \frac{\pi}{2}\) . Text Transcription: d^2 y / dx^2 x=2 sin^2 theta, y=2 sin^2 theta tan theta 0 leq theta leq pi/2

For the following exercises, find the area of the regions bounded by the parametric curves and the indicated values of the parameter. \(x=2 \cot \theta, y=2 \sin ^{2} \theta, 0 \leq \theta \leq \pi\) Text Transcription: x=2 cot theta, y=2 sin^2 theta, 0 leq theta leq pi

For the following exercises, find the area of the regions bounded by the parametric curves and the indicated values of the parameter. [T] x = 2a cos t ? a cos(2t), y = 2a sin t ? a sin(2t), \(0 \leq t<2 \pi\) Text Transcription: 0 leq t < 2pi

For the following exercises, find the area of the regions bounded by the parametric curves and the indicated values of the parameter. [T] x = a sin(2t), y = b sin(t), \(0 \leq t<2 \pi\) (the “hourglass”) Text Transcription: 0 leq t < 2pi

For the following exercises, find the area of the regions bounded by the parametric curves and the indicated values of the parameter. [T] x = 2a cos t ? a sin(2t), y = b sin t, \(0 \leq t<2 \pi\) (the “teardrop”) Text Transcription: 0 leq t < 2pi

For the following exercises, find the arc length of the curve on the indicated interval of the parameter. x = 4t + 3, y = 3t ? 2, \(0 \leq t \leq 2 \pi\) Text Transcription: 0 leq t leq 2pi

For the following exercises, find the arc length of the curve on the indicated interval of the parameter. \(x=\frac{1}{3} t^{3}, \quad y=\frac{1}{2} t^{2}, \quad 0 \leq t \leq 1\) Text Transcription: x=1/3 t^3, y=1/2 t^2, 0 leq t leq 1

For the following exercises, find the arc length of the curve on the indicated interval of the parameter. x = cos(2t), y = sin(2t), \(0 \leq t \leq \frac{\pi}{2}\) Text Transcription: 0 leq t leq pi/2

For the following exercises, find the arc length of the curve on the indicated interval of the parameter. \(x=1+t^{2}, \quad y=(1+t)^{3}, \quad 0 \leq t \leq 1\) Text Transcription: x=1+t^2, y=(1+t)^3, 0 leq t leq 1

For the following exercises, find the arc length of the curve on the indicated interval of the parameter. \(x=e^{t} \cos t, \quad y=e^{t} \sin t, \quad 0 \leq t \leq \frac{\pi}{2}\) Text Transcription: x=e^t cos t, y=e^t sin t, 0 leq t leq pi/2

For the following exercises, find the arc length of the curve on the indicated interval of the parameter. \(x=a \cos ^{3} \theta, y=a \sin ^{3} \theta\) on the interval \((0,2 \pi)\) (the hypocycloid) Text Transcription: x=a cos^3 theta, y=a sin^3 theta (0,2pi)

For the following exercises, find the arc length of the curve on the indicated interval of the parameter. Find the length of one arch of the cycloid x = 4(t ? sin t), y = 4(1 ? cos t).

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

Find the length of one arch of the cycloid \(x=\theta-\sin \theta\), \(y=1-\cos \theta\). Text Transcription: x = theta - sin theta y = 1 - cos theta

Show that the total length of the ellipse \(x=4 \sin \theta\), \(y=3 \cos \theta\) is \(L=16 \int_{0}^{\pi / 2} \sqrt{1-e^{2} \sin ^{2} \theta} d \theta\), where \(e=\frac{c}{a}\) and \(c=\sqrt{a^{2}-b^{2}}\) . Text Transcription: x = 4 sin theta y = 3 cos theta L = 16 int_0^pi/2 sqrt 1-e^2sin^2theta dtheta e = c/a c = sqrt a^2 - b^2

Find the length of the curve \(x=e^{t}-t\), \(y=4 e^{t / 2}\) , \(-8 \leq t \leq 3\). Text Transcription: x = e^t - t y = 4e^t/2 -8 leq t leq 3

For the following exercises, find the area of the surface obtained by rotating the given curve about the x-axis. \(x=t^{3}\), \(y=t^{2}\), \(0 \leq t \leq 1\) Text Transcription: x = t^3 y = t^2 0 leq t leq 1

For the following exercises, find the area of the surface obtained by rotating the given curve about the x-axis. \(x=a \cos ^{3} \theta\), \(y=a \sin ^{3} \theta\), \(0 \leq \theta \leq \frac{\pi}{2}\) Text Transcription: x = a cos^3 theta y = a sin^3 theta 0 leq theta leq pi/2

[T] Use a CAS to find the area of the surface generated by rotating \(x=t+t^{3}\) , \(y=t-\frac{1}{t^{2}}\) , \(1 \leq t \leq 2\) about the x-axis. (Answer to three decimal places.) Text Transcription: x = t + t^3 y = t - 1/t^2 1 leq t leq 2

Find the surface area obtained by rotating \(x=3 t^{2}\) , \(y=2 t^{3}\) , \(0 \leq t \leq 5\) about the y-axis. Text Transcription: x = 3t^2 y = 2t^3 0 leq t leq 5

Find the area of the surface generated by revolving \(x=t^{2}\) , y = 2t, \(0 \leq t \leq 4\) about the x-axis. Text Transcription: x = t^2 0 leq t leq 4

Find the area of the surface generated by revolving \(x=t^{2}\) , \(y=2 t^{2}\), \(0 \leq t \leq 1\) about the y-axis. Text Transcription: x = t^2 y = 2t^2 0 leq t leq 1