Solved: What is estimation When is it helpful to use estimation

Consider the parametric equations \(x=\sqrt{t}\) and y = 3 - t. (a) Construct a table of values for t = 0, 1, 2, 3, and 4. (b) Plot the points (x, y) generated in the table, and sketch a graph of the parametric equations. Indicate the orientation of the graph. (c) Use a graphing utility to confirm your graph in part (b). (d) Find the rectangular equation by eliminating the parameter, and sketch its graph. Compare the graph in part (b) with the graph of the rectangular equation. Text Transcription: x=sqrtt

Consider the parametric equations \(x=4\ \cos^2 \theta\ \text{and}\ y=2\ \sin \theta\). (a) Construct a table of values of \(\theta=-\frac{\pi}{2},\ -\frac{\pi}{4},\ 0,\ \frac{\pi}{4},\ \text{and}\ \frac{\pi}{2}\). (b) Plot the points (x, y) generated in the table, and sketch a graph of the parametric equations. Indicate the orientation of the graph. (c) Use a graphing utility to confirm your graph in part (b). (d) Find the rectangular equation by eliminating the parameter, and sketch its graph. Compare the graph in part (b) with the graph of the rectangular equation. (e) If values of \(\theta\) were selected from the interval \([\pi/2,\ 3\pi/2]\) for the table in part (a), would the graph in part (b) be different? Explain. Text Transcription: x=4 cos^2 theta and y = 2 sin theta theta = -pi/2 , -pi/4 , 0, pi/4 , and pi/2 theta [pi/2, 3pi/2]

In Exercises 3–20, sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. x = 2t - 3, y = 3t + 1

In Exercises 3–20, sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. x = 5 - 4t, y = 2 + 5t

In Exercises 3–20, sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. \(x=t+1,\ \ \ y=t^2\) Text Transcription: x=t+1, y=t^2

In Exercises 3–20, sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. \(x=2t^2,\ \ \ y=t^4+1\) Text Transcription: x=2t^2 , y=t^4 +1

In Exercises 3–20, sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. \(x=t^3,\ \ \ y=\frac{t^2}{2}\) Text Transcription: x=t^3 , y=t^2 /2

In Exercises 3–20, sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. \(x=t^2+t,\ \ \ y=t^2-t\) Text Transcription: x=t^2 +t , y=t^2 -t

In Exercises 3–20, sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. \(x=\sqrt{t},\ \ \ y=t-5\) Text Transcription: x=sqrtt , y=t-5

In Exercises 3–20, sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. \(x=\sqrt[4]{t},\ \ \ y=8-t\) Text Transcription: x=4th root t, y=8-t

In Exercises 3–20, sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. \(x=t-3,\ \ \ y=\frac{t}{t-3}\) Text Transcription: x=t-3, y=t/t-3

In Exercises 3–20, sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. \(x=1+\frac{1}{t},\ \ \ y=t-1\) Text Transcription: x=1+1/t , y=t-1

In Exercises 3–20, sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. x = 2t, y = | t - 2 |

In Exercises 3–20, sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. x = | t - 1 |, y = t + 2

In Exercises 3–20, sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. \(x=e^t,\ \ \ y=e^{3t}+1\) Text Transcription: x=e^t , y = e^3t +1

In Exercises 3–20, sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. \(x=e^{-t},\ \ \ y=e^{2t}-1\) Text Transcription: x=e^-t , y=e^2t -1

In Exercises 3–20, sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. \(x=\sec\ \theta,\ \ \ y=\cos \theta,\ \ \ 0\ \leq\ \theta\ <\ \pi/2,\ \ \ \pi/2\ <\ \theta\ \leq\ \pi\) Text Transcription: x = sec theta, y=cos theta, 0 leq theta < pi/2, pi/2 < theta leq pi

In Exercises 3–20, sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. \(x=\tan^2 \theta,\ \ \ y=8 \sin \theta\) Text Transcription: x=tan^2 theta, y = sec^2 theta

In Exercises 3–20, sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. \(x=8 \cos\ \theta,\ \ \ y=8 \sin \theta\) Text Transcription: x=8 cos theta, y = 8 sin theta

In Exercises 3–20, sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. \(x=3 \cos \theta,\ \ \ y=7 \sin \theta\) Text Transcription: x=3 cos theta, y = 7 sin theta

In Exercises 21–32, use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. \(x=6 \sin 2\theta,\ \ y=4 \cos 2\theta\) Text Transcription: x=6 sin 2 theta, y = 4 cos 2 theta

In Exercises 21–32, use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. \(x=\cos \theta,\ \ y=2 \sin 2\theta\) Text Transcription: x=cos theta, y = 2 sin 2 theta

In Exercises 21–32, use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. \(x=4+2 \cos \theta\) \(y=-1+ \sin \theta\) Text Transcription: x=4 + 2 cos theta y=-1 + sin theta

In Exercises 21–32, use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. \(x=-2+3 \cos \theta\) \(y=-5+3 \sin \theta\) Text Transcription: x=-2+3 cos theta y=-5 + 3 sin theta

In Exercises 21–32, use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. \(x=-3+4 \cos \theta\) \(y=2+5 \sin \theta\) Text Transcription: x=-3+4 cos theta y=2+5 sin theta

In Exercises 21–32, use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. \(x=\sec \theta\) \(y=\tan \theta\) Text Transcription: x=sec theta y=tan theta

In Exercises 21–32, use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. \(x=4 \sec \theta,\ \ \ y=3 \tan \theta\) Text Transcription: x=4 sec theta, y = 3 tan theta

In Exercises 21–32, use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. \(x=\cos^3 \theta,\ \ \ y=\sin^3 \theta\) Text Transcription: x=cos^3 theta, y=sin^3 theta

In Exercises 21–32, use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. \(x=t^3,\ \ \ y=3 \ln t\) Text Transcription: x=t^3 , y=3 ln t

In Exercises 21–32, use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. \(x=\ln 2t,\ \ \ y=t^2\) Text Transcription: x=ln 2t, y=t^2

In Exercises 21–32, use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. \(x=e^{-t},\ \ \ y=e^{3t}\) Text Transcription: x=e^-t , y=e^3t

In Exercises 21–32, use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. \(x=e^{2t},\ \ \ y=e^t\) Text Transcription: x=e^2t , y=e^t

Comparing Plane Curves In Exercises 33–36, determine any differences between the curves of the parametric equations. Are the graphs the same? Are the orientations the same? Are the curves smooth? Explain. (a) \(x=t\) (b) \(x=\cos \theta\) \(y=2t+1\) \(y=2 \cos \theta +1\) (c) \(x=e^{-t}\) (d) \(x=e^t\) \(y=2e^{-t}+1\) \(y=2e^t+1\) Text Transcription: x=t y=2t+1 x=cos theta y=2 cos theta+1 x=e^-t y=2e^-t +1 x=e^t y=2e^t +1

Comparing Plane Curves In Exercises 33–36, determine any differences between the curves of the parametric equations. Are the graphs the same? Are the orientations the same? Are the curves smooth? Explain. (a) \(x=2 \cos \theta\) (b) \(x=\sqrt{4t^2 -1}/|t|\) \(y = 2 \sin \theta\) \(y=1/t\) (c) \(x=\sqrt{t}\) (d) \(x=-\sqrt{4-e^{2t}}\) \(y=\sqrt{4-t}\) \(y=e^t\) Text Transcription: x=2 cos theta y=2 sin theta x=sqrt4t^2 -1 / |t| y=1/t x=sqrtt y=sqrt4-t x=-sqrt4-e^2t y=e^t

Comparing Plane Curves In Exercises 33–36, determine any differences between the curves of the parametric equations. Are the graphs the same? Are the orientations the same? Are the curves smooth? Explain. (a) \(x=\cos \theta\) (b) \(x=\cos (-\theta)\) \(y = 2 \sin^2 \theta\) \(y=2 \sin^2(-\theta)\) \(0\ <\ \theta\ <\ \pi\) \(0\ <\ \theta\ <\ \pi\) Text Transcription: x=cos theta y=2 sin^2 theta 0 < theta < pi x=cos(-theta) y=2 sin^2 (-theta) 0 < theta < pi

Comparing Plane Curves In Exercises 33–36, determine any differences between the curves of the parametric equations. Are the graphs the same? Are the orientations the same? Are the curves smooth? Explain. (a) \(x=t+1,\ \ y=t^3\) (b) \(x=-t+1,\ \ y=(-t)^3\) Text Transcription: x=t+1, y=t^3 x=-t+1, y=(-t)^3

Conjecture (a) Use a graphing utility to graph the curves represented by the two sets of parametric equations. x = 4 cos t x = 4 cos(-t) y = 3 sin t y = 3 sin(-t) (b) Describe the change in the graph when the sign of the parameter is changed. (c) Make a conjecture about the change in the graph of parametric equations when the sign of the parameter is changed. (d) Test your conjecture with another set of parametric equations.

Writing Review Exercises 33–36 and write a short paragraph describing how the graphs of curves represented by different sets of parametric equations can differ even though eliminating the parameter from each yields the same rectangular equation.

In Exercises 39–42, eliminate the parameter and obtain the standard form of the rectangular equation. Line through \((x_1,\ y_1)\ \text{and}\ (x_2,\ y_2)\) \(x=x_1+t(x_2-x_1),\ \ \ y=y_1+t(y_2-y_1)\) Text Transcription: (x_1 , y_1) and (x_2 , y_2) x = x_1 +t(x_2 - x_1), y = y_1 +t(y_2 - y_1)

In Exercises 39–42, eliminate the parameter and obtain the standard form of the rectangular equation. Circle: \(x=h+r \cos \theta,\ \ y=k+r \sin \theta\) Text Transcription: x=h+r cos theta, y=k+r sin theta

In Exercises 39–42, eliminate the parameter and obtain the standard form of the rectangular equation. Ellipse: \(x=h+a \cos \theta,\ \ y=k+b \sin \theta\) Text Transcription: Ellipse: x=h+a cos theta, y=k+b sin theta

In Exercises 39–42, eliminate the parameter and obtain the standard form of the rectangular equation. Hyperbola: \(x=h+a \sec \theta,\ \ y=k+b \tan \theta\) Text Transcription: x=h+a sec theta, y=k+b tan theta

In Exercises 43–50, use the results of Exercises 39–42 to find a set of parametric equations for the line or conic. Line: passes through (0, 0) and (4, -7)

In Exercises 43–50, use the results of Exercises 39–42 to find a set of parametric equations for the line or conic. Line: passes through (1, 4) and (5, -2)

In Exercises 43–50, use the results of Exercises 39–42 to find a set of parametric equations for the line or conic. Circle: center: (3, 1); radius: 2

In Exercises 43–50, use the results of Exercises 39–42 to find a set of parametric equations for the line or conic. Circle:center: (-6, 2); radius: 4

In Exercises 43–50, use the results of Exercises 39–42 to find a set of parametric equations for the line or conic. Ellipse: vertices: \((\pm10,\ 0)\); foci: \((\pm8,\ 0)\) Text Transcription: (pm10, 0) (pm8, 0)

In Exercises 43–50, use the results of Exercises 39–42 to find a set of parametric equations for the line or conic. Ellipse: vertices: (4, 7), (4, -3); foci: (4, 5), (4, -1)

In Exercises 43–50, use the results of Exercises 39–42 to find a set of parametric equations for the line or conic. Hyperbola: vertices: \((\pm4,\ 0)\); foci: \((\pm5,\ 0)\) Text Transcription: (pm4, 0) (pm5, 0)

In Exercises 43–50, use the results of Exercises 39–42 to find a set of parametric equations for the line or conic. Hyperbola: vertices: \((0,\ \pm1)\); foci: \((0,\ \pm2)\) Text Transcription: (0, pm1) (0, pm2)

In Exercises 51–54, find two different sets of parametric equations for the rectangular equation. y = 6x - 5

In Exercises 51–54, find two different sets of parametric equations for the rectangular equation. y = 4/(x - 1)

In Exercises 51–54, find two different sets of parametric equations for the rectangular equation. \(y=x^3\) Text Transcription: y=x^3

In Exercises 51–54, find two different sets of parametric equations for the rectangular equation. \(y=x^2\) Text Transcription: y=x^2

In Exercises 55–58, find a set of parametric equations for the rectangular equation that satisfies the given condition. y = 2x - 5, t = 0 at the point (3, 1)

In Exercises 55–58, find a set of parametric equations for the rectangular equation that satisfies the given condition. y = 4x + 1, t = -1 at the point (-2, -7)

In Exercises 55–58, find a set of parametric equations for the rectangular equation that satisfies the given condition. \(y=x^2,\ \ t=4\) at the point (4, 16) Text Transcription: y=x^2 , t=4

In Exercises 55–58, find a set of parametric equations for the rectangular equation that satisfies the given condition. \(y=4-x^2,\ \ t=1\) at the point (1, 3) Text Transcription: y=4-x^2 , t=1

In Exercises 59–66, use a graphing utility to graph the curve represented by the parametric equations. Indicate the direction of the curve. Identify any points at which the curve is not smooth. Cycloid: \(x=2(\theta-\sin \theta),\ \ \ y=2(1-\cos \theta)\) Text Transcription: x=2(theta - sin theta), y = 2(1 - cos theta)

In Exercises 59–66, use a graphing utility to graph the curve represented by the parametric equations. Indicate the direction of the curve. Identify any points at which the curve is not smooth. Cycloid: \(x=\theta+\sin \theta,\ \ \ y=1-\cos \theta\) Text Transcription: x=theta + sin theta, y=1-cos theta

In Exercises 59–66, use a graphing utility to graph the curve represented by the parametric equations. Indicate the direction of the curve. Identify any points at which the curve is not smooth. Prolate cycloid: \(x=\theta-\frac{3}{2} \sin \theta,\ \ \ y=1-\frac{3}{2} \cos \theta\) Text Transcription: x=theta - 3/2 sin theta, y=1-3/2 cos theta

In Exercises 59–66, use a graphing utility to graph the curve represented by the parametric equations. Indicate the direction of the curve. Identify any points at which the curve is not smooth. Prolate cycloid: \(x=2\theta-4 \sin \theta,\ \ \ y=2-4 \cos \theta\) Text Transcription: x=2theta - 4 sin theta, y=2-4 cos theta

In Exercises 59–66, use a graphing utility to graph the curve represented by the parametric equations. Indicate the direction of the curve. Identify any points at which the curve is not smooth. Hypocycloid: \(x=3 \cos^3 \theta,\ \ \ y=3 \sin^3 \theta\) Text Transcription: x=3 cos^3 theta, y=3 sin^3 theta

In Exercises 59–66, use a graphing utility to graph the curve represented by the parametric equations. Indicate the direction of the curve. Identify any points at which the curve is not smooth. Curtate cycloid: \(x=2\theta-\sin \theta,\ \ \ y=2-\cos \theta\) Text Transcription: x=2 theta - sin theta, y = 2 - cos theta

In Exercises 59–66, use a graphing utility to graph the curve represented by the parametric equations. Indicate the direction of the curve. Identify any points at which the curve is not smooth. Witch of Agnesi: \(x=2 \cot \theta,\ \ \ y=2 \sin^2 \theta\) Text Transcription: x=2 cot theta, y=2 sin^2 theta

In Exercises 59–66, use a graphing utility to graph the curve represented by the parametric equations. Indicate the direction of the curve. Identify any points at which the curve is not smooth. Folium of Descartes: \(x=3t/(1+t^3),\ \ \ y=3t^2/(1+t^3)\) Text Transcription: x=3t/(1+t^3), y=3t^2 /(1+t^3)

Explain the process of sketching a plane curve given by parametric equations. What is meant by the orientation of the curve?

Match each set of parametric equations with the correct graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] Explain your reasoning. (i) \(x=t^2-1,\ \ \ y=t+2\) (ii) \(x=\sin^2 \theta -1,\ \ \ y=\sin \theta +2\) (iii) Lissajous curve: \(x=4 \cos \theta,\ \ \ y=2 \sin 2 \theta\) (iv) Evolute of ellipse: \(x= \cos^3 \theta,\ \ \ y=2 \sin^3 \theta\) (v) Involute of circle: \(x= \cos \theta + \theta \sin \theta,\ \ \ y= \sin \theta - \theta \cos \theta\) (vi) Serpentine curve: \(x=\cot \theta,\ \ \ y=4 \sin \theta \cos \theta\) Text Transcription: x=t^2 -1, y=t+2 x=sin^2 theta-1, y=sin theta + 2 x=4 cos theta, y=2 sin 2 theta x=cos^3 theta, y=2 sin^3 theta x=cos theta + theta sin theta, y=sin theta - theta cos theta x=cot theta, y=4 sin theta cos theta

Curtate Cycloid A wheel of radius a rolls along a line without slipping. The curve traced by a point P that is b units from the center (b < a) is called a curtate cycloid (see figure). Use the angle \(\theta\) to find a set of parametric equations for this curve. Text Transcription: theta

Epicycloid A circle of radius 1 rolls around the outside of a circle of radius 2 without slipping. The curve traced by a point on the circumference of the smaller circle is called an epicycloid (see figure). Use the angle \(\theta\) to find a set of parametric equations for this curve. Text Transcription: theta

True or False? In Exercises 71–73, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The graph of the parametric equations \(x=t^2\ \text{and}\ y=t^2\) is the line y = x. Text Transcription: x=t^2 and y=t^2

True or False? In Exercises 71–73, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If y is a function of t and x is a function of t, then y is a function of x.

True or False? In Exercises 71–73, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The curve represented by the parametric equations x = t and y = cos t can be written as an equation of the form y = f(x).

Consider the parametric equations x = 8 cos t and y = 8 sin t. (a) Describe the curve represented by the parametric equations. (b) How does the curve represented by the parametric equations x = 8 cos t + 3 and y = 8 sin t + 6 compare to the curve described in part (a)? (c) How does the original curve change when cosine and sin are interchanged?

Project Motion In Exercises 75 and 76, consider a projectile launched at a height h feet above the ground and at an angle \(\theta\) with the horizontal. If the initial velocity is \(v_0\) feet per second, the path of the projectile is modeled by the parametric equations \(x=(v_0 \cos \theta)t\ \text{and}\ y=h+(v_0 \sin \theta)t-16t^2\). The center field fence in a ballpark is 10 feet high and 400 feet from home plate. The ball is hit 3 feet above the ground. It leaves the bat at an angle of \(\theta\) degrees with the horizontal at a speed of 100 miles per hour (see figure). (a) Write a set of parametric equations for the path of the ball. (b) Use a graphing utility to graph the path of the ball when \(\theta=15^{\circ}\). Is the hit a home run? (c) Use a graphing utility to graph the path of the ball when \(\theta=23^{\circ}\). Is the hit a home run? (d) Find the minimum angle at which the ball must leave the bat in order for the hit to be a home run. Text Transcription: theta v_0 x=(v_0 cos theta)t and y=h+(v_0 sin theta)t-16t^2 theta=15 degrees theta=23 degrees

Project Motion In Exercises 75 and 76, consider a projectile launched at a height h feet above the ground and at an angle \(\theta\) with the horizontal. If the initial velocity is \(v_0\) feet per second, the path of the projectile is modeled by the parametric equations \(x=(v_0 \cos \theta)t\ \text{and}\ y=h+(v_0 \sin \theta)t-16t^2\). A rectangular equation for the path of a projectile is \(y=5+x-0.005x^2\). (a) Eliminate the parameter t from the position function for the motion of a projectile to show that the rectangular equation is \(y=-\frac{16 \sec^2 \theta}{v_0 ^2}x^2+(\tan \theta)\ x+h\). (b) Use the result of part (a) to find h, \(v_0\), and \(\theta\). Find the parametric equations of the path. (c) Use a graphing utility to graph the rectangular equation for the path of the projectile. Confirm your answer in part (b) by sketching the curve represented by the parametric equations. (d) Use a graphing utility to approximate the maximum height of the projectile and its range. Text Transcription: theta v_0 x=(v_0 cos theta)t and y=h+(v_0 sin theta)t-16t^2 y=5+x-0.005x^2 y=-16 sec^2 theta /v_0 ^2 x^2 + (tan theta) x+h