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All Textbook Solutions for Precalculus
For Exercises 17-22, an equation of a degenerate conic section is given. Complete the square and describe the graph of each equation. x2+y24x+2y+5=018PRE19PRE20PRE21PRE22PRE23PRE24PREGiven Ax2+Cy2+Dx+Ey+F=0, show that the equation represents a parabola if A=0 butC0 and D0 .1SPSuppose that the x- and y-axes are rotated 45 about the origin to form the x- and y-axes . Given P4,2 in the xy-plane, find the coordinate of P in the xy-plane.Write the equation xy=8 in terms of x and y given that the angle of rotation of the x- and y-axes is 45 from the x- and y-axes .4SP5SP6SP1PE2PESuppose that the equation Ax2+Bxy+Cy2+Dx+Ey+F=0 represents a conic section (nondegenerative case). Which term determines whether the conic section is rotated?The conic section defined by Ax2+Bxy+Cy2+Dx+Ey+F=0 will have no rotation relative to the new coordinate axes x and y if the x- and y-axes are rotated through an angle satisfying the relationship cot2= (answer in terms of A,B, and C ).For Exercises 5-8, the given equation represents a conic section (nondegenerative case). Identify the type of conic section. (See Example 1) a. 8x22y23x+2y6=0 b. 6y2+4x12y24=0 c. 9x24y218x+12y=0For Exercises 5-8, the given equation represents a conic section (nondegenerative case). Identify the type of conic section. (See Example 1) a. 5x2+8y+4=0 b. 3x2+3y24x+2y8=0 c. 2x2+y24y+1=07PEFor Exercises 5-8, the given equation represents a conic section (nondegenerative case). Identify the type of conic section. (See Example 1) a. 6x2+3x+10y=10y2+8 b. 3x2+18xy=5x+2y+9 c. 4x2+8x5=y2+6y+39PEFor Exercises 9-14, assume that the x- and y-axes are rotated through angle about the origin to form the x- and y-axes . (See Example 2) Given =30 and P4,8 in the xy-plane, find the coordinates of P in the xy-plane .11PEFor Exercises 9-14, assume that the x- and y-axes are rotated through angle about the origin to form the x- and y-axes . (See Example 2) Given =45 and P2,52 in the xy-plane, find the coordinates of P in the xy-plane .13PEFor Exercises 9-14, assume that the x- and y-axes are rotated through angle about the origin to form the x- and y-axes . (See Example 2) Given cos=513 and sin=1213 and P39,26 in the xy-plane, find the coordinates of P in the xy-plane .15PEFor Exercises 15-18, assume that the x- and y-axes are rotated through angle about the origin to form the x- and y-axes . Write the equation in xy-coordinates . (See Example 3) xy4=0,=4517PE18PE19PE20PEFor Exercises 19-22, assume that 02 . Find the exact values of cos and sin . Then approximate the value of to the nearest tenth of a degree if necessary. cot2=24722PEFor Exercises 23-28, a. Determine an acute angle of rotation to eliminate the xy-term . (See Example 4) b. Use a rotation of axes to eliminate the xy-term to write the equation in the form Ax2+Cy2+Dx+Ey+F=0 . c. Identify the type of curve represented by the equation. d. Write the equation in standard form and sketch the graph. 3x22xy+3y216=024PEFor Exercises 23-28, a. Determine an acute angle of rotation to eliminate the xy-term . (See Example 4) b. Use a rotation of axes to eliminate the xy-term to write the equation in the form Ax2+Cy2+Dx+Ey+F=0 . c. Identify the type of curve represented by the equation. d. Write the equation in standard form and sketch the graph. 11x2103xy+y216=0For Exercises 23-28, a. Determine an acute angle of rotation to eliminate the xy-term . (See Example 4) b. Use a rotation of axes to eliminate the xy-term to write the equation in the form Ax2+Cy2+Dx+Ey+F=0 . c. Identify the type of curve represented by the equation. d. Write the equation in standard form and sketch the graph. 3x2103xy+13y2+18=0For Exercises 23-28, a. Determine an acute angle of rotation to eliminate the xy-term . (See Example 4) b. Use a rotation of axes to eliminate the xy-term to write the equation in the form Ax2+Cy2+Dx+Ey+F=0 . c. Identify the type of curve represented by the equation. d. Write the equation in standard form and sketch the graph. x223xy+3y216316y=0For Exercises 23-28, a. Determine an acute angle of rotation to eliminate the xy-term . (See Example 4) b. Use a rotation of axes to eliminate the xy-term to write the equation in the form Ax2+Cy2+Dx+Ey+F=0 . c. Identify the type of curve represented by the equation. d. Write the equation in standard form and sketch the graph. x2+23xy+3y283x+8y=029PE30PEFor Exercises 31-34, the given equation represents a conic section (nondegenerative case). Identify the type of conic section. (See Examples 5 and 6) a. 4x212xy+6y2+2x3y8=0 b. 4x212xy+9y2+2x3y8=032PE33PE34PEFor Exercises 35-42, the equation represents a conic section (nondegenerative case). a. Use the discriminant to determine the type of conic section represented. (See Example 5) b. Use a rotation of axes to eliminate the xy-term to write the equation in the form Ax2+Cy2+Dx+Ey+F=0 . c. Write the equation in standard form and sketch the graph. 3x2+263xy23y2144=0For Exercises 35-42, the equation represents a conic section (nondegenerative case). a. Use the discriminant to determine the type of conic section represented. (See Example 5) b. Use a rotation of axes to eliminate the xy-term to write the equation in the form Ax2+Cy2+Dx+Ey+F=0 . c. Write the equation in standard form and sketch the graph. 13x2+63xy+7y264=0For Exercises 35-42, the equation represents a conic section (nondegenerative case). a. Use the discriminant to determine the type of conic section represented. (See Example 5) b. Use a rotation of axes to eliminate the xy-term to write the equation in the form Ax2+Cy2+Dx+Ey+F=0 . c. Write the equation in standard form and sketch the graph. 16x224xy+9y2+140x230y+625=0For Exercises 35-42, the equation represents a conic section (nondegenerative case). a. Use the discriminant to determine the type of conic section represented. (See Example 5) b. Use a rotation of axes to eliminate the xy-term to write the equation in the form Ax2+Cy2+Dx+Ey+F=0 . c. Write the equation in standard form and sketch the graph. 4x26xy4y2+10x+310y25=0For Exercises 35-42, the equation represents a conic section (nondegenerative case). a. Use the discriminant to determine the type of conic section represented. (See Example 5) b. Use a rotation of axes to eliminate the xy-term to write the equation in the form Ax2+Cy2+Dx+Ey+F=0 . c. Write the equation in standard form and sketch the graph. 40x2+20xy+25y2+245x485y=0For Exercises 35-42, the equation represents a conic section (nondegenerative case). a. Use the discriminant to determine the type of conic section represented. (See Example 5) b. Use a rotation of axes to eliminate the xy-term to write the equation in the form Ax2+Cy2+Dx+Ey+F=0 . c. Write the equation in standard form and sketch the graph. x2+6xy+9y2910x1710y+90=0For Exercises 35-42, the equation represents a conic section (nondegenerative case). a. Use the discriminant to determine the type of conic section represented. (See Example 5) b. Use a rotation of axes to eliminate the xy-term to write the equation in the form Ax2+Cy2+Dx+Ey+F=0 . c. Write the equation in standard form and sketch the graph. 5x2+6xy+5y2122x42y16=0For Exercises 35-42, the equation represents a conic section (nondegenerative case). a. Use the discriminant to determine the type of conic section represented. (See Example 5) b. Use a rotation of axes to eliminate the xy-term to write the equation in the form Ax2+Cy2+Dx+Ey+F=0 . c. Write the equation in standard form and sketch the graph. 7x2+50xy+7y2+782x+1142y18=043PE44PE45PE46PE47PE48PEGiven y=33x a. What angle does the line y=33x make with the positive x-axis ? b. Suppose that the x- and y-axes are rotated through an angle of 30 to form the x- and y-axes . Write the equation y=33x in terms of x and y . c. Suppose that the x- and y-axes are rotated through an angle of 60 to form the x- and y-axes . Write the equation y=33x in terms of x and y .Given y=22x a. What angle does the line y=22x make with the positive x-axis ? b. Suppose that the x- and y-axes are rotated through an angle of 45 to form the x'- and y-axes . Write the equation y=22x in terms of x and y . c. Suppose that the x- and y-axes are rotated through an angle of 90 to form the x- and y-axes . Write the equation y=22x in terms of x and y .51PE52PEDiscuss the graph of x22xy+y21=0.Discuss the graph of x28xy+y2=0.55PE56PE57PEGiven an equation in x and y, many graphing utilities require that they variable be isolated. Explain how to isolate the y variable in the equation 2x2+3xy+5y24x+2y6=0.Use the relationships between A,B,andC and A,B,andC on page 960 to show that A+C is invariant under rotation. That is, show that A+C=A+C .60PEFor a positive real number r, explain why the equation x2+y2=r2 is or is not invariant under rotation.Show that.x2+y2=r2 is invariant under rotation.Recall from Section 9.3 that a point Px,y can be represented by the matrix xy . By applying matrix addition, subtraction, or multiplication we can translate the point or rotate the point to a new location. Given Px,y , the product cossinsincosxy gives the coordinates of the point rotated by an angle about the origin. a. Write a product of matrices that rotates the point 6,2 counterclockwise 60 about the origin. b. Compute the product in part (a). c. Graph the point 6,2 and the point found in part (b) to illustrate the rotation.Round the coordinates of the rotated point to 1 decimal place.The set of pointsx1,y1,x2,y2,xn,yn can be represented as the matrix x1x2...xny1y2...yn a. Write a matrix to represent the points defining the given figure. b. Suppose a computer programmer wants to rotate the matrix 90 counterclockwise. Write a product of matrices that will perform this rotation and compute the result. c. Plot the original points and arrow along with the rotated points and arrow.Graph the equation. r=822cos2SPGraph the equation. r=1653cosRotate the ellipse r=1653cos and graph the result on a graphing utility. a. Counterclockwise 23 b. Clockwise 41PE2PE3PE4PEFor Exercises 5-10, a. Determine the value of e and d . b. Identify the type of conic represented by the equation. c. Write an equation for the directrix in rectangular coordinates. d. Match the equation with its graph. Choose from A-F . r=105+5sinFor Exercises 5-10, a. Determine the value of e and d . b. Identify the type of conic represented by the equation. c. Write an equation for the directrix in rectangular coordinates. d. Match the equation with its graph. Choose from A-F . r=1025sinFor Exercises 5-10, a. Determine the value of e and d . b. Identify the type of conic represented by the equation. c. Write an equation for the directrix in rectangular coordinates. d. Match the equation with its graph. Choose from A-F . r=1052cos8PE9PE10PE11PE12PEFor Exercises 11-22, graph the conic. (See Examples 1-3) r=52+3sin14PE15PE16PE17PE18PE19PE20PE21PE22PE23PEFor Exercises 23-24, refer to the polar equation of the hyperbola. a. Determine the center in rectangular coordinates. b. Determine equations of the asymptotes in rectangular coordinates. (See Example 2) r=835cos (Exercise 14)25PEFor Exercises 25-26, refer to the polar equation of the ellipse. a. Determine the center in rectangular coordinates. b. Determine the length of the major axis. c. Determine the length of the minor axis. r=10csc4csc3 (Exercise 20)For Exercises 27-30, write an equation of the conic rotated about the origin through the given angle. Then graph the result on a graphing utility. (See Example 4) r=11sin;Clockwise3For Exercises 27-30, write an equation of the conic rotated about the origin through the given angle. Then graph the result on a graphing utility. (See Example 4) r=85+6cos;Clockwise4For Exercises 27-30, write an equation of the conic rotated about the origin through the given angle. Then graph the result on a graphing utility. (See Example 4) r=1243cos;Counterclockwise76For Exercises 27-30, write an equation of the conic rotated about the origin through the given angle. Then graph the result on a graphing utility. (See Example 4) r=52+2sin;Counterclockwise43a. Write a polar equation to represent the vertical line defined by x=2 in rectangular form. b. Use the polar equation from part (a) to write a polar equation of the line rotated about the origin 20 clockwise. c. Graph the equations from parts (a) and (b).a. Write a polar equation to represent the horizontal line defined by y=4 in rectangular form. b. Use the polar equation from part (a) to write a polar equation of the line rotated about the origin 15 counterclockwise. c. Graph the equations from parts (a) and (b).For Exercises 33-36, convert the equation to rectangular coordinates. Compare the result to the indicated example or exercise. r=95+4sin; Example 3For Exercises 33-36, convert the equation to rectangular coordinates. Compare the result to the indicated example or exercise. r=1236cos; Example 235PE36PE37PE38PE39PE40PE41PEDiscuss the difference between the graphs of r1=84cos and r2=84sin .43PE44PEFor Exercises 45-46, use the results of Exercises 43-44 to a. Find a polar equation of the planet's orbit. b. Find the distances Rp and Ra between the planet and Sun at perihelion and aphelion, respectively. Round to the nearest million kilometers. Mercury:a=5.790107km,e=0.2056For Exercises 45-46, use the results of Exercises 43-44 to a. Find a polar equation of the planet's orbit. b. Find the distances Rp and Ra between the planet and Sun at perihelion and aphelion, respectively. Round to the nearest million kilometers. Mars: a=2.279108km,e=0.0935Graph the equations r1=210.6cos and r2=210.6cos on the same viewing window. How does the negative value of d affect the graph of the second equation?48PEGraph the equations r=11esin for e=0.8,1,and1.2 on a viewing window with 16.1x16.1 and 10y10 . Comment on the effect of e on the graph.1SPRepeat Example 2 with the equations x=t1 and y=t1 .Repeat Example 3 with the equations x=4cos and y=4sin .Write parametric equations for the curve defined by y=2x3 with the given definition for x . a. x=t b. x=4t c. x=t5Repeat Example 5 with an ant that begins at A72,10 and walks to B20,36 in 13sec .Repeat Example 6 with a cannon ball launched from a cannon at an initial height of 1m , at an angle of 30 to the horizontal, with an initial speed of 200m/sec .Round values to 1 decimal place.If f and g are continuous functions of t over an interval l, then the set of points ft,gt is called a curve. The equations x=ft and y=gt are called equations and t is called the .2PEIdentify the endpoints of the curve defined by x=3t and y=2t+1 on the interval 2t1 .Suppose that an object travels along a straight path in a rectangular coordinate system with units of distance in feet. If the object moves from the point 12,18 to 36,16 in 4 sec, the horizontal component of velocity is and the vertical component of velocity is .5PE6PEFor Exercises 7-10, sketch the plane curve by plotting points. Indicate the orientation of the curve. (See Example 1) x=t2 and y=t+3 for 1t3For Exercises 7-10, sketch the plane curve by plotting points. Indicate the orientation of the curve. (See Example 1) x=t24 and y=t1 for 2t29PEFor Exercises 7-10, sketch the plane curve by plotting points. Indicate the orientation of the curve. (See Example 1) x=t3 and y=tFor Exercises 11-26, a. Eliminate the parameter and write an equation in rectangular coordinates. b. Sketch the curve and indicate its orientation. (See Examples 2-3) x=t+2 and y=t2+tFor Exercises 11-26, a. Eliminate the parameter and write an equation in rectangular coordinates. b. Sketch the curve and indicate its orientation. (See Examples 2-3) x=t1 and y=t22tFor Exercises 11-26, a. Eliminate the parameter and write an equation in rectangular coordinates. b. Sketch the curve and indicate its orientation. (See Examples 2-3) x=t1 and y=tt1For Exercises 11-26, a. Eliminate the parameter and write an equation in rectangular coordinates. b. Sketch the curve and indicate its orientation. (See Examples 2-3) x=t+2 and y=tt+215PEFor Exercises 11-26, a. Eliminate the parameter and write an equation in rectangular coordinates. b. Sketch the curve and indicate its orientation. (See Example 2-3) x=t3 and y=t217PEFor Exercises 11-26, a. Eliminate the parameter and write an equation in rectangular coordinates. b. Sketch the curve and indicate its orientation. (See Examples 2-3) x=2et and y=et219PEFor Exercises 11-26, a. Eliminate the parameter and write an equation in rectangular coordinates. b. Sketch the curve and indicate its orientation. (See Examples 2-3) x=9t and y=t21PE22PE23PE24PE25PE26PE27PE28PE29PEFor Exercises 27-30, eliminate the parameter and write an equation in rectangular coordinates to represent the given curve. Hyperbola: x=h+asec and y=k+btan31PEFor Exercises 31-34, write parametric equations for the given curve for the given definitions of x . (See Example 4) y=4x+1 a. x=t b. x=t2 c. x=4t33PEFor Exercises 31-34, write parametric equations for the given curve for the given definitions of x . (See Example 4) y=2x,x0 a. x=t,t0 b. x=t235PE36PEFor Exercises 35-42, use the results of Exercises 27-30 and use the parameter t to write parametric equations representing the given curve. Answers may vary. Circle with center 4,3 and radius 538PEFor Exercises 35-42, use the results of Exercises 27-30 and use the parameter t to write parametric equations representing the given curve. Answers may vary. Hyperbola with center: 0,0 , vertices 0,2, and asymptotes y=23xFor Exercises 35-42, use the results of Exercises 27-30 and use the parameter t to write parametric equations representing the given curve. Answers may vary. Hyperbola with center 1,3, vertices 2,3 and 4,3, and foci 4,3 and 6,3For Exercises 35-42, use the results of Exercises 27-30 and use the parameter t to write parametric equations representing the given curve. Answers may vary. Ellipse with center 2,1, vertices 1,1 and 5,1, and endpoints of the minor axis 2,3 and 2,1For Exercises 35-42, use the results of Exercises 27-30 and use the parameter t to write parametric equations representing the given curve. Answers may vary. Ellipse with center 0,0, vertices 0,5, and foci 0,4For Exercises 43-46, write parametric equations on a restricted interval for t to define the given graph.For Exercises 43-46, write parametric equations on a restricted interval for t to define the given graph.For Exercises 43-46, write parametric equations on a restricted interval for t to define the given graph. Right half of an ellipseFor Exercises 43-46, write parametric equations on a restricted interval for t to define the given graph. Top branch of a hyperbolaFor Exercises 47-48, an object undergoes uniform linear motion on a straight path from point A to point B in tsec . Write parametric equations over an interval l that describe the motion along the path. A=1,5,B=7,3, and t=2sec48PE49PEA hospital is located 3mi west and 4mi north of the center of town. Suppose that a Medi-Vac helicopter flies at a constant speed from the hospital to the location of an accident 15mi east and 20mi south of the center of town in 15min14hr . Choose a coordinate system with the origin at the center of town. a. Write parametric equations to represent the path of the helicopter as a function of the time t (in hr) after the helicopter leaves the hospital. b. Where is the helicopter located 10 min after leaving the hospital?51PEA golfer tees off on level ground, and hits the ball with an initial speed of 52m/sec at an angle of 34 above the horizontal. Choose a coordinate system with the origin at the point where the ball is struck. a. Write parametric equations that model the path of the ball as a function of time t (in sec). b. For how long will the ball be in the air before it hits the ground? Round to the nearest hundredth of a second. c. Approximate the horizontal distance that the ball travels before it hits the ground. Round to the nearest foot. d. When is the ball at its maximum height? Find the exact value and an approximation to the nearest hundredth of a second. e. What is the maximum height? Round to the nearest foot.Anna hits a softball at a height of 3ft from the ground. The softball leaves her bat traveling with an initial speed of 80ft/sec , at an angle of 30 from the horizontal. Choose a coordinate system with the origin at ground level directly under the point where the ball is struck. a. Write parametric equations that model the path of the ball as a function of time t (in sec). b. When is the ball at its maximum height? c. What is the maximum height? Round to the nearest foot. d. If an outfielder catches the ball at a height of 5ft, how long was the ball in the air after being struck? Give the exact answer and the answer rounded to the nearest hundredth of a second. e. How far is the outfielder from home plate when she catches the ball? Round to the nearest foot.Tony hits a baseball at a height of 3ft from the ground. The ball leaves his bat traveling with an initial speed of 120ft/sec at an angle of 45 from the horizontal. Choose a coordinate system with the origin at ground level directly under the point where the ball is struck. a. Write parametric equations that model the path of the ball as a function of time t (in sec). b. When is the ball at its maximum height? Give the exact value and round to the nearest hundredth of a second. c. What is the maximum height? d. If an outfielder catches the ball at a height of 6ft, for how long was the ball in the air after being struck? Give the exact answer and the answer rounded to the nearest hundredth of a second. e. How far is the outfielder from home plate when he catches the ball? Round to the nearest foot.55PETwo planes flying at the same altitude are on a course to fly over a control tower. Plane A is 50mi east of the tower flying 125mph . Plane B is 90mi south of the tower flying 200mph . Place the origin of a rectangular coordinate system at the intersection. a. Write parametric equations that model the path of each plane as a function of the time t0 (in hr). b. Determine the times required for each plane to reach a point directly above the tower. Based on these results, will the planes crash? c. Write the distance between the planes as a function of the time t . d. How close do the planes pass? Round to the nearest tenth of a mile.Suppose that a mortar is fired from ground level at an angle of 45 with an initial speed of 200ft/sec . Choose a coordinate system with the origin at the point of launch. a. Write parametric equations to define the path of the mortar as a function of the time t (in sec). b. What is the range of the mortar? That is, what is the horizontal distance traveled from the point of launch to the point where the mortar lands? c. What are the coordinates of the mortar at its maximum height? d. Eliminate the parameter and write an equation in rectangular coordinates to represent the path. e. If a drone will be at position 848.5,272.5 at a time 6sec after the mortar is launched, will the mortar hit the drone? Assume a 1-ft margin of error.58PE59PE60PE61PEFor Exercises 61-62, describe the differences in the graphs of C1 through C3 . C1:x=t and y=25t2 C2:x=et and y=25e2t C3:x=cost and y=25cos2tDo the parametric equations x=t and y=t+5 define a parabola? Why or why not?Is the point 16,12 on the curve defined by x=2t and y=t+1 ? Why or why not?For a short interval of time, a military supply plane flies on a hyperbolic path given by the equation y2122.52x21002=1;y0, where x and y are measured in meters. a. What are the coordinates of the point on the flight path closest to the ground? b. When the plane reaches its closest point to the ground, it drops a bag of supplies to people on the ground. Assuming that the plane is traveling 200m/sec due east at the time of the drop, write parametric equations representing the path of the bag as a function of the time t (in sec) after the drop. c. Determine the coordinates of the point where the bag hits the ground.66PESuppose that an imaginary string of unlimited length is attached to a point on a circle and pulled taut so that it is tangent to the circle. Keeping the string pulled tight, wind (or unwind) the string around the circle. The path traced out by the end of the string as it moves is called an involute of the circle (shown in blue). Given a circle of radius r with parametric equations x=rcos, y=rsin, show that the parametric equations of the involute of the circle are x=rcos+sin and y=rsincos .For Exercises 68-70, use a graphing utility to graph the given curve on the recommended viewing window. Witch of Agnesi: x=2cott and y=2sin2t t:0,2,0.1, x:6.4,6.4,1 , y:4,4,169PE70PEA cycloid is a curve created by rolling a circle along a line. We can also create families of curves by rolling a circle around another circle. If a circle of radius r rolls around the interior of a larger circle of radius R, a fixed point on the smaller circle traces out a curve called a hypocycloid. For Exercises 71-72, use the following parametric equations for a hypocycloid. x=Rrcost+rcosRrrt and y=RrsintrsinRrrt a. Write parametric equations for a hypocycloid with R=3 and r=1 . The curve defined by these equations is called a deltoid. b. Graph the exterior circle given by x=3cost and y=3sint and the deltoid from part (a). Use 0t2 and a viewing window of 4.8,4.8,1 by 3,3,1 . c. For what values of t on the interval 0,2 do the deltoid curve and circle intersect?A cycloid is a curve created by rolling a circle along a line. We can also create families of curves by rolling a circle around another circle. If a circle of radius r rolls around the interior of a larger circle of radius R, a fixed point on the smaller circle traces out a curve called a hypocycloid. For Exercises 71-72, use the following parametric equations for a hypocycloid. x=Rrcost+rcosRrrt and y=RrsintrsinRrrt a. Write parametric equations for a hypocycloid with R=4 and r=1 . The curve defined by these equations is called an astroid (not to be confused with asteroid). b. Graph the exterior circle given by x=4cost and y=4sint and the astroid from part (a). Use 0t2 and a viewing window of 6.4,6.4,1 by 4,4,1 . c. For what values of t on the interval 0,2 do the astroid curve and circle intersect?If a circle of radius r rolls around the exterior of a circle of radius Rr, a fixed point on the outer circle traces out a curve called an epicycloid. For Exercises 73-74, use the following parametric equations for an epicycloid. x=R+rcostrcosR+rrt and y=R+rsintrsinR+rrt a. Write parametric equations for a epicycloid with R=1 and r=1 . The curve defined by these equations is called a cardioid. b. Graph the circle given by x=cost and y=sint and the cardioid from part (a). Use 0t2 and a viewing window of 6.4,6.4,1 by 4,4,1 .If a circle of radius r rolls around the exterior of a circle of radius Rr, a fixed point on the outer circle traces out a curve called an epicycloid. For Exercises 73-74, use the following parametric equations for an epicycloid x=R+rcostrcosR+rrt and y=R+rsintrsinR+rrt a. Write parametric equations for a epicycloid with R=2 and r=1 . The curve defined by these equations is called a nephroid meaning “kidney shaped.� b. Graph the circle given by x=2cost and y=2sint and the nephroid from part (a). Use 0t2 and a viewing window of 6.4,6.4,1 by 4,4,1 .For Exercises 1-2, simplify the expression. 8!3!5!2REFor Exercises 3-4, write the first five terms of the sequence. an=n+1!n!For Exercises 3-4, write the first five terms of the sequence. c1=5;cn=2cn1+1Given the sequence defined by bn=1n1n , which terms are positive and which are negative?Given bn=n+43n , find b45 .a. Write the first five terms of the sequence defined by an=2n+3 . b. Evaluate i=152i+3 . c. Use the formula Sn=n2a1+an to verify the fifth partial sum of the arithmetic sequence from part (a).Write the sum 443+4547+ using summation notation with n as the index of summation. (Using techniques from calculus, we can show that this sum converges to .)For Exercises 9-11, find the sum. i=382iFor Exercises 9-11, find the sum. i=13410For Exercises 9-11, find the sum. i=1601i+1Write an expression for the apparent nth term an for the sequence. 10,30,90,270,For Exercises 13-14, write the sum using summation notation. Use i as the index of summation. 32+44+58+616+73214RERewrite the series as an equivalent series with the new index of summation. i=110i3=j=0=k=2Determine if the statement is true or false. i=11004i2=200+4i=1100iSuppose that a single cell of bacteria divides every 20min for 4hr . Write a formula for the sequence an representing the number of cells after the nth cell division.For Exercises 18-20, determine whether the sequence is arithmetic. If so, identify the common difference d . 11,10.2,9.4,8.6,7.8,For Exercises 18-20, determine whether the sequence is arithmetic. If so, identify the common difference d . 4,214,132,314,For Exercises 18-20, determine whether the sequence is arithmetic. If so, identify the common difference d . 9,12,16,21,27,Determine the first five terms of the arithmetic sequence an with a1=4 , and d=8 .a. Write an expression for the nth term of the arithmetic sequence an with a1=19 , and d=5 . b. Find a36 .Find the 23rd term of an arithmetic sequence with a1=15 and a57=239 .Given an arithmetic sequence with a15=86 and a37=240 , find the 104th term.Find the number of terms of the arithmetic sequence. 11,14,17,20,23,122A sales person working for a heating and air-conditioning company earns an annual base salary of 30,000 plus 500 on every new system he sells. Suppose that an is a sequence representing the sales person’s total yearly income based on the number of units sold n . a. Write a formula for the nth term of a sequence that represents the sales person's total income for n units sold. In this case, define an with domain n0 to allow for the possibility of 0 units sold. b. How much will the sales person earn in a year for selling 42 new units?Find the sum of the first 35 terms of the arithmetic sequence 1,9,17,25,For Exercises 28-30, find the sum. 3+10+17+24++437For Exercises 28-30, find the sum. n=13625nFor Exercises 28-30, find the sum. i=1685+12iHow long will it take to pay off a debt of 3960 if 50 is paid off the first month, 60 is paid off the second month, 70 is paid off the third month, and so on?For Exercises 32-34, determine whether the sequence is geometric. If so, find the value of r . 310,31000,3100,000,For Exercises 32-34, determine whether the sequence is geometric. If so, find the value of r . 3,9,36,180,For Exercises 32-34, determine whether the sequence is geometric. If so, find the value of r . 5p35p5,5p7,5p9,Write the first five terms of a geometric sequence with a1=120 , and r=23 .Write a formula for the nth term of the geometric sequence. 40,20,10,5,For Exercises 37-39, find the indicated term of a geometric sequence from the given information. a1=4 and a2=12 . Find a6 .For Exercises 37-39, find the indicated term of a geometric sequence from the given information. a1=15 and a4=59 . Find a7 .For Exercises 37-39, find the indicated term of a geometric sequence from the given information. a7=164 and r=14 . Find a1 .Find a1 and r for a geometric sequence given that a3=18 and a6=486 .For Exercises 41-46, find the sum of the geometric series, if possible. n=1753n1For Exercises 41-46, find the sum of the geometric series, if possible. i=161223i1For Exercises 41-46, find the sum of the geometric series, if possible. k=1556k1For Exercises 41-46, find the sum of the geometric series, if possible. i=1234i1For Exercises 41-46, find the sum of the geometric series, if possible. n=3612n1For Exercises 41-46, find the sum of the geometric series, if possible. 3612+443+Write 0.87 as a fraction.An estimated 150,000 people attended the Coconut Grove art festival over a 3-day period. Admission to the event is 10 per person. In addition, suppose that each person spends an average of 100 on art, drinks, and food. a. How much money is initially infused into the local economy during the festival for admission, art, drinks, and food. b. If the money is later respent in the community over and over again at a rate of 70 , determine the total amount spent. Assume that the money is respent an infinite number of times.For Exercise 49-50, find the value of an ordinary annuity in which regular payments of P dollars are made at the end of each compounding period, n times per year, at an interest rate r for t years. P=$150,n=12,r=4,t=16yrFor Exercise 49-50, find the value of an ordinary annuity in which regular payments of P dollars are made at the end of each compounding period, n times per year, at an interest rate r for t years. P=$300,n=12,r=4,t=32yra. At age 28 , an employee begins investing 100 each pay period (twice per month) in an ordinary annuity. If the interest rate is 5.5 , find the value of the annuity when the employee retires at age 62 . b. Determine the value of the annuity if the employee waits to retire at age 65 .52REFor Exercises 53-56, use mathematical induction to prove the given statement for all positive integers n . 3+7+11++4n1=n2n+1For Exercises 53-56, use mathematical induction to prove the given statement for all positive integers n . 5+2+9++7n12=n27n17For Exercises 53-56, use mathematical induction to prove the given statement for all positive integers n . 1+4+16++4n1=134n1For Exercises 53-56, use mathematical induction to prove the given statement for all positive integers n . 10n1 is divisible by 3 .Use mathematical induction to show that 4nn+2! for integers n2 .58REEvaluate the given expression. Compare the results to the result of Exercise 58. a. 30 b. 31 c. 32 d. 33For Exercises 60-64, expand the binomial by using the binomial theorem. 4y+34For Exercises 60-64, expand the binomial by using the binomial theorem. 2x35For Exercises 60-64, expand the binomial by using the binomial theorem. 5c3d24For Exercises 60-64, expand the binomial by using the binomial theorem. t5+u36For Exercises 60-64, expand the binomial by using the binomial theorem. x2+2y3For Exercises 65-67, find the indicated term of the binomial expansion. 5x+4y6 ; fifth termFor Exercises 65-67, find the indicated term of the binomial expansion. 3x428 ; middle termFor Exercises 65-67, find the indicated term of the binomial expansion. 2c2d59 ; Term containing d25 .Given fx=4x3+2x , find the difference quotient.Use the binomial theorem to find the value of 3+2i4 where i is the imaginary unit.Gaynelle can travel one of 3 roads from her home to school. From school to work there are 4 different routes. How many different routes are available for Gaynelle to travel from home to school to work?A disc jockey has 7 songs that he must play in a half-hour period. In how many different ways can he arrange the 7 songs?In how many ways can the letters in the word SHUFFLE be arranged?In how many ways can the word SPACE be misspelled?A quiz consists of 6 true/false questions and 4 multiple-choice questions. The multiple-choice questions each have 5 possible responses a,b,c,d,e of which only one is correct. In how many different ways can a student fill out the answers to the quiz?A 3-digit code is to be made from the set of digits 4,5,6,7,8 . a. How many codes can be formed if there are no restrictions? b. How many codes can be formed if the corresponding 3-digit number is to be an even number? c. How many codes can be formed if the corresponding 3-digit number is to be a multiple of 5 and there can be no repetition of digits?Evaluate 21P4 and interpret its meaning.Evaluate 24C4 and interpret its meaning.Evaluate 10P3 and 10C3 and compare the results.In how many ways can a statistician select a sample of 15 people from a population of 90 people?How many triangles can be made if the vertices are from three of the six points on the circle? One such triangle is shown in the figure.The Daytona 500 auto race has 40 cars that initially start the race. How many first-, second- and third-place finishes can occur?Suppose that a drama class has 22 students. a. In how many ways can four students be selected to take part in a survey? b. In how many ways can four students be selected to act out a scene from a play involving 4 different parts?To meet the graduation requirements, a student must take 2 English classes out of 10 available, 3 math classes out of 6 available, and 2 history classes out of 7 available. Assuming that the student has met the prerequisites of each course and that there are no scheduling conflicts, determine the number of ways in which the student can select these courses.Which of the following can represent the probability of an event? a. 0 b. 1 c. 1.2 d. 0.12 e. 1.2 f. 120 g. 0.12 h. 45If the probability of an event is 0.0042 , is the event likely to occur or not likely to occur?If the probability of an event is 8790 , is the event likely to occur or not likely to occur?If PE=0.73 , what is the probability of PE ?Suppose that a box containing music CDs has 4 with country music, 6 with rock music, 3 with jazz music, and 7 with rap music. If one CD is selected at random from the box determine the probability that a. The CD has rap music. b. The CD has jazz music. c. The CD does not have jazz music. d. The CD has classical music.