 9.9.1.1: Sketch the curve by using the parametric equations to plot points. ...
 9.9.3.1: Plot the point whose polar coordinates are given. Then find two oth...
 9.9.5.1: Write a polar equation of a conic with the focus at the origin and ...
 9.1: . (a) What is a parametric curve? (b) How do you sketch a parametri...
 9.9.1.2: Sketch the curve by using the parametric equations to plot points. ...
 9.9.3.2: Plot the point whose polar coordinates are given. Then find two oth...
 9.9.5.2: Write a polar equation of a conic with the focus at the origin and ...
 9.2: (a) How do you find the slope of a tangent to a parametric curve? (...
 9.9.1.3: Sketch the curve by using the parametric equations to plot points. ...
 9.9.3.3: Plot the point whose polar coordinates are given. Then find the Car...
 9.9.4.3: Find the area of the region that is bounded by the given curve and ...
 9.9.5.3: Write a polar equation of a conic with the focus at the origin and ...
 9.3: Write an expression for the length of a parametric curve.
 9.9.1.4: Sketch the curve by using the parametric equations to plot points. ...
 9.9.3.4: Plot the point whose polar coordinates are given. Then find the Car...
 9.9.4.4: Find the area of the region that is bounded by the given curve and ...
 9.9.5.4: Write a polar equation of a conic with the focus at the origin and ...
 9.4: (a) Use a diagram to explain the meaning of the polar coordinates o...
 9.9.1.5: (a) Sketch the curve by using the parametric equations to plot poin...
 9.9.4.5: Find the area of the shaded region.
 9.9.5.5: Write a polar equation of a conic with the focus at the origin and ...
 9.5: (a) How do you find the slope of a tangent line to a polar curve? (...
 9.9.1.6: (a) Sketch the curve by using the parametric equations to plot poin...
 9.9.2.6: Find an equation of the tangent to the curve at the point correspon...
 9.9.3.6: The Cartesian coordinates of a point are given. (i) Find polar coor...
 9.9.4.6: Find the area of the shaded region.
 9.9.5.6: Write a polar equation of a conic with the focus at the origin and ...
 9.6: (a) What is the eccentricity of a conic section? (b) What can you s...
 9.9.1.7: (a) Sketch the curve by using the parametric equations to plot poin...
 9.9.2.7: Find an equation of the tangent to the curve , at the point by two ...
 9.9.3.7: Sketch the region in the plane consisting of points whose polar coo...
 9.9.4.7: Find the area of the shaded region.
 9.9.5.7: Write a polar equation of a conic with the focus at the origin and ...
 9.7: The parametric equations , have the same graph as , .
 9.9.1.8: (a) Sketch the curve by using the parametric equations to plot poin...
 9.9.2.8: Find equations of the tangents to the curve , at the origin. Then g...
 9.9.3.8: Sketch the region in the plane consisting of points whose polar coo...
 9.9.4.8: Find the area of the shaded region.
 9.9.5.8: Write a polar equation of a conic with the focus at the origin and ...
 9.8: A hyperbola never intersects its directrix.
 9.9.1.9: (a) Eliminate the parameter to find a Cartesian equation of the cur...
 9.9.2.9: Find and . For which values of is the curve concave upward?3 x 4 t ...
 9.9.3.9: Sketch the region in the plane consisting of points whose polar coo...
 9.9.4.9: Sketch the curve and find the area that it encloses.r r 31 cos 9. 2...
 9.9.5.9: (a) Find the eccentricity, (b) identify the conic, (c) give an equa...
 9.9: r = 1 + cos 20
 9.9.1.10: (a) Eliminate the parameter to find a Cartesian equation of the cur...
 9.9.2.10: Find and . For which values of is the curve concave upward?y t x t ...
 9.9.3.10: Sketch the region in the plane consisting of points whose polar coo...
 9.9.4.10: Sketch the curve and find the area that it encloses.r 31 cos 9. 2
 9.9.5.10: (a) Find the eccentricity, (b) identify the conic, (c) give an equa...
 9.10: r = 1 + cos 30
 9.9.1.11: (a) Eliminate the parameter to find a Cartesian equation of the cur...
 9.9.2.11: Find and . For which values of is the curve concave upward?y t et x...
 9.9.3.11: Sketch the region in the plane consisting of points whose polar coo...
 9.9.4.11: Sketch the curve and find the area that it encloses.r 2 cos 3 r
 9.9.5.11: (a) Find the eccentricity, (b) identify the conic, (c) give an equa...
 9.11: r2 = sec 20
 9.9.1.12: (a) Eliminate the parameter to find a Cartesian equation of the cur...
 9.9.2.12: Find and . For which values of is the curve concave upward?x t ln t...
 9.9.3.12: Sketch the region in the plane consisting of points whose polar coo...
 9.9.4.12: Sketch the curve and find the area that it encloses.r 2 cos 2 r
 9.9.5.12: (a) Find the eccentricity, (b) identify the conic, (c) give an equa...
 9.12: r = 2 cos (0/2)
 9.9.1.13: (a) Eliminate the parameter to find a Cartesian equation of the cur...
 9.9.2.13: Find the points on the curve where the tangent is horizontal or ver...
 9.9.3.13: Identify the curve by finding a Cartesian equation for the curve.r ...
 9.9.4.13: Graph the curve and find the area that it encloses.r 1 2 sin 6 r
 9.9.5.13: (a) Find the eccentricity, (b) identify the conic, (c) give an equa...
 9.13: r = 1 1 + cos 0
 9.9.1.14: (a) Eliminate the parameter to find a Cartesian equation of the cur...
 9.9.2.14: Find the points on the curve where the tangent is horizontal or ver...
 9.9.3.14: Identify the curve by finding a Cartesian equation for the curve.r ...
 9.9.4.14: Graph the curve and find the area that it encloses.r 2 sin 3 sin 9 r
 9.9.5.14: (a) Find the eccentricity, (b) identify the conic, (c) give an equa...
 9.14: r = 8 4 + 3 sin 0
 9.9.1.15: Describe the motion of a particle with position as varies in the gi...
 9.9.2.15: Find the points on the curve where the tangent is horizontal or ver...
 9.9.3.15: Identify the curve by finding a Cartesian equation for the curve.r ...
 9.9.4.15: Find the area of the region enclosed by one loop of the curve.r sin...
 9.9.5.15: (a) Find the eccentricity, (b) identify the conic, (c) give an equa...
 9.15: Find a polar equation for the curve represented by the given Cartes...
 9.9.1.16: Describe the motion of a particle with position as varies in the gi...
 9.9.2.16: Find the points on the curve where the tangent is horizontal or ver...
 9.9.3.16: Identify the curve by finding a Cartesian equation for the curve.r ...
 9.9.4.16: Find the area of the region enclosed by one loop of the curve.r 4 s...
 9.9.5.16: (a) Find the eccentricity, (b) identify the conic, (c) give an equa...
 9.16: Find a polar equation for the curve represented by the given Cartes...
 9.9.1.17: Describe the motion of a particle with position as varies in the gi...
 9.9.2.17: Use a graph to estimate the coordinates of the leftmost point on th...
 9.9.3.17: Find a polar equation for the curve represented by the given Cartes...
 9.9.4.17: Find the area of the region enclosed by one loop of the curve.r 1 2...
 9.9.5.17: Graph the conics with , , , and on a common screen. How does the va...
 9.17: The curve with polar equation is called a cochleoid. Use a graph of...
 9.9.1.18: Describe the motion of a particle with position as varies in the gi...
 9.9.2.18: Try to estimate the coordinates of the highest point and the leftmo...
 9.9.3.18: Find a polar equation for the curve represented by the given Cartes...
 9.9.4.18: Find the area of the region enclosed by one loop of the curve.r 2 c...
 9.9.5.18: (a) Graph the conics for and various values of . How does the value...
 9.18: Graph the ellipse and its directrix.
 9.9.1.19: Use the graphs of and to sketch the parametric curve , . Indicate w...
 9.9.2.19: Graph the curve in a viewing rectangle that displays all the import...
 9.9.3.19: Find a polar equation for the curve represented by the given Cartes...
 9.9.4.19: Find the area of the region that lies inside the first curve and ou...
 9.9.5.19: Show that a conic with focus at the origin, eccentricity , and dire...
 9.19: Find the slope of the tangent line to the given curve at the point ...
 9.9.1.20: Use the graphs of and to sketch the parametric curve , . Indicate w...
 9.9.2.20: Graph the curve in a viewing rectangle that displays all the import...
 9.9.3.20: Find a polar equation for the curve represented by the given Cartes...
 9.9.4.20: Find the area of the region that lies inside the first curve and ou...
 9.9.5.20: Show that a conic with focus at the origin, eccentricity , and dire...
 9.20: Find the slope of the tangent line to the given curve at the point ...
 9.9.1.21: Use the graphs of and to sketch the parametric curve , . Indicate w...
 9.9.2.21: Show that the curve , has two tangents at and find their equations....
 9.9.3.21: For each of the described curves, decide if the curve would be more...
 9.9.4.21: Find the area of the region that lies inside the first curve and ou...
 9.9.5.21: Show that a conic with focus at the origin, eccentricity , and dire...
 9.21: Find the slope of the tangent line to the given curve at the point ...
 9.9.1.22: Match the parametric equations with the graphs labeled IVI. Give re...
 9.9.2.22: At what point does the curve , cross itself? Find the equations of ...
 9.9.3.22: For each of the described curves, decide if the curve would be more...
 9.9.4.22: Find the area of the region that lies inside the first curve and ou...
 9.9.5.22: Show that the parabolas and intersect at right angles.
 9.22: Find the slope of the tangent line to the given curve at the point ...
 9.9.1.23: Graph the curve .
 9.9.2.23: (a) Find the slope of the tangent line to the trochoid , in terms o...
 9.9.4.23: Find the area of the region that lies inside both curves.r sin r cos r
 9.9.5.23: (a) Show that the polar equation of an ellipse with directrix can b...
 9.23: Find and . x t cos t y t sin t d
 9.24: Find and . 3 x 1 t 2 y t t 3
 9.9.1.24: Graph the curves and and find their points of intersection correct ...
 9.9.2.24: (a) Find the slope of the tangent to the astroid , in terms of . (b...
 9.9.3.24: Sketch the curve with the given polar equation.r 6 2 3r 2 0 2, 3
 9.9.4.24: Find the area of the region that lies inside both curves.r sin 2 r ...
 9.9.5.24: (a) The planets move around the Sun in elliptical orbits with the S...
 9.25: Use a graph to estimate the coordinates of the lowest point on the ...
 9.9.1.25: (a) Show that the parametric equations where , describe the line se...
 9.9.2.25: At what points on the curve , is the tangent parallel to the line w...
 9.9.3.25: Sketch the curve with the given polar equation.r sin r
 9.9.4.25: Find the area of the region that lies inside both curves.r sin 2 r ...
 9.9.5.25: The orbit of Halleys comet, last seen in 1986 and due to return in ...
 9.26: . Find the area enclosed by the loop of the curve in Exercise 25.
 9.9.1.26: Use a graphing device and the result of Exercise 25(a) to draw the ...
 9.9.2.26: Find equations of the tangents to the curve , that pass through the...
 9.9.3.26: Sketch the curve with the given polar equation.r 3 cos r
 9.9.4.26: Find the area of the region that lies inside both curves.r2 = 2 sin...
 9.9.5.26: The HaleBopp comet, discovered in 1995, has an elliptical orbit wi...
 9.27: At what points does the curve have vertical or horizontal tangents?...
 9.9.1.27: Find parametric equations for the path of a particle that moves alo...
 9.9.2.27: Use the parametric equations of an ellipse, , , , to find the area ...
 9.9.3.27: Sketch the curve with the given polar equation.r 21 sin 0 r 1
 9.9.4.27: Find the area inside the larger loop and outside the smaller loop o...
 9.9.5.27: The planet Mercury travels in an elliptical orbit with eccentricity...
 9.28: Find the area enclosed by the curve in Exercise 27.
 9.9.1.28: (a) Find parametric equations for the ellipse . [Hint: Modify the e...
 9.9.2.28: Find the area bounded by the curve , and the line .
 9.9.3.28: Sketch the curve with the given polar equation.r 1 3 cos r
 9.9.4.28: When recording live performances, sound engineers often use a micro...
 9.9.5.28: The distance from the planet Pluto to the Sun is km at perihelion a...
 9.29: Find the area enclosed by the curve .
 9.9.1.29: Use a graphing calculator or computer to reproduce the picture.
 9.9.2.29: Find the area bounded by the curve , , , and the lines and .
 9.9.3.29: Sketch the curve with the given polar equation.r 0 r
 9.9.4.29: Find all points of intersection of the given curves.r cos r 1 cos r
 9.9.5.29: Using the data from Exercise 27, find the distance traveled by the ...
 9.30: Find the area enclosed by the inner loop of the curve r 1 3 sin . r
 9.9.1.30: Use a graphing calculator or computer to reproduce the picture.
 9.9.2.30: Find the area of the region enclosed by the astroid , .
 9.9.3.30: Sketch the curve with the given polar equation.r ln 1 r
 9.9.4.30: Find all points of intersection of the given curves.r cos 3 r sin 3 r
 9.31: Find the points of intersection of the curves and .
 9.9.1.31: Compare the curves represented by the parametric equations. How do ...
 9.9.2.31: Find the area under one arch of the trochoid of Exercise 34 in Sect...
 9.9.3.31: Sketch the curve with the given polar equation.r sin 2 r
 9.9.4.31: Find all points of intersection of the given curves.31. r sin r sin...
 9.32: Find the points of intersection of the curves and .
 9.9.1.32: Compare the curves represented by the parametric equations. How do ...
 9.9.2.32: Let be the region enclosed by the loop of the curve in Example 1. (...
 9.9.3.32: Sketch the curve with the given polar equation.r 2 cos 3
 9.9.4.32: Find all points of intersection of the given curves.r r 2 cos 2 2 s...
 9.33: Find the area of the region that lies inside both of the circles and .
 9.9.1.33: Derive Equations 1 for the case .
 9.9.2.33: Set up, but do not evaluate, an integral that represents the length...
 9.9.3.33: Sketch the curve with the given polar equation. r 2 cos 4 r
 9.9.4.33: Find the exact length of the polar curver 3 sin 0 3 g
 9.34: Find the area of the region that lies inside the curve but outside ...
 9.9.1.34: Let be a point at a distance from the center of a circle of radius ...
 9.9.2.34: Set up, but do not evaluate, an integral that represents the length...
 9.9.3.34: Sketch the curve with the given polar equation. r sin 5 5
 9.9.4.34: Find the exact length of the polar curver e 0 2 2 r
 9.35: Find the length of the curve.y 2t 0 t 2 3 x 3t 2
 9.9.1.35: If and are fixed numbers, find parametric equations for the curve t...
 9.9.2.35: Set up, but do not evaluate, an integral that represents the length...
 9.9.3.35: Sketch the curve with the given polar equation.r 2 sin 2 2 4 cos 2 3
 9.9.4.35: Find the exact length of the polar curver 0 2 2 35. r
 9.36: Find the length of the curve.x 2 3t y cosh 3t 0 t 1 y
 9.9.1.36: A curve, called a witch of Maria Agnesi, consists of all possible p...
 9.9.2.36: Set up, but do not evaluate, an integral that represents the length...
 9.9.3.36: Sketch the curve with the given polar equation. r r 2 sin 2 2
 9.9.4.36: Find the exact length of the polar curver 0 2 r
 9.37: Find the length of the curve.r 1 2 x
 9.9.1.37: Suppose that the position of one particle at time is given by and t...
 9.9.3.37: Sketch the curve with the given polar equation.r 2 cos32
 9.9.4.37: Use a calculator to find the length of the curve correct to four de...
 9.38: Find the length of the curve.r sin 0 3
 9.9.1.38: If a projectile is fired with an initial velocity of meters per sec...
 9.9.3.38: Sketch the curve with the given polar equation.r 2 r 2 cos32 1 r r
 9.9.4.38: Use a calculator to find the length of the curve correct to four de...
 9.39: The curves defined by the parametric equations are called strophoid...
 9.9.1.39: Investigate the family of curves defined by the parametric equation...
 9.9.2.39: Find the length of the curve.x y ln1 t 0 t 2 t 1
 9.9.3.39: Sketch the curve with the given polar equation.r 1 2 cos 2 r
 9.40: A family of curves has polar equations where is a positive number. ...
 9.9.1.40: The swallowtail catastrophe curves are defined by the parametric eq...
 9.9.2.40: Find the length of the curve.x e y 5 2t 0 t 3 t
 9.9.3.40: Sketch the curve with the given polar equation.r 1 2 cos2 r 2
 9.41: Find a polar equation for the ellipse with focus at the origin, ecc...
 9.9.1.41: The curves with equations , are called Lissajous figures. Investiga...
 9.9.2.41: Graph the curve and find its length.y e 0 t t x e sin t t 41. cos t x
 9.9.3.41: The figure shows the graph of as a function of in Cartesian coordin...
 9.42: Show that the angles between the polar axis and the asymptotes of t...
 9.9.1.42: Investigate the family of curves defined by the parametric equation...
 9.9.2.42: Graph the curve and find its length.x cos t ln(tan y sin t 4 t 34 1...
 9.9.3.42: The figure shows the graph of as a function of in Cartesian coordin...
 9.43: In the figure the circle of radius is stationary, and for every , t...
 9.9.2.43: Graph the curve and find its length.y 4e 8 t 3 t2 x et t x c
 9.9.3.43: Show that the polar curve (called a conchoid) has the line as a ver...
 9.9.2.44: Find the length of the loop of the curve , .
 9.9.3.44: Sketch the curve .
 9.9.2.45: Use Simpsons Rule with to estimate the length of the curve , , .
 9.9.3.45: Show that the curve (called a cissoid of Diocles) has the line as a...
 9.9.2.46: In Exercise 36 in Section 9.1 you were asked to derive the parametr...
 9.9.3.46: Match the polar equations with the graphs labeled IVI. Give reasons...
 9.9.2.47: Find the distance traveled by a particle with position as varies in...
 9.9.3.47: Find the slope of the tangent line to the given polar curve at the ...
 9.9.2.48: Find the distance traveled by a particle with position as varies in...
 9.9.3.48: Find the slope of the tangent line to the given polar curve at the ...
 9.9.2.49: Show that the total length of the ellipse , , , is where is the ecc...
 9.9.3.49: Find the slope of the tangent line to the given polar curve at the ...
 9.9.2.50: Find the total length of the astroid , , where a 0.
 9.9.3.50: Find the slope of the tangent line to the given polar curve at the ...
 9.9.2.51: (a) Graph the epitrochoid with equations What parameter interval gi...
 9.9.3.51: Find the points on the given curve where the tangent line is horizo...
 9.9.2.52: A curve called Cornus spiral is defined by the parametric equations...
 9.9.3.52: Find the points on the given curve where the tangent line is horizo...
 9.9.2.53: A string is wound around a circle and then unwound while being held...
 9.9.3.53: Find the points on the given curve where the tangent line is horizo...
 9.9.2.54: A cow is tied to a silo with radius by a rope just long enough to r...
 9.9.3.54: Find the points on the given curve where the tangent line is horizo...
 9.9.3.55: Show that the polar equation , where , represents a circle, and fin...
 9.9.3.56: Show that the curves and intersect at right angles.
 9.9.3.57: Use a graphing device to graph the polar curve. Choose the paramete...
 9.9.3.58: Use a graphing device to graph the polar curve. Choose the paramete...
 9.9.3.59: Use a graphing device to graph the polar curve. Choose the paramete...
 9.9.3.60: Use a graphing device to graph the polar curve. Choose the paramete...
 9.9.3.61: How are the graphs of and related to the graph of ? In general, how...
 9.9.3.62: Use a graph to estimate the coordinate of the highest points on th...
 9.9.3.63: (a) Investigate the family of curves defined by the polar equations...
 9.9.3.64: A family of curves is given by the equations , where is a real numb...
 9.9.3.65: A family of curves has polar equations Investigate how the graph ch...
 9.9.3.66: The astronomer Giovanni Cassini (16251712) studied the family of cu...
 9.9.3.67: Let be any point (except the origin) on the curve . If is the angle...
 9.9.3.68: (a) Use Exercise 67 to show that the angle between the tangent line...
Solutions for Chapter 9: PARAMETRIC EQUATIONS AND POLAR COORDINATES
Full solutions for Essential Calculus (Available Titles CengageNOW)  1st Edition
ISBN: 9780495014423
Solutions for Chapter 9: PARAMETRIC EQUATIONS AND POLAR COORDINATES
Get Full SolutionsChapter 9: PARAMETRIC EQUATIONS AND POLAR COORDINATES includes 263 full stepbystep solutions. Essential Calculus (Available Titles CengageNOW) was written by Patricia and is associated to the ISBN: 9780495014423. This expansive textbook survival guide covers the following chapters and their solutions. Since 263 problems in chapter 9: PARAMETRIC EQUATIONS AND POLAR COORDINATES have been answered, more than 7561 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Essential Calculus (Available Titles CengageNOW), edition: 1.

Amplitude
See Sinusoid.

Arctangent function
See Inverse tangent function.

Base
See Exponential function, Logarithmic function, nth power of a.

De Moivre’s theorem
(r(cos ? + i sin ?))n = r n (cos n? + i sin n?)

Discriminant
For the equation ax 2 + bx + c, the expression b2  4ac; for the equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, the expression B2  4AC

Equation
A statement of equality between two expressions.

Graph of an inequality in x and y
The set of all points in the coordinate plane corresponding to the solutions x, y of the inequality.

Inverse cotangent function
The function y = cot1 x

Inverse of a matrix
The inverse of a square matrix A, if it exists, is a matrix B, such that AB = BA = I , where I is an identity matrix.

Irrational zeros
Zeros of a function that are irrational numbers.

Logistic growth function
A model of population growth: ƒ1x2 = c 1 + a # bx or ƒ1x2 = c1 + aekx, where a, b, c, and k are positive with b < 1. c is the limit to growth

Measure of center
A measure of the typical, middle, or average value for a data set

Onetoone function
A function in which each element of the range corresponds to exactly one element in the domain

Parabola
The graph of a quadratic function, or the set of points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).

Quotient identities
tan ?= sin ?cos ?and cot ?= cos ? sin ?

Quotient of functions
a ƒ g b(x) = ƒ(x) g(x) , g(x) ? 0

Randomization
The principle of experimental design that makes it possible to use the laws of probability when making inferences.

Rectangular coordinate system
See Cartesian coordinate system.

Standard representation of a vector
A representative arrow with its initial point at the origin

Transpose of a matrix
The matrix AT obtained by interchanging the rows and columns of A.
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