 8.1: In Exercises 1 to 12, if the equation is that of an ellipse or a hy...
 8.2: In Exercises 1 to 12, if the equation is that of an ellipse or a hy...
 8.3: In Exercises 1 to 12, if the equation is that of an ellipse or a hy...
 8.4: In Exercises 1 to 12, if the equation is that of an ellipse or a hy...
 8.5: In Exercises 1 to 12, if the equation is that of an ellipse or a hy...
 8.6: In Exercises 1 to 12, if the equation is that of an ellipse or a hy...
 8.7: In Exercises 1 to 12, if the equation is that of an ellipse or a hy...
 8.8: In Exercises 1 to 12, if the equation is that of an ellipse or a hy...
 8.9: In Exercises 1 to 12, if the equation is that of an ellipse or a hy...
 8.10: In Exercises 1 to 12, if the equation is that of an ellipse or a hy...
 8.11: In Exercises 1 to 12, if the equation is that of an ellipse or a hy...
 8.12: In Exercises 1 to 12, if the equation is that of an ellipse or a hy...
 8.13: In Exercises 13 and 14, find the eccentricity
 8.14: In Exercises 13 and 14, find the eccentricity
 8.15: In Exercises 15 to 22, find the equation of the conic that satisfie...
 8.16: In Exercises 15 to 22, find the equation of the conic that satisfie...
 8.17: In Exercises 15 to 22, find the equation of the conic that satisfie...
 8.18: In Exercises 15 to 22, find the equation of the conic that satisfie...
 8.19: In Exercises 15 to 22, find the equation of the conic that satisfie...
 8.20: In Exercises 15 to 22, find the equation of the conic that satisfie...
 8.21: In Exercises 15 to 22, find the equation of the conic that satisfie...
 8.22: In Exercises 15 to 22, find the equation of the conic that satisfie...
 8.23: Telescope Design The parabolic mirror of a telescope has a diameter...
 8.24: Arched Door Design The top of an arched door has a semielliptical s...
 8.25: In Exercises 25 to 28, write the equation of the conic in a rotated...
 8.26: In Exercises 25 to 28, write the equation of the conic in a rotated...
 8.27: In Exercises 25 to 28, write the equation of the conic in a rotated...
 8.28: In Exercises 25 to 28, write the equation of the conic in a rotated...
 8.29: In Exercises 29 to 32, use the Conic Identification Theorem to iden...
 8.30: In Exercises 29 to 32, use the Conic Identification Theorem to iden...
 8.31: In Exercises 29 to 32, use the Conic Identification Theorem to iden...
 8.32: In Exercises 29 to 32, use the Conic Identification Theorem to iden...
 8.33: In Exercises 33 and 34, determine whether the graph of the given eq...
 8.34: In Exercises 33 and 34, determine whether the graph of the given eq...
 8.35: In Exercises 35 to 46, graph each polar equation
 8.36: In Exercises 35 to 46, graph each polar equation
 8.37: In Exercises 35 to 46, graph each polar equation
 8.38: In Exercises 35 to 46, graph each polar equation
 8.39: In Exercises 35 to 46, graph each polar equation
 8.40: In Exercises 35 to 46, graph each polar equation
 8.41: In Exercises 35 to 46, graph each polar equation
 8.42: In Exercises 35 to 46, graph each polar equation
 8.43: In Exercises 35 to 46, graph each polar equation
 8.44: In Exercises 35 to 46, graph each polar equation
 8.45: In Exercises 35 to 46, graph each polar equation
 8.46: In Exercises 35 to 46, graph each polar equation
 8.47: In Exercises 47 to 50, find a polar form of each equation.
 8.48: In Exercises 47 to 50, find a polar form of each equation.
 8.49: In Exercises 47 to 50, find a polar form of each equation.
 8.50: In Exercises 47 to 50, find a polar form of each equation.
 8.51: In Exercises 51 to 54, find a rectangular form of each equation.
 8.52: In Exercises 51 to 54, find a rectangular form of each equation.
 8.53: In Exercises 51 to 54, find a rectangular form of each equation.
 8.54: In Exercises 51 to 54, find a rectangular form of each equation.
 8.55: In Exercises 55 to 58, graph the conic given by each polar equation.
 8.56: In Exercises 55 to 58, graph the conic given by each polar equation.
 8.57: In Exercises 55 to 58, graph the conic given by each polar equation.
 8.58: In Exercises 55 to 58, graph the conic given by each polar equation.
 8.59: In Exercises 59 to 65, eliminate the parameter and graph the curve ...
 8.60: In Exercises 59 to 65, eliminate the parameter and graph the curve ...
 8.61: In Exercises 59 to 65, eliminate the parameter and graph the curve ...
 8.62: In Exercises 59 to 65, eliminate the parameter and graph the curve ...
 8.63: In Exercises 59 to 65, eliminate the parameter and graph the curve ...
 8.64: In Exercises 59 to 65, eliminate the parameter and graph the curve ...
 8.65: In Exercises 59 to 65, eliminate the parameter and graph the curve ...
 8.66: In Exercises 66 and 67, the parameter t represents time and the par...
 8.67: In Exercises 66 and 67, the parameter t represents time and the par...
 8.68: In Exercises 68 to 70, use a graphing utility.
 8.69: In Exercises 68 to 70, use a graphing utility.
 8.70: In Exercises 68 to 70, use a graphing utility.
 8.71: Path of a Projectile The path of a projectile (assume air resistanc...
Solutions for Chapter 8: College Algebra and Trigonometry 7th Edition
Full solutions for College Algebra and Trigonometry  7th Edition
ISBN: 9781439048603
Solutions for Chapter 8
Get Full SolutionsCollege Algebra and Trigonometry was written by and is associated to the ISBN: 9781439048603. Since 71 problems in chapter 8 have been answered, more than 5947 students have viewed full stepbystep solutions from this chapter. Chapter 8 includes 71 full stepbystep solutions. This textbook survival guide was created for the textbook: College Algebra and Trigonometry, edition: 7. This expansive textbook survival guide covers the following chapters and their solutions.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.