 9.5.9.5.1: Fill in the blanks. If f and g are continuous functions of t on an ...
 9.5.9.5.2: Fill in the blanks. The _______ of a curve is the direction in whic...
 9.5.9.5.3: Fill in the blanks. The process of converting a set of parametric e...
 9.5.9.5.4: In Exercises 16, match the set of parametric equations with its gra...
 9.5.9.5.5: In Exercises 16, match the set of parametric equations with its gra...
 9.5.9.5.6: In Exercises 16, match the set of parametric equations with its gra...
 9.5.9.5.7: Consider the parametric equations and (a) Create a table of and val...
 9.5.9.5.8: Consider the parametric equations and (a) Create a table of and val...
 9.5.9.5.9: Library of Parent Functions In Exercises 9 and 10, determine the pl...
 9.5.9.5.10: Library of Parent Functions In Exercises 9 and 10, determine the pl...
 9.5.9.5.11: In Exercises 1126, sketch the curve represented by the parametric e...
 9.5.9.5.12: In Exercises 1126, sketch the curve represented by the parametric e...
 9.5.9.5.13: In Exercises 1126, sketch the curve represented by the parametric e...
 9.5.9.5.14: In Exercises 1126, sketch the curve represented by the parametric e...
 9.5.9.5.15: In Exercises 1126, sketch the curve represented by the parametric e...
 9.5.9.5.16: In Exercises 1126, sketch the curve represented by the parametric e...
 9.5.9.5.17: In Exercises 1126, sketch the curve represented by the parametric e...
 9.5.9.5.18: In Exercises 1126, sketch the curve represented by the parametric e...
 9.5.9.5.19: In Exercises 1126, sketch the curve represented by the parametric e...
 9.5.9.5.20: In Exercises 1126, sketch the curve represented by the parametric e...
 9.5.9.5.21: In Exercises 1126, sketch the curve represented by the parametric e...
 9.5.9.5.22: In Exercises 1126, sketch the curve represented by the parametric e...
 9.5.9.5.23: In Exercises 1126, sketch the curve represented by the parametric e...
 9.5.9.5.24: In Exercises 1126, sketch the curve represented by the parametric e...
 9.5.9.5.25: In Exercises 1126, sketch the curve represented by the parametric e...
 9.5.9.5.26: In Exercises 1126, sketch the curve represented by the parametric e...
 9.5.9.5.27: In Exercises 2732, use a graphing utility to graph the curve repres...
 9.5.9.5.28: In Exercises 2732, use a graphing utility to graph the curve repres...
 9.5.9.5.29: In Exercises 2732, use a graphing utility to graph the curve repres...
 9.5.9.5.30: In Exercises 2732, use a graphing utility to graph the curve repres...
 9.5.9.5.31: In Exercises 2732, use a graphing utility to graph the curve repres...
 9.5.9.5.32: In Exercises 2732, use a graphing utility to graph the curve repres...
 9.5.9.5.33: In Exercises 33 and 34, determine how the plane curves differ from ...
 9.5.9.5.34: In Exercises 33 and 34, determine how the plane curves differ from ...
 9.5.9.5.35: In Exercises 3538, eliminate the parameter and obtain the standard ...
 9.5.9.5.36: In Exercises 3538, eliminate the parameter and obtain the standard ...
 9.5.9.5.37: In Exercises 3538, eliminate the parameter and obtain the standard ...
 9.5.9.5.38: In Exercises 3538, eliminate the parameter and obtain the standard ...
 9.5.9.5.39: In Exercises 3942, use the results of Exercises 37 40 to find a set...
 9.5.9.5.40: In Exercises 3942, use the results of Exercises 37 40 to find a set...
 9.5.9.5.41: In Exercises 3942, use the results of Exercises 37 40 to find a set...
 9.5.9.5.42: In Exercises 3942, use the results of Exercises 37 40 to find a set...
 9.5.9.5.43: In Exercises 43 48, find two different sets of parametric equations...
 9.5.9.5.44: In Exercises 43 48, find two different sets of parametric equations...
 9.5.9.5.45: In Exercises 43 48, find two different sets of parametric equations...
 9.5.9.5.46: In Exercises 43 48, find two different sets of parametric equations...
 9.5.9.5.47: In Exercises 43 48, find two different sets of parametric equations...
 9.5.9.5.48: In Exercises 43 48, find two different sets of parametric equations...
 9.5.9.5.49: In Exercises 49 and 50, use a graphing utility to graph the curve r...
 9.5.9.5.50: In Exercises 49 and 50, use a graphing utility to graph the curve r...
 9.5.9.5.51: In Exercises 5154, match the parametric equations with the correct ...
 9.5.9.5.52: In Exercises 5154, match the parametric equations with the correct ...
 9.5.9.5.53: In Exercises 5154, match the parametric equations with the correct ...
 9.5.9.5.54: In Exercises 5154, match the parametric equations with the correct ...
 9.5.9.5.55: Projectile Motion In Exercises 5558, consider a projectile launched...
 9.5.9.5.56: Projectile Motion In Exercises 5558, consider a projectile launched...
 9.5.9.5.57: Projectile Motion In Exercises 5558, consider a projectile launched...
 9.5.9.5.58: Projectile Motion In Exercises 5558, consider a projectile launched...
 9.5.9.5.59: True or False? In Exercises 5962, determine whether the statement i...
 9.5.9.5.60: True or False? In Exercises 5962, determine whether the statement i...
 9.5.9.5.61: True or False? In Exercises 5962, determine whether the statement i...
 9.5.9.5.62: True or False? In Exercises 5962, determine whether the statement i...
 9.5.9.5.63: As increases, the ellipse given by the parametric equations and is ...
 9.5.9.5.64: Think About It The graph of the parametric equations and is shown b...
 9.5.9.5.65: In Exercises 6568, check for symmetry with respect to both axes and...
 9.5.9.5.66: In Exercises 6568, check for symmetry with respect to both axes and...
 9.5.9.5.67: In Exercises 6568, check for symmetry with respect to both axes and...
 9.5.9.5.68: In Exercises 6568, check for symmetry with respect to both axes and...
Solutions for Chapter 9.5: Parametric Equations
Full solutions for Precalculus With Limits A Graphing Approach  5th Edition
ISBN: 9780618851522
Solutions for Chapter 9.5: Parametric Equations
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Precalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522. This textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5. Chapter 9.5: Parametric Equations includes 68 full stepbystep solutions. Since 68 problems in chapter 9.5: Parametric Equations have been answered, more than 33339 students have viewed full stepbystep solutions from this chapter.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Column space C (A) =
space of all combinations of the columns of A.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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).

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).