- 4.1: f x 3x x 10
- 4.2: f x 4x3 2 5x
- 4.3: f x 8 x2 10x 24
- 4.4: f x x2 x 2 x2 4 f
- 4.5: f x 4 x 3
- 4.6: f x 2x2 5x 3 x2 2 f
- 4.7: g x x2 x2 4
- 4.8: g x 1 x 32 g
- 4.9: h x 5x 20 x2 2x 24 g
- 4.10: h x x3 4x2 x2 3x 2 h
- 4.11: AVERAGE COST A business has a production cost of for producing unit...
- 4.12: SEIZURE OF ILLEGAL DRUGS The cost (in millions of dollars) for the ...
- 4.13: f x 3 2x 2
- 4.14: f x 4 x
- 4.15: g x 2 x 1 x
- 4.16: h x x 4 x 7 g
- 4.17: p x 5x2 4x2 1
- 4.18: f x 2x x 2 4
- 4.19: f x x x 2 1
- 4.20: h x 9 x 32 f
- 4.21: f x 6x2 x 2 1
- 4.22: y 2x 2 x 2 4 f
- 4.23: f x 6x2 11x 3 3x2 x y
- 4.24: f x 6x2 7x 2 4x2 1 f x
- 4.25: f x 2x3 x2 1
- 4.26: f x x2 1 x 1
- 4.27: f x x2 3x 10 x 2
- 4.28: f x x3 x2 25 f
- 4.29: f x 3x3 2x2 3x 2 3x2 x 4 f x
- 4.30: f x 3x3 4x2 12x 16 3x2 5x 2 f
- 4.31: AVERAGE COST The cost of producing units of a product is and the av...
- 4.32: PAGE DESIGN A page that is inches wide and inches high contains 30 ...
- 4.33: PHOTOSYNTHESIS The amount of uptake (in milligrams per square decim...
- 4.34: MEDICINE The concentration of a medication in the bloodstream hours...
- 4.35: y 2 16 2 16x
- 4.36: 16x2 y y 2 16
- 4.37: x2 64 y2 4 1
- 4.38: x2 1 y2 36 1 x
- 4.39: x 2 400 2 20y 0
- 4.40: x2 y x 2 400
- 4.41: y2 49 x2 144 1
- 4.42: x2 49 y2 144 1 y
- 4.43: Passes through the point horizontal axis
- 4.44: Passes through the point vertical axis
- 4.45: Focus:
- 4.46: Focus
- 4.47: Directrix:
- 4.48: Directrix:
- 4.49: SATELLITE ANTENNA A cross section of a large parabolic antenna (see...
- 4.50: SUSPENSION BRIDGE Each cable of a suspension bridge is suspended (i...
- 4.51: Vertices: ; minor axis of length 6
- 4.52: Vertices: ; minor axis of length 2
- 4.53: Vertices: ; passes through the point
- 4.54: Vertices: ; foci:
- 4.55: Foci: ; minor axis of length 10
- 4.56: Foci: ; major axis of length 12
- 4.57: ARCHITECTURE A semielliptical archway is to be formed over the entr...
- 4.58: WADING POOL You are building a wading pool that is in the shape of ...
- 4.59: Vertices: ; foci:
- 4.60: Vertices: ; foci
- 4.61: Vertices: ; asymptotes
- 4.62: Vertices: ; asymptotes:
- 4.63: Vertex: directrix:
- 4.64: Focus: directrix:
- 4.65: Vertex: focus:
- 4.66: Vertex: focus:
- 4.67: Vertices: passes through the point
- 4.68: Center: vertices:
- 4.69: Vertices: foci:
- 4.70: Vertices: foci:
- 4.71: Vertices: asymptotes:
- 4.72: Vertices: passes through the point
- 4.73: Vertices: foci:
- 4.74: Vertices: foci:
- 4.75: Foci: asymptotes:
- 4.76: Foci: asymptotes:
- 4.77: 2 6x 2y 9 0
- 4.78: y 2 12y 8x 20 0 x
- 4.79: x 2 y 2 2x 4y 5 0 y
- 4.80: 16x 2 16y 2 16x 24y 3 0 x
- 4.81: x 2 9y 2 10x 18y 25 0
- 4.82: 4x 2 y 2 16x 15 0
- 4.83: 9x2 y2 72x 8y 119 0 4
- 4.84: x 2 9y 2 10x 18y 7 0
- 4.85: ARCHITECTURE A parabolic archway is 12 meters high at the vertex. A...
- 4.86: ARCHITECTURE A church window (see figure) is bounded above by a par...
- 4.87: RUNNING PATH Let represent a water fountain located in a city park....
- 4.88: The domain of a rational function can never be the set of all real ...
- 4.89: The graph of the equation can be a single point. Ax Ey F 0
Solutions for Chapter 4: Rational Functions and Conics
Full solutions for Algebra and Trigonometry | 8th Edition
A = CTC = (L.J]))(L.J]))T for positive definite A.
Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
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.
Invert A by row operations on [A I] to reach [I A-I].
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
A symmetric matrix with eigenvalues of both signs (+ and - ).
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. 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.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
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.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).