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 9.1.9.1.83: In Exercises 83 and 84, the equations of a parabola and a tangent l...
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 9.1.9.1.85: In Exercises 8588, find an equation of the tangent line to the para...
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 9.1.9.1.89: Revenue The revenue R (in dollars) generated by the sale of x 32in...
 9.1.9.1.90: Beam Deflection A simply supported beam is 64 feet long and has a l...
 9.1.9.1.91: Automobile Headlight The filament of an automobile headlight is at ...
 9.1.9.1.92: Solar Cooker You want to make a solar hot dog cooker using aluminum...
 9.1.9.1.93: Suspension Bridge Each cable of the Golden Gate Bridge is suspended...
 9.1.9.1.94: Road Design Roads are often designed with parabolic surfaces to all...
 9.1.9.1.95: Highway Design Highway engineers design a parabolic curve for an en...
 9.1.9.1.96: Satellite Orbit A satellite in a 100milehigh circular orbit aroun...
 9.1.9.1.97: Path of a Projectile The path of a softball is modeled by The coord...
 9.1.9.1.98: Projectile Motion Consider the path of a projectile projected horiz...
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 9.1.9.1.109: Writing Cross sections of television antenna dishes are parabolic i...
 9.1.9.1.110: Think About It The equation is a degenerate conic. Sketch the graph...
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Solutions for Chapter 9.1: Circles and Parabolas
Full solutions for Precalculus With Limits A Graphing Approach  5th Edition
ISBN: 9780618851522
Solutions for Chapter 9.1: Circles and Parabolas
Get Full SolutionsChapter 9.1: Circles and Parabolas includes 61 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5. Precalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522. Since 61 problems in chapter 9.1: Circles and Parabolas have been answered, more than 48081 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!

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

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.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Iterative method.
A sequence of steps intended to approach the desired solution.

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).

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

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