 4.4.1: Consider the following problem: Find two numbers whose sum is 23 an...
 4.4.2: Find two numbers whose difference is 100 and whose product is a min...
 4.4.3: Find two positive numbers whose product is 100 and whose sum is a m...
 4.4.4: The sum of two positive numbers is 16. What is the smallest possibl...
 4.4.5: Find the dimensions of a rectangle with perimeter 100 m whose area ...
 4.4.6: Photosynthesis The rate sin mg carbonym3 yhd at which photosynthesi...
 4.4.7: Crop yield A model used for the yield Y of an agricultural crop as ...
 4.4.8: The measles pathogenesis function fstd 2tst 2 21dst 1 1d is used in...
 4.4.9: Consider the following problem: A farmer with 750 ft of fencing wan...
 4.4.10: Consider the following problem: A box with an open top is to be con...
 4.4.11: If 1200 cm2 of material is available to make a box with a square ba...
 4.4.12: A box with a square base and open top must have a volume of 32,000 ...
 4.4.13: (a) Show that of all the rectangles with a given area, the one with...
 4.4.14: A rectangular storage container with an open top is to have a volum...
 4.4.15: ind the point on the line y 2x 1 3 that is closest to the origin.
 4.4.16: Find the point on the curve y sx that is closest to the point s3, 0d.
 4.4.17: Age and size at maturity Most organisms grow for a period of time b...
 4.4.18: Enzootic stability Suppose the rate at which people of age a get in...
 4.4.19: If Csxd is the cost of producing x units of a commodity, then the a...
 4.4.20: If Rsxd is the revenue that a company receives when it sells x unit...
 4.4.21: Sustainable harvesting Example 5 was based on the assumption that w...
 4.4.22: The von Bertalanffy model for the growth of an individual fish assu...
 4.4.23: Suppose that, instead of the specific nectar function in Example 2,...
 4.4.24: Suppose that, instead of the specific oxygen function in Example 4,...
 4.4.25: Let v1 be the velocity of light in air and v2 the velocity of light...
 4.4.26: The graph shows the fuel consumption c of a car (measured in gallon...
 4.4.27: Bof the apex angle is amazingly consistent. Based on the geometry o...
 4.4.28: Swimming speed of fish For a fish swimming at a speed v relative to...
 4.4.29: t paths Ornithologists have determined that some species of birds t...
 4.4.30: Crows and whelks Crows on the west coast of Canada feed on whelks b...
Solutions for Chapter 4.4: Optimization Problems
Full solutions for Biocalculus: Calculus for Life Sciences  1st Edition
ISBN: 9781133109631
Solutions for Chapter 4.4: Optimization Problems
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Biocalculus: Calculus for Life Sciences was written by and is associated to the ISBN: 9781133109631. Chapter 4.4: Optimization Problems includes 30 full stepbystep solutions. This textbook survival guide was created for the textbook: Biocalculus: Calculus for Life Sciences , edition: 1. Since 30 problems in chapter 4.4: Optimization Problems have been answered, more than 27471 students have viewed full stepbystep solutions from this chapter.

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.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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

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).

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.

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].

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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 .

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.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)ยท(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

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

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.