 Chapter 1: Functions and Sequences
 Chapter 1.1: Four Ways to Represent a Function
 Chapter 1.2: A Catalog of Essential Functions
 Chapter 1.3: New Functions from Old Functions
 Chapter 1.4: Exponential Functions
 Chapter 1.5: Logarithms; Semilog and LogLog Plots
 Chapter 1.6: Sequences and Difference Equations
 Chapter 10: Systems of Linear Differential Equations
 Chapter 10.1: Qualitative Analysis of Linear Systems
 Chapter 10.2: Qualitative Analysis of Linear Systems
 Chapter 10.3: Applications
 Chapter 10.4: Systems of Nonlinear Differential Equations
 Chapter 2: Limits
 Chapter 2.1: Limits of Sequences
 Chapter 2.2: Limits of Functions at Infinity
 Chapter 2.3: Limits of Functions at Finite Numbers
 Chapter 2.4: Limits: Algebraic Methods
 Chapter 2.5: Continuity
 Chapter 3: Derivatives
 Chapter 3.1: Derivatives and Rates of Change
 Chapter 3.2: The Derivative as a Function
 Chapter 3.3: Basic Differentiation Formulas
 Chapter 3.4: The Chain Rule
 Chapter 3.5: The Chain Rule
 Chapter 3.6: Exponential Growth and Decay
 Chapter 3.7: Derivatives of the Logarithmic and Inverse Tangent Functions
 Chapter 3.8: Linear Approximations and Taylor Polynomials
 Chapter 4: Applications of Derivatives
 Chapter 4.1: Maximum and Minimum Values
 Chapter 4.2: How Derivatives Affect the Shape of a Graph
 Chapter 4.3: LHospitals Rule: Comparing Rates of Growth
 Chapter 4.4: Optimization Problems
 Chapter 4.5: Recursions: Equilibria and Stability
 Chapter 4.6: Antiderivatives
 Chapter 5: Integrals
 Chapter 5.1: Areas, Distances, and Pathogenesis
 Chapter 5.2: The Definite Integral
 Chapter 5.3: The Fundamental Theorem of Calculus
 Chapter 5.4: The Substitution Rule
 Chapter 5.5: Integration by Parts
 Chapter 5.6: Partial Fractions
 Chapter 5.7: Integration Using Tables and Computer Algebra Systems
 Chapter 5.8: Improper Integrals
 Chapter 6: Applications of Integrals
 Chapter 6.1: Areas Between Curves
 Chapter 6.2: Average Values
 Chapter 6.3: Further Applications to Biology
 Chapter 6.4: Volumes
 Chapter 7: Differential Equations
 Chapter 7.1: Modeling with Differential Equations
 Chapter 7.2: Phase Plots, Equilibria, and Stability
 Chapter 7.3: Direction Fields and Eulers Method
 Chapter 7.4: Separable Equations
 Chapter 7.5: Phase Plane Analysis
 Chapter 7.6: Phase Plane Analysis
 Chapter 8: Vectors and Matrix Models
 Chapter 8.1: Coordinate Systems
 Chapter 8.2: Vectors
 Chapter 8.3: The Dot Product
 Chapter 8.4: Matrix Algebra
 Chapter 8.5: Matrices and the Dynamics of Vectors
 Chapter 8.6: Eigenvectors and Eigenvalues
 Chapter 8.7: Eigenvectors and Eigenvalues
 Chapter 8.8: Iterated Matrix Models
 Chapter 9: Multivariable Calculus
 Chapter 9.1: Functions of Several Variables
 Chapter 9.2: Partial Derivatives
 Chapter 9.3: Tangent Planes and Linear Approximations
 Chapter 9.4: The Chain Rule
 Chapter 9.5: Directional Derivatives and the Gradient Vector
 Chapter 9.6: Maximum and Minimum Values
Biocalculus: Calculus for Life Sciences 1st Edition  Solutions by Chapter
Full solutions for Biocalculus: Calculus for Life Sciences  1st Edition
ISBN: 9781133109631
Biocalculus: Calculus for Life Sciences  1st Edition  Solutions by Chapter
Get Full SolutionsBiocalculus: Calculus for Life Sciences was written by Patricia and is associated to the ISBN: 9781133109631. This textbook survival guide was created for the textbook: Biocalculus: Calculus for Life Sciences , edition: 1. Since problems from 71 chapters in Biocalculus: Calculus for Life Sciences have been answered, more than 5563 students have viewed full stepbystep answer. This expansive textbook survival guide covers the following chapters: 71. The full stepbystep solution to problem in Biocalculus: Calculus for Life Sciences were answered by Patricia, our top Math solution expert on 03/08/18, 08:15PM.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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!

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.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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

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.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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.

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

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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

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

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
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