 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
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Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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!

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Kirchhoff's Laws.
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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)ยท(b  Ax) = o.

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

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.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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

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

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).