 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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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!

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.

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

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Elimination.
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.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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].

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Lucas numbers
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.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(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 .

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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

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.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.