 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 7728 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.

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

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.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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.

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.

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

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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.
I don't want to reset my password
Need help? Contact support
Having trouble accessing your account? Let us help you, contact support at +1(510) 9441054 or support@studysoup.com
Forgot password? Reset it here