 1.1: (a) What is a function? What are its domain and range? (b) What is ...
 1.2: Discuss four ways of representing a function. Illustrate your discu...
 1.3: (a) What is an even function? How can you tell if a function is eve...
 1.4: What is an increasing function?
 1.5: What is a mathematical model?
 1.6: Give an example of each type of function. (a) Linear function (b) P...
 1.7: Sketch by hand, on the same axes, the graphs of the following funct...
 1.8: Draw, by hand, a rough sketch of the graph of each function. (a) y ...
 1.9: Suppose that f has domain A and t has domain B. (a) What is the dom...
 1.10: How is the composite function f 8 t defined? What is its domain?
 1.11: Suppose the graph of f is given. Write an equation for each of the ...
 1.12: (a) What is a onetoone function? How can you tell if a function i...
 1.13: a) What is a semilog plot? (b) If a semilog plot of your data lies ...
 1.14: (a) What is a loglog plot? (b) If a loglog plot of your data lies...
 1.15: (a) What is a sequence? (b) What is a recursive sequence?
 1.16: Discretetime models (a) If there are Nt cells at time t and they d...
 1.17: 1319 Use transformations to sketch the graph of the function. 17. f...
 1.18: 1319 Use transformations to sketch the graph of the function. 18. f...
 1.19: 1319 Use transformations to sketch the graph of the function. 19. f...
 1.20: Determine whether f is even, odd, or neither even nor odd. (a) fsxd...
 1.21: If fsxd ln x and tsxd x 2 2 9, find the functions (a) f 8 t, (b) t ...
 1.22: Express the function Fsxd 1ysx 1sx as a composition of three functi...
 1.23: Life expectancy Life expectancy improved dramatically in the 20th c...
 1.24: A smallappliance manufacturer finds that it costs $9000 to produce...
 1.25: If fsxd 2x 1 ln x, find f 21 s2d.
 1.26: Find the inverse function of fsxd x 1 1 2x
 1.27: Find the exact value of each expression. (a) e 2 ln 3 (b) log10 25 1 l
 1.28: Solve each equation for x. (a) ex 5 (b) ln x 2 (c) ee x 2
 1.29: The halflife of palladium100, 100Pd, is four days. (So half of an...
 1.30: Population growth The population of a certain species in a limited ...
 1.31: Graph members of the family of functions fsxd lnsx 2 2 cd for sever...
 1.32: Graph the three functions y x b , y bx , and y logb x on the same s...
 1.33: 3334 Data points sx, yd are given. (a) Draw a scatter plot of the d...
 1.34: 3334 Data points sx, yd are given. (a) Draw a scatter plot of the d...
 1.35: Nigerian population The table gives the midyear population of Niger...
 1.36: 3637 Find the first six terms of the sequence. 36. an sinsny3d
 1.37: 3637 Find the first six terms of the sequence. 37. a1 3, an11 n 1 2...
 1.38: Find a formula for the general term of the sequence 23, 5 4 , 2 7 9...
 1.39: If x0 0.9 and xt11 2.7xts1 2 xtd, calculate xt to four decimal plac...
 1.40: BevertonHolt model An alternative to the logistic model for restri...
Solutions for Chapter 1: Functions and Sequences
Full solutions for Biocalculus: Calculus for Life Sciences  1st Edition
ISBN: 9781133109631
Solutions for Chapter 1: Functions and Sequences
Get Full SolutionsSince 40 problems in chapter 1: Functions and Sequences have been answered, more than 26254 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Biocalculus: Calculus for Life Sciences , edition: 1. Chapter 1: Functions and Sequences includes 40 full stepbystep solutions. Biocalculus: Calculus for Life Sciences was written by and is associated to the ISBN: 9781133109631. This expansive textbook survival guide covers the following chapters and their solutions.

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.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (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.

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 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.

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

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.

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.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

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

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Outer product uv T
= column times row = rank one matrix.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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

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

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.