 1.4.1: 14 Use the Law of Exponents to rewrite and simplify the expression....
 1.4.2: 14 Use the Law of Exponents to rewrite and simplify the expression....
 1.4.3: 14 Use the Law of Exponents to rewrite and simplify the expression....
 1.4.4: 14 Use the Law of Exponents to rewrite and simplify the expression....
 1.4.5: (a) Write an equation that defines the exponential function with ba...
 1.4.6: . (a) How is the number e defined? (b) What is an approximate value...
 1.4.7: 710 Graph the given functions on a common screen. How are these gra...
 1.4.8: 710 Graph the given functions on a common screen. How are these gra...
 1.4.9: 710 Graph the given functions on a common screen. How are these gra...
 1.4.10: 710 Graph the given functions on a common screen. How are these gra...
 1.4.11: 1116 Make a rough sketch of the graph of the function. Do not use a...
 1.4.12: 1116 Make a rough sketch of the graph of the function. Do not use a...
 1.4.13: 1116 Make a rough sketch of the graph of the function. Do not use a...
 1.4.14: 1116 Make a rough sketch of the graph of the function. Do not use a...
 1.4.15: 1116 Make a rough sketch of the graph of the function. Do not use a...
 1.4.16: 1116 Make a rough sketch of the graph of the function. Do not use a...
 1.4.17: Starting with the graph of y ex , write the equation of the graph t...
 1.4.18: Starting with the graph of y ex , find the equation of the graph th...
 1.4.19: 1920 Find the domain of each function. 19. (a) fsxd 1 2 ex 2 1 2 e1...
 1.4.20: 1920 Find the domain of each function. 19. (a) fsxd 1 2 ex 2 1 2 e1...
 1.4.21: 2122 Find the exponential function fsxd Cbx whose graph is given. 2...
 1.4.22: 2122 Find the exponential function fsxd Cbx whose graph is given. 2...
 1.4.23: . If fsxd 5x , show that fsx 1 hd 2 fsxd h 5x S 5h 2 1 h D
 1.4.24: Suppose you are offered a job that lasts one month. Which of the fo...
 1.4.25: Suppose the graphs of fsxd x 2 and tsxd 2x are drawn on a coordinat...
 1.4.26: Compare the functions fsxd x 5 and tsxd 5x by graphing both functio...
 1.4.27: Compare the functions fsxd x 10 and tsxd ex by graphing both f and ...
 1.4.28: Use a graph to estimate the values of x such that ex . 1,000,000,000.
 1.4.29: Use a graph to estimate the values of x such that ex . 1,000,000,000.
 1.4.30: A bacteria culture starts with 500 bacteria and doubles in size eve...
 1.4.31: The halflife of bismuth210, 210Bi, is 5 days. (a) If a sample has...
 1.4.32: An isotope of sodium, 24Na, has a halflife of 15 hours. A sample o...
 1.4.33: Halflife of HIV Use the graph of V in Figure 12 to estimate the ha...
 1.4.34: Blood alcohol concentration After alcohol is fully absorbed into th...
 1.4.35: World population Use a calculator with exponential regression capab...
 1.4.36: US population The table gives the population of the United States, ...
 1.4.37: If you graph the function fsxd 1 2 e 1yx 1 1 e 1yx youll see that f...
 1.4.38: Graph several members of the family of functions fsxd 1 1 1 ae bx w...
Solutions for Chapter 1.4: Exponential Functions
Full solutions for Biocalculus: Calculus for Life Sciences  1st Edition
ISBN: 9781133109631
Solutions for Chapter 1.4: Exponential Functions
Get Full SolutionsThis textbook survival guide was created for the textbook: Biocalculus: Calculus for Life Sciences , edition: 1. Since 38 problems in chapter 1.4: Exponential Functions have been answered, more than 26147 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Biocalculus: Calculus for Life Sciences was written by and is associated to the ISBN: 9781133109631. Chapter 1.4: Exponential Functions includes 38 full stepbystep solutions.

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!

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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).

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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.

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.

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

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).

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·