- 5.1.1: (a) By reading values from the given graph of f, use four rectangle...
- 5.1.2: (a) Use six rectangles to find estimates of each type for the area ...
- 5.1.3: . (a) Estimate the area under the graph of fsxd cos x from x 0 to x...
- 5.1.4: a) Estimate the area under the graph of fsxd sx from x 0 to x 4 usi...
- 5.1.5: (a) Estimate the area under the graph of fsxd 1 1 x 2 from x 21 to ...
- 5.1.6: a) Graph the function fsxd x 2 2 ln x 1 < x < 5 (b) Estimate the ar...
- 5.1.7: The speed of a runner increased steadily during the first three sec...
- 5.1.8: Speedometer readings for a motorcycle at 12-second intervals are gi...
- 5.1.9: Measles pathogenesis The function fstd 2tst 2 21dst 1 1d can be use...
- 5.1.10: Measles pathogenesis If a patient has had previous exposure to meas...
- 5.1.11: SARS incidence The table shows the number of people per day who die...
- 5.1.12: Niche overlap The extent to which species compete for resources is ...
- 5.1.13: The velocity graph of a braking car is shown. Use it to estimate th...
- 5.1.14: The velocity graph of a car accelerating from rest to a speed of 12...
- 5.1.15: 1517 Use Definition 2 to find an expression for the area under the ...
- 5.1.16: 1517 Use Definition 2 to find an expression for the area under the ...
- 5.1.17: 1517 Use Definition 2 to find an expression for the area under the ...
- 5.1.18: (a) Use Definition 2 to find an expression for the area under the c...
- 5.1.19: Let A be the area under the graph of an increasing continuous funct...
- 5.1.20: If A is the area under the curve y ex from 1 to 3, use Exercise 19 ...
- 5.1.21: (a) Express the area under the curve y x 5 from 0 to 2 as a limit. ...
- 5.1.22: Find the exact area of the region under the graph of y e2x from 0 t...
- 5.1.23: Find the exact area under the cosine curve y cos x from x 0 to x b,...
- 5.1.24: (a) Let An be the area of a polygon with n equal sides inscribed in...
Solutions for Chapter 5.1: Areas, Distances, and Pathogenesis
Full solutions for Biocalculus: Calculus for Life Sciences | 1st Edition
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
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.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
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.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
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.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
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.
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).
= Xl (column 1) + ... + xn(column n) = combination of columns.
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •
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.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).