 15.1: Find f0 (x) given that f(x) is: a x3 b 2x3 c 7x2 d 6 px e 3 p3 x f ...
 15.2: Find dy dx for: a y = 2:5x3 1:4x2 1:3 b y = x2 c y = 1 5x2 d y = 10...
 15.3: Differentiate with respect to x: a 6x + 2 b x px c (5 x)2 d 6x2 9x4...
 15.4: Find the gradient of the tangent to: a y = x2 at x = 2 b y = 8 x2 a...
 15.5: Suppose f(x) = x2 + (b + 1)x + 2c, f(2) = 4, and f0 (1) = 2. Find t...
 15.6: Find the gradient function of f(x) where f(x) is: a 4 px + x b p3 x...
 15.7: a If y = 4x 3 x , find dy dx and interpret its meaning. b The posit...
 15.8: Suppose f(x) = 2 sin3 x 3 sin x. a Show that f0 (x) = 3 cos x cos 2...
 15.9: Find d2y dx2 given: a y = ln x b y = x ln x c y = (ln x)2
 15.10: Given f(x) = x2 1 x , find: a f(1) b f0 (1) c f00(1) d f(3)(1)
 15.11: If y = 2e3x + 5e4x, show that d2y dx2 7 dy dx + 12y = 0.
 15.12: If y = sin(2x + 3), show that d2y dx2 + 4y = 0
 15.13: If y = sin x, show that d4y dx4 = y.
 15.14: If y = 2 sin x + 3 cos x, show that y00 + y = 0 where y00 represent...
Solutions for Chapter 15: Rules of differentiation
Full solutions for Mathematics for the International Student: Mathematics SL  3rd Edition
ISBN: 9781921972089
Solutions for Chapter 15: Rules of differentiation
Get Full SolutionsMathematics for the International Student: Mathematics SL was written by and is associated to the ISBN: 9781921972089. This textbook survival guide was created for the textbook: Mathematics for the International Student: Mathematics SL, edition: 3. Since 14 problems in chapter 15: Rules of differentiation have been answered, more than 10597 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 15: Rules of differentiation includes 14 full stepbystep solutions.

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

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

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.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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

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).

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).