 16.1: Find the equation of the tangent to: a y = x 2x2 + 3 at x = 2 b y =...
 16.2: Find the equation of the normal to: a y = x2 at the point (3, 9) b ...
 16.3: a Find the equations of the horizontal tangents to y = 2x3 + 3x2 12...
 16.4: Find the equation of the tangent to: a y = p2x + 1 at x = 4 b y = 1...
 16.5: Find the equation of the tangent to: a y = p2x + 1 at x = 4 b y = 1...
 16.6: The curve y = a p1 bx where a and b are constants, has a tangent wi...
 16.7: The curve y = a p1 bx where a and b are constants, has a tangent wi...
 16.8: Find the equation of: a the tangent to the function f : x 7! ex at ...
 16.9: Show that the curve with equation y = cos x 1 + sin x does not have...
 16.10: Find the equation of: a the tangent to y = sin x at the origin b th...
 16.11: a Find where the tangent to the curve y = x3 at the point where x =...
 16.12: Consider the function f(x) = x2 + 4 x2 . a Find f0 (x). b Find the ...
 16.13: The tangent to y = x2ex at x = 1 cuts the x and yaxes at A and B r...
 16.14: a Find the equation of the tangent to y = x2 x + 9 at the point whe...
 16.15: Find the equation of the tangent to y = ex at the point where x = a...
 16.16: Consider f(x) = 8 x2 . a Sketch the graph of the function. b Find t...
 16.17: Find, correct to 2 decimal places, the angle between the tangents t...
 16.18: A quadratic of the form y = ax2, a > 0, touches the logarithmic fun...
Solutions for Chapter 16: Properties of curves
Full solutions for Mathematics for the International Student: Mathematics SL  3rd Edition
ISBN: 9781921972089
Solutions for Chapter 16: Properties of curves
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. Chapter 16: Properties of curves includes 18 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 18 problems in chapter 16: Properties of curves have been answered, more than 12512 students have viewed full stepbystep solutions from this chapter.

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.

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).

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.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

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.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

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.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

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