MCD4500 Test 1 Key points PDF

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Matrix 1. Properties of Transpose ● (AB )T = AT B T ● (AT )T = A

Solve linear equations 2. For system of equation: a1x + b1 y + c1 z = d1 a2x + b2 y + c2 z = d2 a3x + b3 y + c3 z = d3 Coefficient matrix: [a1 b1 c1 ] [a2 b2 c2 ]

Augmented matrix: [ a1 b1 c1 | d1 ] [ a2 b2 c2 | d2 ]

[a3 b3 c3 ]

[ a3 b3 c3 | d3 ]

Matrix equation notation Ax=b: [ a1 b1 c1 ] [x]

[d1]

[ a2 b2 c2 ] [y] = [d2] [ a3 b3 c3 ] [z] [d3] [ a1 b1 c1 ] Determinant of Matrix A: det( [ a2 b2 c2 ] ) [ a3 b3 c3 ]

Det = 0

Many sets of solutions; No set of solution

Det ≠ 0

Unique set of solution

Eigenvalues and Eigenvectors [ a1 b1 c1 ] 3. For matrix A = [ a2 b2 c2 ] [ a3 b3 c3 ] To find eigenvalues: det (A − λI) = 0 λ1 = ... λ2 = ... To find eigenvectors: Substitute λ into A − λI , then (A − λI) * V = 0 For λ=..., eigenvector is v=...

Euclidean Geometry Lines

4. Vector parametric equation: r = a (or b) + tv t ∈ R ⇓ （x, y , z) =< a, b, c >+ t < p, q, r > { x = a + tp Cartesian parametric equation: { y = b + tq t∊R { z = c + tr 5. Two lines are parallel: v 1 = λ v 2 Two lines are intersect: L1 = (x 1, y 1, z 1) = < a1, b1, c1 > + t < p1, q 1, r1 > L2 = (x 2, y 2, z 2) = < a2, b2, c2 > + s < p2, q 2, r2 > Found t,s through: x1=x2, y1=y2 Then substitute t,s into z1 and z2, if z1=z2, intersect. Vector (n) perpendicular to both L1, L2: v1 ☓ v2 = | i j k | | x1 y 1 z 1 | | x2 y 2 z 2 | Shortest distance between L1(AB), L2(CD): d = | AC ☓ n | / |n| Shortest distance between a point (P) to a line (A is point on the line): d = | AP ☓ V | / |V |

Area = | w x v |

S=½|axb|

Euclidean Geometry Surfaces

6. Cartesian equation: px + qy + rz = d Then substitute x,y,z with A to find d. Vector equation: (x − a) * n = 0 => [< x, y , z > − < a, b, c >] * < p, q, r > = 0 n = (p,q,r) is the normal of plane Parametric equations (three points A,B,C are given): r = A + t AB + s AC t, s ∈ R 7. Shortest distance between two planes (P on plane 1, Q on plane 2): d = |P Q * n | / |n| Shortest distance from a point (P) to a plane (A is a point on the plane)： h = |AP * n| / |n|

Tangent vectors and lines and Normal planes 8. For a curve r(t) = x(t) i + y(t) j + z(t) k , find the tangent line at point t = a Direction of the tangent line: r′(t) = < x′(t), y ′(t), z ′(t) > At t = a : r′(a) = < x′(a), y ′(a), z ′(a) > So the equation of the tangent line @ t = a: L = r(a) + s r ′(a) s ∈ R  ( Tangent is also the normal of normal plane )

Hyperbolic Functions 9. Definition: sinh(u) = ½( eu − e−u ), cosh(u) = ½( eu + e−u ); written in formula sheet

10. Identities: cosh 2(x) − sinh 2(x) = 1 1 − tanh 2x = sech 2x sinh(x + y) = sinh(x) * cosh(y) + cosh(x) * sinh(y) cosh(2x) = cosh 2(x) + sinh 2(x) = 1 + 2sinh 2(x) = 2cosh 2(x) − 1 sinh(2x) = 2sinh(x)cosh(x) 11. Reverse hyperbolic functions

12. Integration requiring hyperbolic functions Original parts

Substitution (x)

Substitution (t)

sqrt(a2 − x2)

x = asin(t)

t = sin −1(x / a)

sqrt(a2 + x2)

x = asinh(t)

t = sinh−1 (x / a)

sqrt(x2 − a2)

x = acosh(t)

t = cosh−1(x / a)

x2 + a2

x = atan(t)

t = tan−1 (x / a)

a2 − x2

x = atanh(t) ; x = acoth(t)

t = tanh −1(x / a) ; t = coth−1(x / a)

Limits 13. L’Hopital Rules: For

Type

,

Origin

.

Rearrange

Indeterminate product 0*∞ Then apply L’H Rules Indeterminate difference ∞-∞

To a single fraction, then apply L’H Rules

Indeterminate powers 00 , 1∞ , ∞0

Let Then

Improper Integrals 14. Two types and strategy: Firstly, determine type of improper integrals ●

The domain of integrals is unbounded

Convert origin to

Then

●

, find the limit

The range of integrals is unbounded Find the side that range is unbounded (a or b), Let n = a (or  b, the side found above) Convert origin to Then

, find the limit.

Series 15. Diverges: or

doesn’t exist

Converges: ● Ratio test:

For the infinite series The series converges

The series diverges

Test fail (Use other methods)

●

Integral test: For the infinite series

, let

If f is continuous, decreasing, and positive for all x∊[1, ∞)

●

If

converges

Converges

If

diverges

Diverges

Comparison test (大敛小敛，小散大散） For the infinite series

,

There are two other infinite series ❏ If 0 < an < cn ,

converges.

❏ If 0 < dn < an ,

diverges。

(converges),

(diverges).

Differentiation table f(x)

f’(x)

ex

ex

loga x

1/(xln(a))

ln(x)

1/x

1 sin(x) | sinh(x)

cos(x) | cosh(x)

sin−1 (x) | sinh−1 (x)

｜

cos(x) | cosh(x)

-sin(x) | sinh(x)

cos −1(x) | cosh −1(x)

|

tan(x) | tanh(x)

sec 2(x) | sech2 (x)

tan−1(x) | tanh −1(x) , coth−1 (x)

x2 + 1 | 1 − x2

cot(x) | coth(x)

− csc 2(x) | − csch 2(x)

sec(x) | sech(x)

sec(x)tan(x) | -sech(x)tanh(x)

csc(x) | csch(x)

-csc(x)cot(x) | -csch(x)coth(x)...