Notesale: Turn your study into money

Document Preview

Extracts from the notes are below, to see the PDF you'll receive please use the links above

---

FP3 Revision Notes


Hyperbolic Functions
Key Words: 𝑠𝑖𝑛ℎ, 𝑐𝑜𝑠ℎ, Osborn’s rule


𝑠𝑖𝑛ℎ 𝑥 =

𝑒 𝑥 −𝑒 −𝑥
2

, 𝑐𝑜𝑠ℎ 𝑥 =

o
o

𝑥2



Hyperbola:

𝑎2

+

𝑦2
𝑏2
𝑥2
𝑎

𝑦2
𝑏2





Asymptotes at 𝑦 = ±

𝑎

𝑥

Eccentricity, e, describes the ratio of the shortest distance of a general point P
from the Focus to the Directrix 𝑒 =



𝑏

𝐹𝑃
𝐹𝐷



o
If 0 < 𝑒 < 1 = ellipse
o
If 𝑒 = 1 = parabola
o
If 𝑒 > 1 = hyperbola
Each curve has its own definition of e (because of the following above)
o
Ellipse: 𝑏2 = 𝑎2 (1 − 𝑒 2 )
o
Hyperbola: 𝑏2 = 𝑎2 (𝑒 2 − 1)





∫

𝑦𝑏

2

𝑑𝑥

𝑡𝑏

2

2

2

√1 + ( 𝑑𝑥 ) 𝑑𝑦 or ∫ √( 𝑑𝑥) + ( 𝑑𝑦) 𝑑𝑡
𝑑𝑦

𝑑𝑡

𝑦𝑎

𝑑𝑡

𝑡𝑎

Surface Area between 𝐴(𝑥 𝑎 , 𝑦 𝑎 ) and 𝐵(𝑥 𝑏 , 𝑦 𝑏 )
𝑥𝑏

o

x-Axis: 2𝜋 ∫
𝑥𝑎
𝑦𝑏

y-Axis: 2𝜋 ∫

𝑡𝑏

𝑑𝑦 2

𝑑𝑥 2

𝑑𝑦 2

𝑑𝑡

𝑦√1 + ( ) 𝑑𝑥 or 2𝜋 ∫

𝑑𝑡

𝑦√( ) + ( ) 𝑑𝑡

𝑑𝑥

𝑡𝑎
𝑑𝑥 2

𝑥𝑏

𝑥 √1 + ( ) 𝑑𝑦 or 2𝜋 ∫
𝑑𝑦

𝑦𝑎

𝑥𝑎

𝑑𝑦 2

𝑡𝑏

𝑥√1 + ( ) 𝑑𝑥 or 2𝜋 ∫
𝑑𝑥

𝑑𝑥 2

𝑑𝑦 2

𝑑𝑡

𝑑𝑡

𝑥 √( ) + ( ) 𝑑𝑡

∴ 𝑟× 𝑑= 𝑎× 𝑑

𝑦−𝑎2
𝑑2

=

𝑧−𝑎3
𝑑3

Vector equation of a Plane
o
𝑟 = 𝑟 𝑝 + 𝜇𝑑1 + 𝛾𝑑2
o
If n is a vector normal to the plane then (𝑟 − 𝑟 𝑝 ) ∙ 𝑛 = 0
o
Cartesian equation: 𝑛1 𝑥 + 𝑛2 𝑦 + 𝑛3 𝑧 = 𝑝
Intersections (no intersection if parallel or skew)
o
Intersecting lines have a common point
o
Intersecting planes have a commons line
o
An intersecting plane with a line has a common point
Angles (if >90 subtract from 180)

∴ 𝑟∙ 𝑛= 𝑝

𝑎∙𝑏

Angle between 2 lines: 𝑐𝑜𝑠𝜃 = ||𝑎||𝑏| |
Angle between 2 planes: 𝑐𝑜𝑠𝜃 = ||𝑛||𝑚|| (n and m are normal directions vectors to planes)
Angle between a line and a plane: 𝑠𝑖𝑛𝜃 = ||𝑛||𝑑||

𝑛∙𝑚

𝑛∙𝑑

Distances

1
⃗⃗⃗⃗⃗
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑇𝑟𝑖𝑎𝑛𝑔𝑙𝑒 𝐴𝐵𝐶 = |𝐴𝐵 × ⃗⃗⃗⃗⃗ |
𝐴𝐶
2

Shortest distance from point (𝛼, 𝛽, 𝛾) to a plane (𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 = 𝑑):
Shortest distance between skew lines: |

|𝑎𝛼+𝑏𝛽+𝑐𝛾−𝑑|
√𝑎2 +𝑏2 +𝑐 2

(𝑎−𝑎1 )∙(𝑑×𝑑1 )
|
|𝑑×𝑑1 |

Further Matrix Algebra
Key Words: Transpose identity and zero matrix, non-singular, inverse, matrix of minors, E-values and vectors, orthogonal, diagonal

𝐴 𝑇 = Transpose of matrix A (obtained by switching rows and columns)
o
If 𝐴 = 𝐴 𝑇 then A is symmetric

If A and B have dimensions (𝑚 × 𝑝 𝑎𝑛𝑑 𝑝 × 𝑛) then (𝐴𝐵) 𝑇 = 𝐵 𝑇 𝐴 𝑇

If A is non-singular then 𝐴𝐴−1 = 𝐴−1 𝐴 = 𝐼

To find the inverse of a matric A
o
Find 𝑑𝑒𝑡(𝐴)
o
Form M: the matric of the minors of A
o
Form C: by changing signs of elements of M according to the rule of alternating signs
o

𝑡𝑎

Vectors
Key Words: vector, cross, applications, determinant, alternating signs, triple scalar, planes, line, parallel, normal, angles, distances,
⃗⃗⃗⃗⃗ = ⃗⃗⃗⃗⃗ − ⃗⃗⃗⃗⃗

𝐴𝐵
𝑂𝐵
𝑂𝐴

Scalar Product (dot product): 𝑎 ∙ 𝑏 = |𝑎||𝑏| 𝑐𝑜𝑠 𝜃

Vector Product (cross product)
o
Use: to find a vector normal to a plane
o
𝑎 × 𝑏 = |𝑎||𝑏| 𝑠𝑖𝑛 𝜃 ̂ (angle between a and b, unit vector, perpendicular to a and b)
𝑛
𝑖
𝑗
𝑘
𝑎2 𝑎3
𝑎1 𝑎3
𝑎1 𝑎2
o
𝑎 × 𝑏 = | 𝑎1 𝑎2 𝑎3 | = 𝑖 | 𝑏
𝑏3 | − 𝑗 | 𝑏1 𝑏3 | + 𝑘 | 𝑏1 𝑏2 |
2
𝑏1 𝑏2 𝑏3
o
The vector product of any 2 parallel vectors is zero
o
The vector product is anti-commutative as 𝑎 × 𝑏 = −𝑏 × 𝑎

Applications of the Vector Product
⃗⃗⃗⃗⃗
o
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑃𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚 𝐴𝐵𝐶𝐷 = |𝐴𝐵 × ⃗⃗⃗⃗⃗ |
𝐴𝐶
o

𝑑1

=

o

√1 + ( 𝑑𝑦) 𝑑𝑥 or ∫

𝑥𝑎



𝑥−𝑎1

o

o
√𝑥 2 − 1 𝑥 = 𝑐𝑜𝑠ℎ 𝑢
Arc Length between 𝐴(𝑥 𝑎 , 𝑦 𝑎 ) and 𝐵(𝑥 𝑏 , 𝑦 𝑏 )
o

|𝑎 ∙ (𝑏 × 𝑐)|

o

√1 + 𝑥 2 𝑥 = 𝑠𝑖𝑛ℎ 𝑢

𝑥𝑏

Cartesian equation: 𝜇 =

𝑎3
𝑏3 |
𝑐3

o



√1 − 𝑥 2 𝑥 = 𝑠𝑖𝑛 𝑢
1 + 𝑥 2 𝑥 = 𝑡𝑎𝑛 𝑢

o

6

𝑎2
𝑏2
𝑐2

o

Integration
Key Words: learn results, integration by parts x1 trick, reduction formulae, arc length, surface area of revolution

For an integral involving…use…
o
o

1

Vector equation of a Line
o
𝑟 = 𝑎 + 𝜇(𝑏 − 𝑎)
o
Alternative form: (𝑟 − 𝑎) × 𝑑 = 0
o

= 1, 𝑥 = 𝑎𝑐𝑜𝑠ℎ 𝑡 and 𝑦 = 𝑏𝑠𝑖𝑛ℎ 𝑡, 𝑥 = 𝑎𝑠𝑒𝑐 𝜃 and 𝑦 =

𝑏𝑡𝑎𝑛 𝜃
o



= 1, 𝑥 = 𝑎𝑐𝑜𝑠 𝑡 and 𝑦 = 𝑏𝑠𝑖𝑛 𝑡

−
2

𝑉𝑜𝑙𝑚𝑒 𝑜𝑓 𝑇𝑒𝑡𝑟𝑎ℎ𝑒𝑑𝑟𝑜𝑛 =

2

Osborn’s rule (all trig identities go to hyperbolic except…)
o
𝑠𝑖𝑛2 𝐴 → − 𝑠𝑖𝑛ℎ2 𝐴
Further Coordinate Systems
Key Words: USE FORAMULA BOOKELT!! Ellipse, major/ minor axis, hyperbola, eccentricity, parabola, focus, Directrix, locus
Ellipse:

𝑎1
𝑖
𝑗
𝑘
𝑎 ∙ (𝑏 × 𝑐) = (𝑎1 𝑖 + 𝑎2 𝑗 + 𝑎3 𝑘) ∙ | 𝑏1 𝑏2 𝑏3 | = | 𝑏1
𝑐1
𝑐1 𝑐2 𝑐3
𝑉𝑜𝑙𝑚𝑒 𝑜𝑓 𝑃𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑒𝑝𝑖𝑝𝑒𝑑 = |𝑎 ∙ (𝑏 × 𝑐)|

o

𝑒 𝑥 +𝑒 −𝑥





Triple Scalar Product










𝐴−1 =

1
𝑑𝑒𝑡(𝐴)

𝐶𝑇

If A and B are non-singular then (𝐴𝐵)−1 = 𝐵 −1 𝐴−1
A transformation UT before A represents a transformation of T then a transformation of U on A
Characteristic equation of A is 𝑑𝑒𝑡(𝐴 − 𝜆𝐼)=0 – solve this to find Eigenvalues of A (remember normalising)
For a Matrix 𝐴, a valid eigenvector 𝑥 satisfies 𝐴𝑥 = 𝑘𝑥 (where k is a scalar constant)
If M is square and 𝑀𝑀 𝑇 = 𝐼 then M is orthogonal
o
If orthogonal then 𝑀−1 = 𝑀 𝑇
o
If orthogonal then with normalised column vectors (𝑥1 , 𝑥2 , 𝑥3 ) then: 𝑥1
...
𝑥2 = 𝑥1

---