Establishing the recurrence relation
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This document discusses using recurrence relations to model sequences. It provides examples of finding recurrence relations for sequences like 4, 8, 20, 56 and 2, 6, 26. It then uses a recurrence relation to model a situation where worker ants are dying from disease each week at 20% but they are capturing 100 extra ants each week. Finally, it uses a recurrence relation to determine that a loan shark is charging 10% interest per week on a loan based on the amounts owed after 0 and 2 weeks.
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- PRIMARY KEY and FOREIGN KEY constraints enforce data integrity and relationships between tables.
The Rise and Fall of the Roman Empire
The document provides an overview of the rise and fall of the Roman Empire from 509 BCE to 476 CE. It discusses the transition from Roman Republic to Roman Empire under Augustus, key emperors like Tiberius, Caligula, Nero, Vespasian, and Hadrian, as well as the division of the empire into eastern and western halves under Diocletian. It also examines factors that contributed to the decline of the western empire, including invasions by barbarian tribes and the removal of the capital to Constantinople, while the eastern empire continued on as the Byzantine Empire until 1453.
National 5 Graphic Communication
The document discusses color theory and desktop publishing principles. It describes primary, secondary and tertiary colors. It also discusses tints, shades, warm and cold colors. It outlines design principles like alignment, bleed, captions and drop caps. It provides information on fonts, headers, footers, guides and page layout features.
Vector journeys!
The document discusses vectors and their components. It provides examples of finding the components of vectors using other known vectors, such as finding the components of vector AC given the components of vectors AB and BC. It also discusses using the triangle rule to find unknown vectors. Diagrams are included to illustrate different vector problems and solutions.
Vector multiplication dot product
The document discusses vector multiplication using the dot product. It provides two formulas for calculating the dot product:
1) a.b = |a| |b| cosθ, where |a| and |b| are the magnitudes of the vectors, θ is the angle between them, and cosθ gives the effectiveness of their interaction. The dot product is largest when the vectors are parallel (θ = 0).
2) For vectors in 3D space, a.b = x1x2 + y1y2 + z1z2, where the x, y, z components of each vector are multiplied and summed.
It provides examples of calculating the dot product using both formulas in different vector
Dot product calc angle to finish!
This document discusses using the dot product to calculate the angle between two vectors. It provides the formula for calculating the dot product and shows examples of using the formula to find the angle between vectors. It also discusses how if two vectors are perpendicular, their dot product will equal 0.
Unit vectors 14
This document introduces unit vectors and how they can be used to write the components of a vector. It defines a unit vector as having a magnitude of 1 and lists the standard unit vectors i, j, k. It then shows how any vector can be written as the sum of its components in the direction of each unit vector, using examples like 3i + 4j + 7k. Finally, it provides additional examples of writing vectors in component form using unit vectors and calculating the sum of two vectors.
Vector bits and pieces
This document provides information about vectors, including:
- The relationship between the magnitude of a vector and the cosine of the angle between two vectors in determining their dot product.
- Unit vectors have a magnitude of 1 and are perpendicular to each other.
- Examples are given of calculating the dot product of vectors in different geometric configurations by decomposing into the dot products of the vectors with the unit vectors.
- Key steps are to start from the same point when calculating dot products of vectors in a figure.
Vectors intro
Vectors can represent journeys with distance and direction. They have components that work like coordinates and magnitude that is calculated using Pythagoras. Vectors can be added by placing them nose to tail and adding corresponding components, subtracted by adding the negative vector, and multiplied by a scalar by multiplying each component.
Ratios
The document discusses applying ratios to vectors. It provides examples of using ratios to find unknown points along vectors. The key steps are: 1) Form an equation using the position vector rule relating the unknown point P to the known endpoints A and B; 2) Make P the subject of the equation; 3) Solve the equation to find the coordinates of P. Examples show setting up equations for points dividing vectors in given ratios and solving to find the coordinates of the unknown points.
Parallel + collinear vectors
Vectors AB and CD are parallel if their direction vectors are proportional.
Vectors A, B, and C are collinear if their direction vectors are proportional and they share a common point.
Three examples are given to demonstrate proving vectors are parallel or collinear by showing their direction vectors are proportional.
Position and 3 d vectors amended
This document discusses position vectors in 2D and 3D coordinate systems.
- A position vector starts at the origin and represents the displacement from the origin to a point P defined by its coordinates.
- The position vector rule states that the position vector from point P to point R is equal to the vector from R's position vector subtracted by P's position vector.
- Position vectors and their properties like magnitude, addition, subtraction, and scalar multiplication follow the same rules in 3D as in 2D. Several examples are provided to demonstrate calculating the position vector and magnitude between two points in 3D space.
Solving trig equations higher
The document provides information on solving more difficult trigonometric equations. It discusses finding the acute angle and related angles, using trig function properties to isolate the angle, and considering additional solutions that occur every period of the trig function. Example problems are worked through step-by-step to demonstrate these concepts.
Solving trig equations + double angle formulae
1) The document discusses solving trigonometric equations using compound angle formulas and double angle formulas.
2) It provides examples of using these formulas to solve equations such as sin2x + cosx = 0, 2sinxcosx + cosx = 0, and cos2x - sinx = 0.
3) The key steps are to use a double angle formula to get one side equal to zero, factorize the equation, and then solve the resulting trigonometric mini equations.
Solving exponential equations
This document discusses solving exponential equations by taking the log of both sides. It shows how to solve equations with unknown powers and equations containing e as the base. The key steps are:
1) Isolate the term with the unknown power
2) Take the log of both sides
3) Use properties of logs to isolate the unknown variable
4) Calculate the value of the variable
Examples are provided to illustrate solving equations with unknown powers, equations containing e, and applying the method to context problems. The overall approach is to take logs to transform exponential equations into linear equations that can be easily solved.
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Establishing the recurrence relation
- 1. Block 1 Establishing the Recurrence Relation
- 2. What is to be learned? • How to find the formulae for linear recurrence relations that are not arithmetic or geometric
- 3. 4 8 20 56…. Is of form Un+1 = mUn + c so U2 = mU1 + c U3 = mU2 + c 4m + c = 8 8m + c = 20 U1 U2 U3 U4 8 = m(4) + c 20= m(8) + c 4m = 12 m=3 c=-4 Un+1 = 3Un – 4 U5 = 3(56) – 4 Substituting U5 = 164 Rearranging Sim. Equations
- 4. Finding Recurrence Relation 2 6 26…. Is of form Un+1 = mUn + c so U2 = mU1 + c U3 = mU2 + c 2m + c = 6 6m + c = 26 U1 U2 U3 6 = m(2) + c 26= m(6) + c 4m = 20 m=5 c=-4 Un+1 = 5Un – 4 Substituting Rearranging Sim. Equations
- 5. 3 11 27 59…. Is of form Un+1 = mUn + c so U2 = mU1 + c U3 = mU2 + c 3m + c = 11 11m + c = 27 U1 U2 U3 U4 11 = m(3) + c 27 = m(11) + c 8m = 16 m=2 c=5 Un+1 = 2Un + 5 U5 = 2(59) + 5 Substituting U5 = 123 Rearranging Sim. Equations Key Question Make a recurrence relation to fit this sequence and use it to find the next term
- 6. An ant colony are suffering from an outbreak of disease and worker ants are reducing by a certain percentage weekly. Being resourceful creatures they are capturing an extra ants to work for them each week There were initially 1400 worker ants. A week later there are 1220. The following week, there are 1076. What percentage of ants are dying? How many ants are they capturing? AAn+1n+1 = mA= mAnn + c+ c A0 A1 A2
- 7. AAn+1n+1 = mA= mAnn + c+ c A0 = 1400 A1 = 1220 A2 = 1076 AA11 = mA= mA00 + c+ c AA22 = mA= mA11 + c+ c 1220 = 1400m + c1220 = 1400m + c 1076 = 1220m + c1076 = 1220m + c 144 = 180m m = 0.8 c = 100 AAn+1n+1 = 0.8A= 0.8Ann + 100+ 100 20% decrease in worker ants 100 ants captured weekly
- 8. Sandra has borrowed money from Lenny the loan shark. The original loan was £3000. She is paying back £200 each week. She is shocked to find that 2 weeks later she now owes £3,210!!!!!! What are Lenny’s rates of interest? Ln+1 = mLn + c L0 L2 – 200 Warning - Very Difficult!!
- 9. Ln+1 = mLn – 200L0 = 3000 L2= 3210 Equation with L0? L1 = mL0 – 200 L1 = m(3000) – 200 Equation with L2? L2 = mL1 – 200 3210 = mL1 – 200 L1 = 3000m – 200 3210 = mL1 – 200
- 10. L1 = 3000m – 200 3210 = mL1 – 200 3210 = m(3000m – 200) – 200 3210 = 3000m2 – 200m – 200 0 = 3000m2 – 200m – 3410 Using Big Nasty Formula m = 1.1 Ln+1 = 1.1Ln – 200 Interest Rate 10% per week.