12X1 T09 01 definitions and theory (2010)Nigel Simmons
The document defines key probability terms like probability, sample space, equally likely events, mutually exclusive events, and non-mutually exclusive events. It also provides the formula for calculating probability as the number of favorable outcomes divided by the total number of possible outcomes. As an example, it calculates the probability of throwing a total of 3 or 7 when rolling a pair of dice.
A combination is a set of objects where the order the objects are arranged is not important. The number of combinations of selecting k objects from n unique objects is calculated as nCk, which is equal to n! / (k!(n-k)!). Combinations count the number of unique sets that can be formed, regardless of order, from a larger set of objects.
11 x1 t05 05 arrangements in a circle (2013)Nigel Simmons
The document discusses the difference between arranging objects in a line versus in a circle. When objects are arranged in a line, there is a definite start and end point, so the first object can be placed in any of n positions. However, in a circular arrangement there is no definite start or end, so the number of arrangements is n! rather than n times the factorial of the remaining objects. Examples are provided to illustrate calculating the number of possible arrangements in different circular seating scenarios.
A combination is a set of objects where the order the objects are arranged is not important. The number of combinations of selecting k objects from n total objects is calculated as nCk, or n!/(k!(n-k)!). This represents the number of unique groups that can be formed without regard to order. Examples are provided calculating the number of combinations for choosing different numbers of objects from a total set.
This document discusses the differences between arranging objects in a line versus arranging them in a circle. When objects are arranged in a line, there is a definite start and end point, so the first object can be placed in any of n positions. However, when objects are arranged in a circle, there is no definite start or end, so the number of arrangements for the first object is always 1, since its position defines where the circle begins. As a result, the total number of arrangements of n objects in a circle is n!, while the number in a line is n*(n-1)*(n-2)...1. Examples are provided to illustrate calculating arrangements of objects in circles.
12X1 T09 06 probability & counting techniquesNigel Simmons
- A chess match between a Home and Away team is played across 4 boards
- The probability of the Home team winning, drawing, or losing on each board is given
- The results are recorded as a string of W, D, L indicating the outcome on each board
- There are 81 possible result recordings as there are 3 possible outcomes on each of the 4 boards
- The probability of the result "WDLD" is calculated as the product of the probabilities of each individual outcome
The document discusses permutations and arrangements of objects where some objects are the same. It provides examples of arranging objects (letters, words) in different scenarios where the number of identical objects varies from 0 to 4. It also includes examples of counting arrangements of letters in words and positions of vowels in words.
The document defines permutations and combinations. A k-permutation is an ordered lineup of k objects taken from a set of n objects. The number of k-permutations of n objects is nPk = n!/(n-k)!. A k-combination is an unordered collection of k objects from a set of n objects. The number of k-combinations of n objects is nCk = n!/(n-k)!k!. Several examples are provided to illustrate calculating the number of permutations and combinations.
12X1 T09 01 definitions and theory (2010)Nigel Simmons
The document defines key probability terms like probability, sample space, equally likely events, mutually exclusive events, and non-mutually exclusive events. It also provides the formula for calculating probability as the number of favorable outcomes divided by the total number of possible outcomes. As an example, it calculates the probability of throwing a total of 3 or 7 when rolling a pair of dice.
A combination is a set of objects where the order the objects are arranged is not important. The number of combinations of selecting k objects from n unique objects is calculated as nCk, which is equal to n! / (k!(n-k)!). Combinations count the number of unique sets that can be formed, regardless of order, from a larger set of objects.
11 x1 t05 05 arrangements in a circle (2013)Nigel Simmons
The document discusses the difference between arranging objects in a line versus in a circle. When objects are arranged in a line, there is a definite start and end point, so the first object can be placed in any of n positions. However, in a circular arrangement there is no definite start or end, so the number of arrangements is n! rather than n times the factorial of the remaining objects. Examples are provided to illustrate calculating the number of possible arrangements in different circular seating scenarios.
A combination is a set of objects where the order the objects are arranged is not important. The number of combinations of selecting k objects from n total objects is calculated as nCk, or n!/(k!(n-k)!). This represents the number of unique groups that can be formed without regard to order. Examples are provided calculating the number of combinations for choosing different numbers of objects from a total set.
This document discusses the differences between arranging objects in a line versus arranging them in a circle. When objects are arranged in a line, there is a definite start and end point, so the first object can be placed in any of n positions. However, when objects are arranged in a circle, there is no definite start or end, so the number of arrangements for the first object is always 1, since its position defines where the circle begins. As a result, the total number of arrangements of n objects in a circle is n!, while the number in a line is n*(n-1)*(n-2)...1. Examples are provided to illustrate calculating arrangements of objects in circles.
12X1 T09 06 probability & counting techniquesNigel Simmons
- A chess match between a Home and Away team is played across 4 boards
- The probability of the Home team winning, drawing, or losing on each board is given
- The results are recorded as a string of W, D, L indicating the outcome on each board
- There are 81 possible result recordings as there are 3 possible outcomes on each of the 4 boards
- The probability of the result "WDLD" is calculated as the product of the probabilities of each individual outcome
The document discusses permutations and arrangements of objects where some objects are the same. It provides examples of arranging objects (letters, words) in different scenarios where the number of identical objects varies from 0 to 4. It also includes examples of counting arrangements of letters in words and positions of vowels in words.
The document defines permutations and combinations. A k-permutation is an ordered lineup of k objects taken from a set of n objects. The number of k-permutations of n objects is nPk = n!/(n-k)!. A k-combination is an unordered collection of k objects from a set of n objects. The number of k-combinations of n objects is nCk = n!/(n-k)!k!. Several examples are provided to illustrate calculating the number of permutations and combinations.
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
The document discusses different methods for factorising expressions:
1) Looking for a common factor and dividing it out of all terms
2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b))
3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.
The document provides information on index laws and the meaning of indices in algebra:
- Index laws state that am × an = am+n, am ÷ an = am-n, and (am)n = amn. Exponents can be added or subtracted when multiplying or dividing terms with the same base.
- Positive exponents indicate a term is raised to a power. Negative exponents indicate a root is being taken. Terms with exponents are evaluated from left to right.
- Examples demonstrate how to simplify expressions using index laws and interpret different types of indices.
12 x1 t01 03 integrating derivative on function (2013)Nigel Simmons
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
The document discusses logarithms and their properties. Logarithms are defined as the inverse of exponentials. If y = ax, then x = loga y. The natural logarithm is log base e, written as ln. Properties of logarithms include: loga m + loga n = loga mn; loga m - loga n = loga(m/n); loga mn = n loga m; loga 1 = 0; loga a = 1. Examples of evaluating logarithmic expressions are provided.
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
P
4
3
2
The document discusses properties of polynomials with multiple roots. It first proves that if a polynomial P(x) has a root x = a of multiplicity m, then the derivative of P(x), P'(x), will have a root x = a of multiplicity m-1. It then provides an example of solving a cubic equation given it has a double root. Finally, it examines a quartic polynomial and shows that its root α cannot be 0, 1, or -1, and that 1/
The document discusses factorizing complex expressions. The main points are:
- If a polynomial's coefficients are real, its roots will appear in complex conjugate pairs.
- Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers, or into n linear factors over complex numbers.
- Odd degree polynomials must have at least one real root.
- Examples of factorizing polynomials over both real and complex numbers are provided.
The document describes the Trapezoidal Rule for approximating the area under a curve between two points. It shows that the area A is estimated by dividing the region into trapezoids with height equal to the function values at the interval endpoints and bases equal to the intervals. In general, the area is approximated as the sum of the areas of each trapezoid, which is equal to the average of the endpoint function values multiplied by the interval length.
The document discusses methods for calculating the volumes of solids of revolution. It provides formulas for finding volumes when an area is revolved around either the x-axis or y-axis. Examples are given for finding volumes of common solids like cones, spheres, and others. Steps are shown for using the formulas to calculate volumes based on given functions and limits of revolution.
The document discusses different methods for calculating the area under a curve or between curves.
(1) The area below the x-axis is given by the integral of the function between the bounds, which can be positive or negative depending on whether the area is above or below the x-axis.
(2) To calculate the area on the y-axis, the function is solved for x in terms of y, then the bounds are substituted into the integral of this new function with respect to y.
(3) The area between two curves is calculated by taking the integral of the upper curve minus the integral of the lower curve, both between the same bounds on the x-axis.
The document discusses 8 properties of definite integrals:
1) Integrating polynomials results in a fraction.
2) Constants can be factored out of integrals.
3) Integrals of sums are equal to the sum of integrals.
4) Splitting an integral range results in the sum of the integrals.
5) Integrals of positive functions over a range are positive, and negative if the function is negative.
6) Integrals can be compared based on the relative values of the integrands.
7) Changing the limits of integration flips the sign of the integral.
8) Integrals of odd functions over a symmetric range are zero, and integrals of even functions are twice the integral over
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
The document discusses different methods for factorising expressions:
1) Looking for a common factor and dividing it out of all terms
2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b))
3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.
The document provides information on index laws and the meaning of indices in algebra:
- Index laws state that am × an = am+n, am ÷ an = am-n, and (am)n = amn. Exponents can be added or subtracted when multiplying or dividing terms with the same base.
- Positive exponents indicate a term is raised to a power. Negative exponents indicate a root is being taken. Terms with exponents are evaluated from left to right.
- Examples demonstrate how to simplify expressions using index laws and interpret different types of indices.
12 x1 t01 03 integrating derivative on function (2013)Nigel Simmons
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
The document discusses logarithms and their properties. Logarithms are defined as the inverse of exponentials. If y = ax, then x = loga y. The natural logarithm is log base e, written as ln. Properties of logarithms include: loga m + loga n = loga mn; loga m - loga n = loga(m/n); loga mn = n loga m; loga 1 = 0; loga a = 1. Examples of evaluating logarithmic expressions are provided.
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
P
4
3
2
The document discusses properties of polynomials with multiple roots. It first proves that if a polynomial P(x) has a root x = a of multiplicity m, then the derivative of P(x), P'(x), will have a root x = a of multiplicity m-1. It then provides an example of solving a cubic equation given it has a double root. Finally, it examines a quartic polynomial and shows that its root α cannot be 0, 1, or -1, and that 1/
The document discusses factorizing complex expressions. The main points are:
- If a polynomial's coefficients are real, its roots will appear in complex conjugate pairs.
- Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers, or into n linear factors over complex numbers.
- Odd degree polynomials must have at least one real root.
- Examples of factorizing polynomials over both real and complex numbers are provided.
The document describes the Trapezoidal Rule for approximating the area under a curve between two points. It shows that the area A is estimated by dividing the region into trapezoids with height equal to the function values at the interval endpoints and bases equal to the intervals. In general, the area is approximated as the sum of the areas of each trapezoid, which is equal to the average of the endpoint function values multiplied by the interval length.
The document discusses methods for calculating the volumes of solids of revolution. It provides formulas for finding volumes when an area is revolved around either the x-axis or y-axis. Examples are given for finding volumes of common solids like cones, spheres, and others. Steps are shown for using the formulas to calculate volumes based on given functions and limits of revolution.
The document discusses different methods for calculating the area under a curve or between curves.
(1) The area below the x-axis is given by the integral of the function between the bounds, which can be positive or negative depending on whether the area is above or below the x-axis.
(2) To calculate the area on the y-axis, the function is solved for x in terms of y, then the bounds are substituted into the integral of this new function with respect to y.
(3) The area between two curves is calculated by taking the integral of the upper curve minus the integral of the lower curve, both between the same bounds on the x-axis.
The document discusses 8 properties of definite integrals:
1) Integrating polynomials results in a fraction.
2) Constants can be factored out of integrals.
3) Integrals of sums are equal to the sum of integrals.
4) Splitting an integral range results in the sum of the integrals.
5) Integrals of positive functions over a range are positive, and negative if the function is negative.
6) Integrals can be compared based on the relative values of the integrands.
7) Changing the limits of integration flips the sign of the integral.
8) Integrals of odd functions over a symmetric range are zero, and integrals of even functions are twice the integral over
4. Surdic Equalities
If a b x A B x
then a A , b B
If x y X Y
5. Surdic Equalities
If a b x A B x
then a A , b B
If x y X Y
then x X , y Y
6. Surdic Equalities
If a b x A B x
then a A , b B
If x y X Y
then x X , y Y
2
e.g. x y 2 5
7. Surdic Equalities
If a b x A B x
then a A , b B
If x y X Y
then x X , y Y
2
e.g. x y 2 5
44 55
94 5
8. Surdic Equalities
If a b x A B x
then a A , b B
If x y X Y
then x X , y Y
2
e.g. x y 2 5
44 55
94 5
9 80
9. Surdic Equalities
If a b x A B x
then a A , b B
If x y X Y
then x X , y Y
2
e.g. x y 2 5
44 55
94 5
9 80 x 9, y 80