12X1 T08 02 general binomial expansions (2011)Nigel Simmons
The document discusses relationships in Pascal's triangle. It shows that the binomial coefficient nCk can be expressed as n-1Ck-1 + n-1Ck. It also shows that Pascal's triangle is symmetrical, with nCk = nCn-k for 1 ≤ k ≤ n-1.
The document discusses how to find the original curve (primitive function) given the derivative (tangent line equation). It states that if the derivative is f'(x)=xn, then the primitive function is f(x)=(x^(n+1))/(n+1)+c. It provides examples such as if f'(x)=3x^4, then f(x)=3x^5/5+c. It also shows how to find the equation of a curve given its gradient function and a point it passes through.
12X1 T08 02 general binomial expansions (2011)Nigel Simmons
The document discusses relationships in Pascal's triangle. It shows that the binomial coefficient nCk can be expressed as n-1Ck-1 + n-1Ck. It also shows that Pascal's triangle is symmetrical, with nCk = nCn-k for 1 ≤ k ≤ n-1.
The document discusses how to find the original curve (primitive function) given the derivative (tangent line equation). It states that if the derivative is f'(x)=xn, then the primitive function is f(x)=(x^(n+1))/(n+1)+c. It provides examples such as if f'(x)=3x^4, then f(x)=3x^5/5+c. It also shows how to find the equation of a curve given its gradient function and a point it passes through.
The document discusses several angle theorems related to circles:
1) Opposite angles of a cyclic quadrilateral are supplementary.
2) The exterior angle of a cyclic quadrilateral is equal to the opposite interior angle.
3) Angles subtended at the circumference by the same or equal arcs are equal. Proofs are provided for each theorem using properties of angles at the center or circumference of a circle.
The document defines and explains key concepts related to polynomial functions. It states that a real polynomial P(x) of degree n is an expression of the form P(x)=p0+p1x+p2x2+...+pn-1xn-1+pxn, where pn≠0 and n≥0 is an integer. It then provides definitions and examples for important polynomial terms like coefficients, degree, leading term, roots, and zeros.
11X1 T14 03 arithmetic & geometric means (2011)Nigel Simmons
The document defines and provides formulas for the arithmetic mean and geometric mean. The arithmetic mean is calculated by summing all values and dividing by the total number of values. The geometric mean is calculated by multiplying all values together and taking the nth root of the product, where n is the number of values. An example is provided to find the arithmetic mean and geometric mean of the values 4 and 25.
A particle undergoing projectile motion obeys two equations of motion: its horizontal acceleration is 0 and its vertical acceleration is -g. The particle has initial conditions when t=0 of an initial horizontal velocity of vcosθ and initial vertical velocity of vsinθ, where v is the initial speed and θ is the launch angle. For example, this can be used to model a ball thrown with an initial speed of 25 m/s at an angle of tan-1(3/4) to the ground to determine properties of its motion.
The document discusses index laws and meanings in algebra. It covers:
- Adding and subtracting like terms, and inability to add unlike terms
- Index laws for multiplication and division of terms with exponents
- Meanings of exponents as they relate to fractions, roots, and powers
- Examples of expanding and simplifying expressions using index laws
- Solutions to exercises involving application of index laws
11X1 T10 07 sum & product of roots (2011)Nigel Simmons
If α and β are the roots of the quadratic equation ax2 + bx + c = 0, then:
(1) The sum of the roots is -b/a;
(2) The product of the roots is c/a;
(3) The roots can be used to form other quadratic equations or to solve for properties of the original quadratic equation.
The document provides instructions for simplifying algebraic fractions. It states that one should always factorize the expression first before cancelling terms. Several worked examples are provided that show the steps to (1) create a common denominator, (2) identify the difference between the old and new denominators, and (3) multiply the numerator by this difference when factorizing.
All straight lines can be written in the form of y = mx + b, where m is the slope and b is the y-intercept. Alternatively, lines can be written in the general form of Ax + By + C = 0, where A, B, and C are integers or surds. Lines parallel to the x-axis have the form y = c, where c is a constant. Lines parallel to the y-axis have the form x = k, where k is a constant. The example shows finding the equation of a line perpendicular to another line in general form.
The document discusses several theorems related to tangents of circles:
1) The angle between a tangent line and radius drawn to the point of contact is 90 degrees.
2) Equal tangents can be drawn from any external point to a circle, and the line joining the point to the center is an axis of symmetry.
3) If two circles share a common tangent, the centers and point of contact must be collinear.
Mathematical induction has the following steps:
1) Prove that the statement is true for the base case (usually n=1).
2) Assume the statement is true for some integer k.
3) Using the assumption from step 2, prove the statement is true for k+1.
4) By proving the statement true for n=1 and showing that if it is true for k then it is true for k+1, the statement is true for all positive integers n.
The document discusses relations and functions. A relation is any set of ordered pairs, while a function assigns each input exactly one output. It also discusses domains and ranges of functions. The domain is the set of all possible inputs, and is found by determining values that would make the function undefined. Examples show how to determine domains based on fractions, roots, and inequality restrictions.
A geometric series is a sequence of numbers where each term is found by multiplying the previous term by a constant value called the common ratio. The document provides examples of finding the common ratio and the general term of geometric series. It also shows how to determine the first term greater than a given value.
The document defines various terms related to circle geometry, including radius, diameter, chord, secant, tangent, arc, sector, segment, concyclic points, cyclic quadrilateral, concentric circles, and theorems about chords and arcs. Specifically, it states that (1) a perpendicular from the circle center to a chord bisects the chord, and the perpendicular bisector of a chord passes through the center, and (2) conversely, the line from the center to the midpoint of a chord is perpendicular to the chord.
The document discusses solving locus problems by eliminating parameters from coordinate points. It provides 3 types of problems: 1) no parameters in coordinates, 2) obvious relationship between coordinates allowing one parameter to be eliminated, 3) no obvious relationship requiring use of a separate proven relationship between parameters. Examples of each type are given along with multi-part word problems involving parabolas, tangents and loci of intersection points.
The document discusses several angle theorems:
1) The angle subtended by an arc at the center of a circle is double the angle subtended by the same arc at the circumference. This is proven using properties of isosceles triangles.
2) The angle in a semicircle is a right angle of 90 degrees. This is proven using the fact that the angle at the center of a full circle is 180 degrees, and the relationship that the central angle is double the circumference angle.
3) Several exercises involving applying these angle theorems are listed at the end.
The document describes how to solve simultaneous equations using three steps: 1) eliminate a variable from the equations, 2) solve for the remaining variable, and 3) substitute back to find the eliminated variable. It provides an example showing these steps to solve two simultaneous equations for x and y. It then shows another example involving eliminating variables to create two new equations that can be solved simultaneously for y and z. The key point is that simultaneous equations can be solved when there are as many equations as unknown variables.
The nth roots of unity are the solutions to the equation zn = ±1.
When placed on an Argand diagram, the nth roots of unity form the vertices of a regular n-sided polygon on the unit circle.
For example, the fifth roots of unity are the solutions to z5 = 1, which are the five regularly spaced points forming a pentagon on the unit circle: 1, cis(2π/5), cis(-2π/5), cis(4π/5), cis(-4π/5).
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 several angle theorems related to circles:
1) Opposite angles of a cyclic quadrilateral are supplementary.
2) The exterior angle of a cyclic quadrilateral is equal to the opposite interior angle.
3) Angles subtended at the circumference by the same or equal arcs are equal. Proofs are provided for each theorem using properties of angles at the center or circumference of a circle.
The document defines and explains key concepts related to polynomial functions. It states that a real polynomial P(x) of degree n is an expression of the form P(x)=p0+p1x+p2x2+...+pn-1xn-1+pxn, where pn≠0 and n≥0 is an integer. It then provides definitions and examples for important polynomial terms like coefficients, degree, leading term, roots, and zeros.
11X1 T14 03 arithmetic & geometric means (2011)Nigel Simmons
The document defines and provides formulas for the arithmetic mean and geometric mean. The arithmetic mean is calculated by summing all values and dividing by the total number of values. The geometric mean is calculated by multiplying all values together and taking the nth root of the product, where n is the number of values. An example is provided to find the arithmetic mean and geometric mean of the values 4 and 25.
A particle undergoing projectile motion obeys two equations of motion: its horizontal acceleration is 0 and its vertical acceleration is -g. The particle has initial conditions when t=0 of an initial horizontal velocity of vcosθ and initial vertical velocity of vsinθ, where v is the initial speed and θ is the launch angle. For example, this can be used to model a ball thrown with an initial speed of 25 m/s at an angle of tan-1(3/4) to the ground to determine properties of its motion.
The document discusses index laws and meanings in algebra. It covers:
- Adding and subtracting like terms, and inability to add unlike terms
- Index laws for multiplication and division of terms with exponents
- Meanings of exponents as they relate to fractions, roots, and powers
- Examples of expanding and simplifying expressions using index laws
- Solutions to exercises involving application of index laws
11X1 T10 07 sum & product of roots (2011)Nigel Simmons
If α and β are the roots of the quadratic equation ax2 + bx + c = 0, then:
(1) The sum of the roots is -b/a;
(2) The product of the roots is c/a;
(3) The roots can be used to form other quadratic equations or to solve for properties of the original quadratic equation.
The document provides instructions for simplifying algebraic fractions. It states that one should always factorize the expression first before cancelling terms. Several worked examples are provided that show the steps to (1) create a common denominator, (2) identify the difference between the old and new denominators, and (3) multiply the numerator by this difference when factorizing.
All straight lines can be written in the form of y = mx + b, where m is the slope and b is the y-intercept. Alternatively, lines can be written in the general form of Ax + By + C = 0, where A, B, and C are integers or surds. Lines parallel to the x-axis have the form y = c, where c is a constant. Lines parallel to the y-axis have the form x = k, where k is a constant. The example shows finding the equation of a line perpendicular to another line in general form.
The document discusses several theorems related to tangents of circles:
1) The angle between a tangent line and radius drawn to the point of contact is 90 degrees.
2) Equal tangents can be drawn from any external point to a circle, and the line joining the point to the center is an axis of symmetry.
3) If two circles share a common tangent, the centers and point of contact must be collinear.
Mathematical induction has the following steps:
1) Prove that the statement is true for the base case (usually n=1).
2) Assume the statement is true for some integer k.
3) Using the assumption from step 2, prove the statement is true for k+1.
4) By proving the statement true for n=1 and showing that if it is true for k then it is true for k+1, the statement is true for all positive integers n.
The document discusses relations and functions. A relation is any set of ordered pairs, while a function assigns each input exactly one output. It also discusses domains and ranges of functions. The domain is the set of all possible inputs, and is found by determining values that would make the function undefined. Examples show how to determine domains based on fractions, roots, and inequality restrictions.
A geometric series is a sequence of numbers where each term is found by multiplying the previous term by a constant value called the common ratio. The document provides examples of finding the common ratio and the general term of geometric series. It also shows how to determine the first term greater than a given value.
The document defines various terms related to circle geometry, including radius, diameter, chord, secant, tangent, arc, sector, segment, concyclic points, cyclic quadrilateral, concentric circles, and theorems about chords and arcs. Specifically, it states that (1) a perpendicular from the circle center to a chord bisects the chord, and the perpendicular bisector of a chord passes through the center, and (2) conversely, the line from the center to the midpoint of a chord is perpendicular to the chord.
The document discusses solving locus problems by eliminating parameters from coordinate points. It provides 3 types of problems: 1) no parameters in coordinates, 2) obvious relationship between coordinates allowing one parameter to be eliminated, 3) no obvious relationship requiring use of a separate proven relationship between parameters. Examples of each type are given along with multi-part word problems involving parabolas, tangents and loci of intersection points.
The document discusses several angle theorems:
1) The angle subtended by an arc at the center of a circle is double the angle subtended by the same arc at the circumference. This is proven using properties of isosceles triangles.
2) The angle in a semicircle is a right angle of 90 degrees. This is proven using the fact that the angle at the center of a full circle is 180 degrees, and the relationship that the central angle is double the circumference angle.
3) Several exercises involving applying these angle theorems are listed at the end.
The document describes how to solve simultaneous equations using three steps: 1) eliminate a variable from the equations, 2) solve for the remaining variable, and 3) substitute back to find the eliminated variable. It provides an example showing these steps to solve two simultaneous equations for x and y. It then shows another example involving eliminating variables to create two new equations that can be solved simultaneously for y and z. The key point is that simultaneous equations can be solved when there are as many equations as unknown variables.
The nth roots of unity are the solutions to the equation zn = ±1.
When placed on an Argand diagram, the nth roots of unity form the vertices of a regular n-sided polygon on the unit circle.
For example, the fifth roots of unity are the solutions to z5 = 1, which are the five regularly spaced points forming a pentagon on the unit circle: 1, cis(2π/5), cis(-2π/5), cis(4π/5), cis(-4π/5).
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
2. Shifting Curves I
y f x c or y c f x
moves f x or c units
3. Shifting Curves I
y f x c or y c f x
moves f x or c units
y f x c
moves f x or c units
4. Shifting Curves I
y f x c or y c f x
moves f x or c units
y f x c
moves f x or c units
y f x
reflects f x in the x axis (i.e. flips upside down)
5. Shifting Curves I
y f x c or y c f x
moves f x or c units
y f x c
moves f x or c units
y f x
reflects f x in the x axis (i.e. flips upside down)
y f x
reflects f x in the y axis
7. e.g. i y x 3 4
2
1. basic curve : y x 2 y
x
8. e.g. i y x 3 4
2
1. basic curve : y x 2 y
2. shift right 3 units
3 x
9. e.g. i y x 3 4
2
1. basic curve : y x 2 y
2. shift right 3 units
4
3. shift up 4 units x
10. e.g. i y x 3 4
2
1. basic curve : y x 2 y
2. shift right 3 units (3,4)
3. shift up 4 units x
11. e.g. i y x 3 4
2
1. basic curve : y x 2 y
2. shift right 3 units (3,4)
3. shift up 4 units x
ii y x 1 2
3
12. e.g. i y x 3 4
2
1. basic curve : y x 2 y
2. shift right 3 units (3,4)
3. shift up 4 units x
ii y x 1 2
3
1. basic curve : y x3 y
x
13. e.g. i y x 3 4
2
1. basic curve : y x 2 y
2. shift right 3 units (3,4)
3. shift up 4 units x
ii y x 1 2
3
1. basic curve : y x3 y
2. reflect in x axis
x
14. e.g. i y x 3 4
2
1. basic curve : y x 2 y
2. shift right 3 units (3,4)
3. shift up 4 units x
ii y x 1 2
3
1. basic curve : y x3 y
2. reflect in x axis
3. shift left 1 unit
x
1
15. e.g. i y x 3 4
2
1. basic curve : y x 2 y
2. shift right 3 units (3,4)
3. shift up 4 units x
ii y x 1 2
3
1. basic curve : y x3 y
2. reflect in x axis
3. shift left 1 unit 2
x
4. shift up 2 units
16. e.g. i y x 3 4
2
1. basic curve : y x 2 y
2. shift right 3 units (3,4)
3. shift up 4 units x
ii y x 1 2
3
1. basic curve : y x3 y
2. reflect in x axis
3. shift left 1 unit (–1,2)
x
4. shift up 2 units
18. iii x 2 2 x y 2 4 y 4 0
x 1 y 2 4 1 4
2 2
9
19. iii x 2 2 x y 2 4 y 4 0
x 1 y 2 4 1 4
2 2
y
9
1. basic curve : x 2 y 2 9
x
20. iii x 2 2 x y 2 4 y 4 0
x 1 y 2 4 1 4
2 2
y
9
1. basic curve : x 2 y 2 9
1 x
2. shift left 1 unit
21. iii x 2 2 x y 2 4 y 4 0
x 1 y 2 4 1 4
2 2
y
9
1. basic curve : x 2 y 2 9 2
x
2. shift left 1 unit
3. shift up 2 units
22. iii x 2 2 x y 2 4 y 4 0
x 1 y 2 4 1 4
2 2
y
9
(–1,2)
1. basic curve : x 2 y 2 9
x
2. shift left 1 unit
3. shift up 2 units
23. iii x 2 2 x y 2 4 y 4 0
x 1 y 2 4 1 4
2 2
y
9
(–1,2)
1. basic curve : x 2 y 2 9
x
2. shift left 1 unit
3. shift up 2 units
iii y 2 x
24. iii x 2 2 x y 2 4 y 4 0
x 1 y 2 4 1 4
2 2
y
9
(–1,2)
1. basic curve : x 2 y 2 9
x
2. shift left 1 unit
3. shift up 2 units
iii y 2 x
y
1. basic curve : y 2 x
1
x
25. iii x 2 2 x y 2 4 y 4 0
x 1 y 2 4 1 4
2 2
y
9
(–1,2)
1. basic curve : x 2 y 2 9
x
2. shift left 1 unit
3. shift up 2 units
iii y 2 x
y
1. basic curve : y 2 x
2. reflect in y axis 1
x
