The document discusses several angle theorems related to cyclic quadrilaterals:
1) Opposite angles of a cyclic quadrilateral are supplementary. This is proven using properties of angles on a circumference.
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. This is proven using properties of angles on a circumference.
The document discusses trigonometric functions, arcs, sectors, and formulas related to them. It defines an arc as a segment of a circle and provides the formula for calculating the length of an arc as l = rθ. It also defines a sector as a region bounded by two radii and an arc and gives the formula for calculating the area of a sector as A = 1/2 r^2θ. An example problem demonstrates using these formulas to calculate the length of an arc and area of a sector for a given diagram.
The document defines key concepts related to quadratic polynomials and parabolas. It states that a quadratic polynomial has the form ax2 + bx + c, and the graph of a quadratic function y = ax2 + bx + c is a parabola. It defines other key terms like the quadratic equation, coefficients, indeterminate, roots/zeroes, and discusses how the values of a, b, and c impact the shape and position of the parabola. An example of graphing the function y = x2 + 8x + 12 is also provided.
The document describes a method for solving equations of the form asinx + bcosx = c. It involves using a trigonometric identity to rewrite the equation in terms of the tangent of an angle, letting t = tan(θ/2). This results in a quadratic equation that can be solved for t, and then the inverse tangent gives the solutions for θ. An example problem is worked through step-by-step to demonstrate the method.
The document discusses two 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, so the angle of a semicircle must be half of that, or 90 degrees.
(1) If the diameter of a circle is the hypotenuse of a right triangle, the third vertex lies on the circle.
(2) If two points on the same side of a line segment subtend the same angle, the four points are concyclic.
(3) If a pair of opposite angles in a quadrilateral are supplementary, the quadrilateral is cyclic.
The document defines various terms related to circle geometry, including radius, diameter, chord, secant, tangent, arc, sector, segment, and concyclic points. It also presents two theorems about chords: 1) a perpendicular from the circle center to a chord bisects the chord, and 2) the converse - a line from the center to the midpoint of a chord is perpendicular to the chord. Diagrams and proofs of the theorems are provided.
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.
The document discusses several angle theorems related to cyclic quadrilaterals:
1) Opposite angles of a cyclic quadrilateral are supplementary. This is proven using properties of angles on a circumference.
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. This is proven using properties of angles on a circumference.
The document discusses trigonometric functions, arcs, sectors, and formulas related to them. It defines an arc as a segment of a circle and provides the formula for calculating the length of an arc as l = rθ. It also defines a sector as a region bounded by two radii and an arc and gives the formula for calculating the area of a sector as A = 1/2 r^2θ. An example problem demonstrates using these formulas to calculate the length of an arc and area of a sector for a given diagram.
The document defines key concepts related to quadratic polynomials and parabolas. It states that a quadratic polynomial has the form ax2 + bx + c, and the graph of a quadratic function y = ax2 + bx + c is a parabola. It defines other key terms like the quadratic equation, coefficients, indeterminate, roots/zeroes, and discusses how the values of a, b, and c impact the shape and position of the parabola. An example of graphing the function y = x2 + 8x + 12 is also provided.
The document describes a method for solving equations of the form asinx + bcosx = c. It involves using a trigonometric identity to rewrite the equation in terms of the tangent of an angle, letting t = tan(θ/2). This results in a quadratic equation that can be solved for t, and then the inverse tangent gives the solutions for θ. An example problem is worked through step-by-step to demonstrate the method.
The document discusses two 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, so the angle of a semicircle must be half of that, or 90 degrees.
(1) If the diameter of a circle is the hypotenuse of a right triangle, the third vertex lies on the circle.
(2) If two points on the same side of a line segment subtend the same angle, the four points are concyclic.
(3) If a pair of opposite angles in a quadrilateral are supplementary, the quadrilateral is cyclic.
The document defines various terms related to circle geometry, including radius, diameter, chord, secant, tangent, arc, sector, segment, and concyclic points. It also presents two theorems about chords: 1) a perpendicular from the circle center to a chord bisects the chord, and 2) the converse - a line from the center to the midpoint of a chord is perpendicular to the chord. Diagrams and proofs of the theorems are provided.
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.
The document discusses trigonometric functions and radian measure. It states that 360 degrees equals 2 pi radians. It then provides a table with common conversions between degrees and radians for angles from 30 to 360 degrees. Examples are also given of converting degrees to radians and radians to degrees.
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
0
The document discusses the relationship between integration and calculating the area under a curve. It shows that the area under a curve from x=a to x=b can be calculated as the integral from a to b of the function f(x) dx. This is equal to the antiderivative F(x) evaluated from b to a. The area under a curve can also be estimated using rectangles, and as the width of the rectangles approaches 0, the estimate becomes the exact area. The derivative of the area function A(x) gives the equation of the curve f(x). Examples are given to calculate the exact area under curves.
5. Equations Reducible To Quadratics
e.g. (i ) x 4 4 x 2 12 0
let m x 2
m2 x 4
m 2 4m 12 0
6. Equations Reducible To Quadratics
e.g. (i ) x 4 4 x 2 12 0
let m x 2
m2 x 4
m 2 4m 12 0
m 6 m 2 0
7. Equations Reducible To Quadratics
e.g. (i ) x 4 4 x 2 12 0
let m x 2
m2 x 4
m 2 4m 12 0
m 6 m 2 0
m6 or m 2
8. Equations Reducible To Quadratics
e.g. (i ) x 4 4 x 2 12 0
let m x 2
m2 x 4
m 2 4m 12 0
m 6 m 2 0
m6 or m 2
x2 6 or x 2 2
9. Equations Reducible To Quadratics
e.g. (i ) x 4 4 x 2 12 0
let m x 2
m2 x 4
m 2 4m 12 0
m 6 m 2 0
m6 or m 2
x2 6 or x 2 2
x 6
10. Equations Reducible To Quadratics
e.g. (i ) x 4 4 x 2 12 0
let m x 2
m2 x 4
m 2 4m 12 0
m 6 m 2 0
m6 or m 2
x2 6 or x 2 2
x 6 no real solutions
11. Equations Reducible To Quadratics
e.g. (i ) x 4 4 x 2 12 0
let m x 2
m2 x 4
m 2 4m 12 0
m 6 m 2 0
m6 or m 2
x2 6 or x 2 2
x 6 no real solutions
x 6
12. Equations Reducible To Quadratics
e.g. (i ) x 4 4 x 2 12 0 (ii ) 9 x 4 3x 3 0
let m x 2
m2 x 4
m 2 4m 12 0
m 6 m 2 0
m6 or m 2
x2 6 or x 2 2
x 6 no real solutions
x 6
13. Equations Reducible To Quadratics
e.g. (i ) x 4 4 x 2 12 0 (ii ) 9 x 4 3x 3 0
let m x 2 let m 3x
m2 x 4
m 2 4m 12 0
m 6 m 2 0
m6 or m 2
x2 6 or x 2 2
x 6 no real solutions
x 6
14. Equations Reducible To Quadratics
e.g. (i ) x 4 4 x 2 12 0 (ii ) 9 x 4 3x 3 0
let m x 2 let m 3x
m2 x 4 m 3 3 3 9x
2 x 2 2x 2 x
m 2 4m 12 0
m 6 m 2 0
m6 or m 2
x2 6 or x 2 2
x 6 no real solutions
x 6
15. Equations Reducible To Quadratics
e.g. (i ) x 4 4 x 2 12 0 (ii ) 9 x 4 3x 3 0
let m x 2 let m 3x
m2 x 4 m 3 3 3 9x
2 x 2 2x 2 x
m 2 4m 12 0 m 2 4m 3 0
m 6 m 2 0
m6 or m 2
x2 6 or x 2 2
x 6 no real solutions
x 6
16. Equations Reducible To Quadratics
e.g. (i ) x 4 4 x 2 12 0 (ii ) 9 x 4 3x 3 0
let m x 2 let m 3x
m2 x 4 m 3 3 3 9x
2 x 2 2x 2 x
m 2 4m 12 0 m 2 4m 3 0
m 6 m 2 0 m 3 m 1 0
m6 or m 2
x2 6 or x 2 2
x 6 no real solutions
x 6
17. Equations Reducible To Quadratics
e.g. (i ) x 4 4 x 2 12 0 (ii ) 9 x 4 3x 3 0
let m x 2 let m 3x
m2 x 4 m 3 3 3 9x
2 x 2 2x 2 x
m 2 4m 12 0 m 2 4m 3 0
m 6 m 2 0 m 3 m 1 0
m6 or m 2 m 3 or m 1
x2 6 or x 2 2
x 6 no real solutions
x 6
18. Equations Reducible To Quadratics
e.g. (i ) x 4 4 x 2 12 0 (ii ) 9 x 4 3x 3 0
let m x 2 let m 3x
m2 x 4 m 3 3 3 9x
2 x 2 2x 2 x
m 2 4m 12 0 m 2 4m 3 0
m 6 m 2 0 m 3 m 1 0
m6 or m 2 m 3 or m 1
x2 6 or x 2 2 3x 3 or 3x 1
x 6 no real solutions
x 6
19. Equations Reducible To Quadratics
e.g. (i ) x 4 4 x 2 12 0 (ii ) 9 x 4 3x 3 0
let m x 2 let m 3x
m2 x 4 m 3 3 3 9x
2 x 2 2x 2 x
m 2 4m 12 0 m 2 4m 3 0
m 6 m 2 0 m 3 m 1 0
m6 or m 2 m 3 or m 1
x2 6 or x 2 2 3x 3 or 3x 1
x 6 no real solutions x 1
x 6
20. Equations Reducible To Quadratics
e.g. (i ) x 4 4 x 2 12 0 (ii ) 9 x 4 3x 3 0
let m x 2 let m 3x
m2 x 4 m 3 3 3 9x
2 x 2 2x 2 x
m 2 4m 12 0 m 2 4m 3 0
m 6 m 2 0 m 3 m 1 0
m6 or m 2 m 3 or m 1
x2 6 or x 2 2 3x 3 or 3x 1
x 6 no real solutions x 1 or x 0
x 6
