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May 2011 schedule
The document contains a schedule for May and June 2011, listing exercises, tests, and exams for different days of the week. It includes Exercise #1-35 on May 11th, a test on May 12th, no school on May 13th for prom, and Victoria Day on May 23rd with no school. The June schedule focuses on provincial exam review from June 6th to 10th, with the provincial exam taking place from 8:45-12:00 on June 16th in the library.
Exercise #23
Calculators can evaluate base 10 logarithms. The change of base theorem, logb a = logc a / logc b, allows evaluating logarithms with non-base 10 bases by relating them to base 10 logarithms. Some useful properties of logarithms are that logb b = 1, logb b^y = y, and b^logb a = a. Students are assigned exercises 23 problems 1-13, 15-16, 18, and 20 to practice evaluating logarithms.
Exercise #21
The document discusses logarithmic functions and their inverses, exponential functions. It provides examples of logarithmic and exponential forms and how to convert between them. It also covers graphing logarithmic functions, their domains and ranges, characteristics like asymptotes. It provides examples of evaluating logarithmic expressions and transformations of logarithmic functions.
Exercise #21
This document discusses logarithmic functions and their inverses, exponential functions. It provides examples of logarithmic and exponential forms and how to convert between them. It also covers graphing logarithmic functions, their domains and ranges, characteristics like asymptotes. It provides examples of evaluating logarithmic expressions and transformations of logarithmic functions.
Exercise #22
This document outlines three theorems for logarithms:
1) The Product Theorem states that the log of a product is equal to the sum of the logs of the factors.
2) The Quotient Theorem states that the log of a quotient is equal to the difference of the logs of the factors.
3) The Power Theorem states that the log of a factor to a given power is equal to the power times the log of the factor.
These theorems only apply when the logarithms have the same base. Examples are provided to demonstrate applying the theorems to simplify logarithmic expressions and evaluate logarithms.
Exercise #19
The document discusses exponential functions of the form f(x) = a^x where a is a constant. It provides examples of exponential functions with a = 2 and a = 1/2. It describes the domain, range, and common point of exponential functions. The document also discusses transformations of exponential functions by adding or subtracting constants, and provides examples of sketching and describing transformed exponential functions. Finally, it lists common exponential expressions.
Exercise #16 notes
This document presents sum and difference identities for trigonometric functions sine and cosine. It gives the formulas for sin(α + β), cos(α + β), sin(α - β) and cos(α - β) in terms of sinα, cosα, sinβ and cosβ. It then provides examples of using these formulas to evaluate trigonometric functions at values that are not covered by special triangles.
Recommended
May 2011 schedule
The document contains a schedule for May and June 2011, listing exercises, tests, and exams for different days of the week. It includes Exercise #1-35 on May 11th, a test on May 12th, no school on May 13th for prom, and Victoria Day on May 23rd with no school. The June schedule focuses on provincial exam review from June 6th to 10th, with the provincial exam taking place from 8:45-12:00 on June 16th in the library.
Exercise #23
Calculators can evaluate base 10 logarithms. The change of base theorem, logb a = logc a / logc b, allows evaluating logarithms with non-base 10 bases by relating them to base 10 logarithms. Some useful properties of logarithms are that logb b = 1, logb b^y = y, and b^logb a = a. Students are assigned exercises 23 problems 1-13, 15-16, 18, and 20 to practice evaluating logarithms.
Exercise #21
The document discusses logarithmic functions and their inverses, exponential functions. It provides examples of logarithmic and exponential forms and how to convert between them. It also covers graphing logarithmic functions, their domains and ranges, characteristics like asymptotes. It provides examples of evaluating logarithmic expressions and transformations of logarithmic functions.
Exercise #21
This document discusses logarithmic functions and their inverses, exponential functions. It provides examples of logarithmic and exponential forms and how to convert between them. It also covers graphing logarithmic functions, their domains and ranges, characteristics like asymptotes. It provides examples of evaluating logarithmic expressions and transformations of logarithmic functions.
Exercise #22
This document outlines three theorems for logarithms:
1) The Product Theorem states that the log of a product is equal to the sum of the logs of the factors.
2) The Quotient Theorem states that the log of a quotient is equal to the difference of the logs of the factors.
3) The Power Theorem states that the log of a factor to a given power is equal to the power times the log of the factor.
These theorems only apply when the logarithms have the same base. Examples are provided to demonstrate applying the theorems to simplify logarithmic expressions and evaluate logarithms.
Exercise #19
The document discusses exponential functions of the form f(x) = a^x where a is a constant. It provides examples of exponential functions with a = 2 and a = 1/2. It describes the domain, range, and common point of exponential functions. The document also discusses transformations of exponential functions by adding or subtracting constants, and provides examples of sketching and describing transformed exponential functions. Finally, it lists common exponential expressions.
Exercise #16 notes
This document presents sum and difference identities for trigonometric functions sine and cosine. It gives the formulas for sin(α + β), cos(α + β), sin(α - β) and cos(α - β) in terms of sinα, cosα, sinβ and cosβ. It then provides examples of using these formulas to evaluate trigonometric functions at values that are not covered by special triangles.
Exercise #18 notes
This document discusses double angle identities for sine, cosine, and tangent. It provides the formulas for sin(2θ), cos(2θ), and tan(2θ) in terms of sin(θ) and cos(θ). The document also includes examples of using the double angle identities to evaluate trigonometric functions and solve equations.
Exercise #17 notes
This document contains examples of using sum and difference identities to express trigonometric functions with combined arguments in terms of single variables. It provides the identity cos(A + B) = cos(A)cos(B) - sin(A)sin(B) and uses it to express cos(π/2 + x) in terms of x alone. It also gives examples of using identities to prove relationships between trigonometric functions with combined arguments.
Exercise #13 notes ~ graphing
This document provides instructions for graphing trigonometric transformations in 3 steps: 1) Determine the a, b, c, and d values from the function's factored form. 2) Draw the median position and amplitude. 3) Determine the period and mark points to graph the wave-like function. Examples graph y=3sin(2x)-1, f(x)=sin(1/2x+1), and f(x)=2cos(3x)-2.
Exercise #11 notes
This document discusses graphing absolute value functions. It provides examples of graphing various absolute value functions, including f(x)=|x|^2, f(x)=2|x|-1, f(x)=|x|^2-1, f(x)=3|x|^2-4, f(x)=|cos(x)|, f(x)=2|x|-3-1, and shows how to write a piecewise function definition for f(x)=|x|^2. The graphs are V-shaped and symmetric about the y-axis, with vertices at the points where the absolute value terms are equal to zero.
Exercise #13 notes ~ equations
This document provides information on writing trigonometric equations from sinusoidal graphs. It outlines the two basic sinusoidal equations as f(x) = a sin(b(x - c)) + d or f(x) = a cos(b(x - c)) + d. The variables a, b, c, and d represent the amplitude, period, start point, and median, respectively. Formulas are given to identify each variable based on properties of the graph like maximum, minimum, and period.
Exercise #15 notes
This document discusses proving trigonometric identities. It lists four common operations for proving identities: 1) adding or subtracting rational expressions, 2) multiplying or dividing rational expressions, 3) factoring, and 4) multiplying by the conjugate. It then provides four examples of identities to prove using these techniques.
Exercise #14 notes
The document discusses trigonometric identities, including 8 fundamental identities. It provides examples of using identities to simplify trigonometric functions and proves identities by showing the left and right hand sides are the same. Examples include simplifying tan θ cos θ, expressing tan θ in terms of sin θ, and proving cot θ sin θ = cosθ.
Exercise #8 notes
This document discusses how to sketch graphs resulting from transformations of basic parent functions, including horizontal and vertical stretches as well as translations. It provides examples of stretching and translating graphs of f(x) = x^2 and generalizes the effects of stretches and translations on basic parent functions. The document concludes with instructions on combining multiple transformations and an example problem determining the x-intercepts resulting from a composite transformation.
Exercise #9 notes
1) Reflections of functions f(x) over the x-axis or y-axis produce new equations by changing the sign of the y-values or x-values respectively.
2) The inverse of a function f(x) is denoted f^-1(x) and is found by switching the x and y values when graphed, effectively reflecting over the line y=x.
3) For example, the inverse of f(x)=2x+5 is found by graphing f^-1(x) which treats x as the output and y as the input.
Exercise #10 notes
This document provides examples for graphing reciprocals of functions. It explains that for reciprocals, smaller numbers on the x-axis become larger on the y-axis, and vice versa. An example graphs the reciprocal of f(x)=1/x^2. It notes that reciprocals can be written in the form f(x)=k/x+h, where k and h are determined from the original function. Further examples graph the reciprocals of f(x)=2/x+4 and f(x)=sin(x).
Exercise #7 notes
This document discusses transformations of graphs including vertical and horizontal shifts. It provides examples of parent functions being shifted vertically by adding or subtracting a number, and horizontally by adding or subtracting inside or outside the function. Examples are given of shifting the graph of f(x)=sinx by 1 unit vertically and horizontally shifting and vertically shifting the graph of f(x)=x^2-3.
Statement and demand assignment
This document provides information for four tax situations including land and building assessments, mill rates, property frontages, improvement taxes, school and education levy taxes, tax credits, and penalties. The questions involve calculating statements and demands for residential properties with varying assessments, mill rates, improvement taxes, and school division taxes.
Statement and demand notes
The document outlines the sections of a Statement and Demand for Taxes notice, including: A) municipal taxes on property classifications and assessed values, B) local improvement costs, C) educational taxes including school division and provincial levy mill rates, D) provincial tax credits, E) arrears and penalties, and F) net taxes due which is the total of taxes and penalties minus credits. Users are prompted to enter property details, mill rates, and other values for each corresponding section of the tax statement.
Statement and demand forms
This document discusses statement and demand forms. Statement forms are used to provide information about an account, transaction, or balance. Demand forms are used to request payment for an amount owed, such as an invoice demanding payment for services or goods that have been delivered. Both statement and demand forms convey important financial information in different ways.
Exercise #6 notes
This document provides instructions for graphing circular functions over an interval from 0 to 2. It defines amplitude and period as key vocabulary terms and has students complete tables to graph sine, cosine, and tangent functions. It then asks students to complete a chart comparing the amplitude, period, domain, range, zeros, y-intercepts, and asymptotes of sine, cosine, and tangent functions.
Circle properties summary
Minor arcs on a circle are labeled with 2 letters and have a measure of less than 180 degrees. The document discusses properties of circles, triangles, and angles including: the Pythagorean theorem, that a radius will meet a tangent at 90 degrees, an inscribed angle is half the measure of the central angle if they share an arc, two inscribed angles subtended by the same arc are congruent, the interior angles of a triangle sum to 180 degrees, and the total measure of angles on a circle is 360 degrees.
Lesson 44
Simple interest is calculated by multiplying the principal amount by the interest rate as a decimal and the time period in years. The principal remains the same and interest is paid at the end of each time interval. The formula is: Interest = Principal x Rate x Time. An example calculates the total repayment amount of borrowing $3,000 from the bank for 2 years at a 4% rate. Another example finds the principal amount if the interest paid after 6 months was $178.50 at a 10.5% annual interest rate. Exercises 1-15 provide additional practice calculating simple interest.
Lesson 41
Budgeting involves planning spending based on earnings. People should save 10% of net pay and spend the rest on expenses, with no more than 30% of take-home pay going to housing costs. When preparing a budget, one determines monthly income and expenses, savings plans, and creates a monthly statement. Income sources include regular pay, bonuses, and benefits, while expenses are fixed costs like rent and variable costs like food and recreation. People should have a reserve of 2-3 months' pay for unexpected costs.
Lesson 39
Local municipalities raise revenue through grants from other levels of government and property taxes. [1] Property taxes are based on the assessed value of a property as determined by an assessor. [2] The municipal government and school division collect taxes at rates based on the portioned assessment and total taxes needed divided by total property value. [3]
Lesson 38
This document discusses graduated commissions, net pay, and deductions from gross pay. It provides an example of a graduated commission structure and calculates earnings. It defines net pay as gross pay minus deductions, which must include CPP, EI, and income taxes, and may include other deductions like pension plans, insurance, and union dues. Current deduction rates and tax brackets are provided for 2008 in Canada. An example calculates net pay for an employee earning $21.75 per hour working 36 hours who pays union dues and contributes to an RRSP.
Chapter 4 - Islamic Financial Institutions in Malaysia.pptx
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Exercise #18 notes
This document discusses double angle identities for sine, cosine, and tangent. It provides the formulas for sin(2θ), cos(2θ), and tan(2θ) in terms of sin(θ) and cos(θ). The document also includes examples of using the double angle identities to evaluate trigonometric functions and solve equations.
Exercise #17 notes
This document contains examples of using sum and difference identities to express trigonometric functions with combined arguments in terms of single variables. It provides the identity cos(A + B) = cos(A)cos(B) - sin(A)sin(B) and uses it to express cos(π/2 + x) in terms of x alone. It also gives examples of using identities to prove relationships between trigonometric functions with combined arguments.
Exercise #13 notes ~ graphing
This document provides instructions for graphing trigonometric transformations in 3 steps: 1) Determine the a, b, c, and d values from the function's factored form. 2) Draw the median position and amplitude. 3) Determine the period and mark points to graph the wave-like function. Examples graph y=3sin(2x)-1, f(x)=sin(1/2x+1), and f(x)=2cos(3x)-2.
Exercise #11 notes
This document discusses graphing absolute value functions. It provides examples of graphing various absolute value functions, including f(x)=|x|^2, f(x)=2|x|-1, f(x)=|x|^2-1, f(x)=3|x|^2-4, f(x)=|cos(x)|, f(x)=2|x|-3-1, and shows how to write a piecewise function definition for f(x)=|x|^2. The graphs are V-shaped and symmetric about the y-axis, with vertices at the points where the absolute value terms are equal to zero.
Exercise #13 notes ~ equations
This document provides information on writing trigonometric equations from sinusoidal graphs. It outlines the two basic sinusoidal equations as f(x) = a sin(b(x - c)) + d or f(x) = a cos(b(x - c)) + d. The variables a, b, c, and d represent the amplitude, period, start point, and median, respectively. Formulas are given to identify each variable based on properties of the graph like maximum, minimum, and period.
Exercise #15 notes
This document discusses proving trigonometric identities. It lists four common operations for proving identities: 1) adding or subtracting rational expressions, 2) multiplying or dividing rational expressions, 3) factoring, and 4) multiplying by the conjugate. It then provides four examples of identities to prove using these techniques.
Exercise #14 notes
The document discusses trigonometric identities, including 8 fundamental identities. It provides examples of using identities to simplify trigonometric functions and proves identities by showing the left and right hand sides are the same. Examples include simplifying tan θ cos θ, expressing tan θ in terms of sin θ, and proving cot θ sin θ = cosθ.
Exercise #8 notes
This document discusses how to sketch graphs resulting from transformations of basic parent functions, including horizontal and vertical stretches as well as translations. It provides examples of stretching and translating graphs of f(x) = x^2 and generalizes the effects of stretches and translations on basic parent functions. The document concludes with instructions on combining multiple transformations and an example problem determining the x-intercepts resulting from a composite transformation.
Exercise #9 notes
1) Reflections of functions f(x) over the x-axis or y-axis produce new equations by changing the sign of the y-values or x-values respectively.
2) The inverse of a function f(x) is denoted f^-1(x) and is found by switching the x and y values when graphed, effectively reflecting over the line y=x.
3) For example, the inverse of f(x)=2x+5 is found by graphing f^-1(x) which treats x as the output and y as the input.
Exercise #10 notes
This document provides examples for graphing reciprocals of functions. It explains that for reciprocals, smaller numbers on the x-axis become larger on the y-axis, and vice versa. An example graphs the reciprocal of f(x)=1/x^2. It notes that reciprocals can be written in the form f(x)=k/x+h, where k and h are determined from the original function. Further examples graph the reciprocals of f(x)=2/x+4 and f(x)=sin(x).
Exercise #7 notes
This document discusses transformations of graphs including vertical and horizontal shifts. It provides examples of parent functions being shifted vertically by adding or subtracting a number, and horizontally by adding or subtracting inside or outside the function. Examples are given of shifting the graph of f(x)=sinx by 1 unit vertically and horizontally shifting and vertically shifting the graph of f(x)=x^2-3.
Statement and demand assignment
This document provides information for four tax situations including land and building assessments, mill rates, property frontages, improvement taxes, school and education levy taxes, tax credits, and penalties. The questions involve calculating statements and demands for residential properties with varying assessments, mill rates, improvement taxes, and school division taxes.
Statement and demand notes
The document outlines the sections of a Statement and Demand for Taxes notice, including: A) municipal taxes on property classifications and assessed values, B) local improvement costs, C) educational taxes including school division and provincial levy mill rates, D) provincial tax credits, E) arrears and penalties, and F) net taxes due which is the total of taxes and penalties minus credits. Users are prompted to enter property details, mill rates, and other values for each corresponding section of the tax statement.
Statement and demand forms
This document discusses statement and demand forms. Statement forms are used to provide information about an account, transaction, or balance. Demand forms are used to request payment for an amount owed, such as an invoice demanding payment for services or goods that have been delivered. Both statement and demand forms convey important financial information in different ways.
Exercise #6 notes
This document provides instructions for graphing circular functions over an interval from 0 to 2. It defines amplitude and period as key vocabulary terms and has students complete tables to graph sine, cosine, and tangent functions. It then asks students to complete a chart comparing the amplitude, period, domain, range, zeros, y-intercepts, and asymptotes of sine, cosine, and tangent functions.
Circle properties summary
Minor arcs on a circle are labeled with 2 letters and have a measure of less than 180 degrees. The document discusses properties of circles, triangles, and angles including: the Pythagorean theorem, that a radius will meet a tangent at 90 degrees, an inscribed angle is half the measure of the central angle if they share an arc, two inscribed angles subtended by the same arc are congruent, the interior angles of a triangle sum to 180 degrees, and the total measure of angles on a circle is 360 degrees.
Lesson 44
Simple interest is calculated by multiplying the principal amount by the interest rate as a decimal and the time period in years. The principal remains the same and interest is paid at the end of each time interval. The formula is: Interest = Principal x Rate x Time. An example calculates the total repayment amount of borrowing $3,000 from the bank for 2 years at a 4% rate. Another example finds the principal amount if the interest paid after 6 months was $178.50 at a 10.5% annual interest rate. Exercises 1-15 provide additional practice calculating simple interest.
Lesson 41
Budgeting involves planning spending based on earnings. People should save 10% of net pay and spend the rest on expenses, with no more than 30% of take-home pay going to housing costs. When preparing a budget, one determines monthly income and expenses, savings plans, and creates a monthly statement. Income sources include regular pay, bonuses, and benefits, while expenses are fixed costs like rent and variable costs like food and recreation. People should have a reserve of 2-3 months' pay for unexpected costs.
Lesson 39
Local municipalities raise revenue through grants from other levels of government and property taxes. [1] Property taxes are based on the assessed value of a property as determined by an assessor. [2] The municipal government and school division collect taxes at rates based on the portioned assessment and total taxes needed divided by total property value. [3]
Lesson 38
This document discusses graduated commissions, net pay, and deductions from gross pay. It provides an example of a graduated commission structure and calculates earnings. It defines net pay as gross pay minus deductions, which must include CPP, EI, and income taxes, and may include other deductions like pension plans, insurance, and union dues. Current deduction rates and tax brackets are provided for 2008 in Canada. An example calculates net pay for an employee earning $21.75 per hour working 36 hours who pays union dues and contributes to an RRSP.
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