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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.
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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.
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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.
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This document outlines three theorems for logarithms:
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2) The Quotient Theorem states that the log of a quotient is equal to the difference of the logs of the factors.
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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
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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
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Exercise #11 notes
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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.
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Chapter 2
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Chapter 6
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Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
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1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
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3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
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