New day 2 examples
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This document provides examples of solving trigonometric equations over the interval [0, 2π]. The first example is to solve sin(x) = 2/3 and 2cos(x) + 1 = 1/3, the second example is to solve sin(x) = 0.6 and cos(x) + 1 = 3/5 to 2 decimals, and the third example is to solve -5tan(x) + 4 = 0.
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This document provides two examples of solving trigonometric equations. The first example solves the equations sin(x)=2/3 and 2cos(x)+1=1 for x between 0 and 2π, finding approximate solutions of 0.6 and 3/5. The second example solves the exact equations -5tan(x)+4=0 and -3+tan(x)=0 for the angle x.
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The document provides examples of solving trigonometric equations over specific intervals. It gives 3 trigonometric equations to solve: sinθ = 0.6 over [0,π], 2cosφ + 1 = 3/5 over [0,2π], and -5tanψ + 4 = 0 over [0,π]. The solutions are given either exactly or to two decimal places where possible.
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This document provides examples of solving trigonometric equations. The first example asks to solve sin(x) = 2/3 and 2cos(x) + 1 = 1/3 for x between 0 and 2pi to 3 decimal places or exactly. The second example asks to solve -5tan(x) + 4 = 0 and tan(x) - 3 = 0 for exact solutions where possible.
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The document provides examples of solving trigonometric equations over specific intervals. It gives 3 trigonometric equations to solve: sinθ = 0.6 over [0,π], 2cosφ + 1 = 3/5 over [0,2π], and -5tanψ + 4 = 0 over [0,π]. The solutions are given either exactly or to two decimal places where possible.
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This document provides examples of solving trigonometric equations. The first example asks to solve sin(x) = 2/3 and 2cos(x) + 1 = 1/3 for x between 0 and 2pi to 3 decimal places or exactly. The second example asks to solve -5tan(x) + 4 = 0 and tan(x) - 3 = 0 for exact solutions where possible.
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The document provides examples to verify trigonometric identities and determine non-permissible values that make identities false. It shows working through identities like sin+cos=1+cot and verifying that θ=30° is a solution. It also gives an example to show that cos(A+B)≠cosA + cosB is not always an identity by providing values where it is false. Finally, it asks to determine non-permissible values that make identities like sin cot /sec = cos^2 and sin+tan/(1+cos)=sin sec false.
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The document discusses how to graph the square root of a function f(x). It explains that the square root of f(x) is below the graph of f(x), except at invariant points where f(x) equals 0 or 1. It provides examples of graphing the square root of functions like f(x), g(x), h(x), and p(x) by locating invariant points, drawing the graph above f(x) between those points, and sketching a smooth curve below f(x) between key points at perfect squares.
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This document introduces using logs on a calculator. It explains that logs cannot be rewritten with the same base but can be solved by taking the log of both sides and solving for the variable. It provides an example of taking the log of both sides of an equation to solve for x, and notes the steps are to expand the logs and then solve for the variable.
Day 1 examples new
1. Reflecting the graph of y = f(x) across the y-axis using the transformation y = f(-x) does not change the range of the graph, which remains [-5, 1].
2. The order of transformations can matter. For example, reflecting a graph across the x-axis before translating it up 4 units results in a different graph than if the transformations were performed in reverse order.
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This document provides instructions for solving log equations in 4 steps: 1) rewrite the equation with a single log, 2) isolate the variable inside the log, 3) take the inverse log of both sides, 4) check that the solution works in the original equation. It cautions that the solution cannot result in a negative log or log of 0.
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This document discusses compound interest problems and provides examples of calculations using the compound interest formula. It defines the general compound interest formula and explains annual and monthly compounding. Examples are given of calculating how long it would take an investment to double at a given interest rate, comparing the future values of different initial investments with different interest rates compounded annually, and calculating monthly investments needed to reach a future value at a given interest rate compounded monthly.
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A box has dimensions of length x, width x + 2, and height x + 10. The volume function is V(x) = x(x + 2)(x + 10), which equals 96. Solving this cubic equation yields the possible values of x as 4, 8. So the dimensions of the box are length 4, width 6, height 14 or length 8, width 10, height 18.
The ice block originally has dimensions of 3 ft wide, 4 ft high, and 5 ft long, with volume of 3 * 4 * 5 = 60 ft^3. If the goal is to reduce the volume to 24 ft^3, then he should remove 0.5 ft from each dimension, making
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This document discusses natural logs (ln) as an alternative to common logs (log). It notes that ln is called the natural log and ln(e) = 1. It states that all log laws still apply when using natural logs, such as ln(M·N) = lnM + lnN. While common logs and natural logs can be used interchangeably, the document suggests ln may be easier for some applications.
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The document asks which of the following binomials are factors of x^4 - 1: (x - 1)(x + 1), (x^2 - 1), (x - 1). The factors of x^4 - 1 are (x - 1)(x + 1)(x^2 - 1).
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Logarithms are defined as the exponent to which a base must be raised to equal the value of the term, such as log base b of N equals the exponent. Logarithms and exponential forms are equivalent ways to express the same concept. Logarithms cannot be taken of zero or negative numbers.
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This document provides information about converting between degrees and radians and finding angles given information about arcs. It includes:
1) Questions asking to convert specific degree measurements to radians and vice versa.
2) An example of being given the radius and arc length of a circle segment and calculating the central angle measure in degrees.
3) Tables showing the relationship between degrees and radians for common angle measures.
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(1) Combinations refer to arrangements where order does not matter, unlike permutations where order does. (2) The formula for calculating the number of combinations of n things taken r at a time is nCr = n! / (n-r)!r!. (3) Examples where order does not matter include card hands, lottery numbers, and committees.
D E M O C R A C Y & S T R E S S M A N A G E M E N T D R S H R I N I W A S...
Democracy and Stress Management
The document discusses democracy and how individual spiritual practices like meditation can help transform selfish interests into a universal perspective of welfare. It states that a healthy democracy is based on the noble aspirations of the people, not their petty greed. Regular spiritual practices like meditation can help leaders and policymakers connect to their true selves and make decisions with benevolent intentions for all. To evolve a healthy global democracy, the document argues that individuals must transform selfishness into a motivation for universal welfare.
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This document discusses vertical shifts of functions. It states that adding or subtracting "k" units to the function f(x) shifts the entire graph up or down by k units without changing the shape of the graph.
New Study Of Gita Nov 11 Dr Shriniwas J Kashalikar
1) Dr. Shriniwas Kashalikar discusses his study of the Bhagavad Gita and how it has helped alleviate anxieties and develop a deeper understanding of truth and virtue.
2) He explains how studying the Gita has helped him realize that we are all moving towards the same goal, and how this realization can help overcome impatience and develop true love.
3) Dr. Kashalikar discusses how properly understanding verses from the Gita, such as about rescuing one's lower consciousness from bondage, has helped him accept that inner conflicts can present opportunities for spiritual growth.
World Famous Prayer Interpretation By Dr
[1] This prayer seeks happiness, health, and well-being for all, with no grief.
[2] While some object to prayer seeing it as weakness, the author explains prayers help connect to infinite cosmic powers and remove "blocks" like ego that develop in humans.
[3] Prayers and chanting help reconnect us to the empowering source of consciousness and remove feelings of isolation, reestablishing our link to the greater whole, like leaves reconnecting to the roots of the tree.
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This document discusses the key characteristics of polynomial functions, including that they can be written in standard form as a sum of terms with non-negative integer exponents, with the highest non-zero term determining the degree of the polynomial. The end behavior of odd-degree polynomials depends on the leading coefficient, with positives falling and negatives rising as x approaches positive or negative infinity. Polynomials have at most as many x-intercepts as their degree and can have up to one fewer local maxima or minima points than their degree.
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Calculating the derivative numerically involves estimating the slope of the tangent line to a function at a given point by taking incremental changes in both the x- and y-values and seeing how the y-value changes in relation to the x-value. The simplest way is to use the definition of the derivative as the limit of the difference quotient as the change in x approaches 0. More advanced techniques include using smaller increments in x to get a more accurate estimate of the derivative.
Day 2 examples
Division statement for Division by x - a involves writing the original polynomial P(x) as the product of the binomial divisor (x - a) and the quotient polynomial Q(x) plus the remainder R, where P(x) is the original polynomial, (x - a) is the binomial divisor, Q(x) is the quotient polynomial, and R is the remainder.
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