Day 1 examples u4f13
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This document discusses key concepts related to the unit circle and circular functions including the initial and terminal arms/sides of an angle, standard position, reference angles, and coterminal angles. Specifically, it defines an angle's initial and terminal arms, notes that the initial arm should lay on the x-axis in standard position, gives examples of reference and coterminal angles, and asks to determine coterminal angles within a given range.
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Day 1a examples
The document discusses the unit circle and circular functions. It defines the standard position of an angle as having its initial arm on the x-axis with the angle formed between the terminal arm and the nearest x-axis known as the reference angle. Coterminal angles are also defined as angles which share the same terminal side or arm, with examples given of coterminal angles for 30 degrees and determining coterminal angles within the range of -400 to 400 degrees for an angle measuring 85 degrees.
Lesson 4.3
The document defines key trigonometric concepts including standard position, quadrantal angles, acute angles, obtuse angles, coterminal angles, reference angles, the unit circle, and the definitions of sine and cosine for an angle in standard position. Standard position defines an angle with its vertex at the origin and initial side along the positive x-axis. Quandrantal, acute, and obtuse angles are defined based on where the terminal side falls. Coterminal angles share the same terminal side and differ by multiples of 360 degrees. The reference angle is the acute angle formed by the terminal side and closest x-axis. The unit circle is a circle with radius 1 centered at the origin, where sine and cosine of an
Calc 1.5a
The document discusses infinite limits and determining vertical asymptotes from rational functions. It provides examples of limits that approach positive or negative infinity as the input increases or decreases without bound. It also gives examples of determining the vertical asymptote(s) of rational functions by finding the values that make the denominator equal to zero. Students are provided with practice problems finding vertical asymptotes of rational functions and evaluating limits that may approach positive or negative infinity.
Materiales y sostenibilidad
Este documento presenta información sobre materiales de ingeniería para la sostenibilidad. Brevemente describe temas relacionados con materiales sostenibles como materiales para energía, medio ambiente, altas temperaturas, aleaciones avanzadas, biomateriales e inteligentes. También menciona la conferencia RIO+20 y principios para trabajar de manera sostenible con materiales como conocer sus propiedades, mitigar daños e innovar.
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El documento habla sobre cómo durante la dictadura militar en Argentina se prohibieron muchos libros y cuentos. Los estudiantes leyeron el cuento "La planta de Bartolo" de Laura Dévetach y discutieron las posibles razones por las que fue censurado. Luego, usando un programa llamado Taxgedo en sus computadoras, los estudiantes trabajaron en grupos para escribir sobre los conceptos de democracia y dictadura, practicando también antónimos y sinónimos. El documento concluye mostrando algunas de las producciones de
Day 2 examples u4f13
This document discusses two systems for measuring angles - one with the unit symbol ° and one without a unit symbol. It defines a central angle as the angle subtended or covered by an arc length given a radius. The document also provides the conversion between radians and degrees, showing that 1 radian is equal to 180/π degrees or approximately 57.3 degrees. It gives examples for converting between radians and degrees.
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Day 1a examples
The document discusses the unit circle and circular functions. It defines the standard position of an angle as having its initial arm on the x-axis with the angle formed between the terminal arm and the nearest x-axis known as the reference angle. Coterminal angles are also defined as angles which share the same terminal side or arm, with examples given of coterminal angles for 30 degrees and determining coterminal angles within the range of -400 to 400 degrees for an angle measuring 85 degrees.
Lesson 4.3
The document defines key trigonometric concepts including standard position, quadrantal angles, acute angles, obtuse angles, coterminal angles, reference angles, the unit circle, and the definitions of sine and cosine for an angle in standard position. Standard position defines an angle with its vertex at the origin and initial side along the positive x-axis. Quandrantal, acute, and obtuse angles are defined based on where the terminal side falls. Coterminal angles share the same terminal side and differ by multiples of 360 degrees. The reference angle is the acute angle formed by the terminal side and closest x-axis. The unit circle is a circle with radius 1 centered at the origin, where sine and cosine of an
Calc 1.5a
The document discusses infinite limits and determining vertical asymptotes from rational functions. It provides examples of limits that approach positive or negative infinity as the input increases or decreases without bound. It also gives examples of determining the vertical asymptote(s) of rational functions by finding the values that make the denominator equal to zero. Students are provided with practice problems finding vertical asymptotes of rational functions and evaluating limits that may approach positive or negative infinity.
Materiales y sostenibilidad
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Day 2 examples u4f13
This document discusses two systems for measuring angles - one with the unit symbol ° and one without a unit symbol. It defines a central angle as the angle subtended or covered by an arc length given a radius. The document also provides the conversion between radians and degrees, showing that 1 radian is equal to 180/π degrees or approximately 57.3 degrees. It gives examples for converting between radians and degrees.
Combs perms review pascals
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive function. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms for those who already suffer from conditions like anxiety and depression.
Day 8 examples u7w14
This document contains 5 problems involving finding terms in expansions of polynomials. The problems involve finding specific terms that contain a given power of x in expansions of polynomials such as (3x^4 - 1)^9, (-x^3 + 2)^6, (x + 1)^3x, (x + 1)^x, and determining the value of m if one term in the expansion of (2x - m)^7 is -262500x^2y^5.
Day 7 examples u7w14
The document discusses binomial expansion, which is the process of multiplying out terms with two variables according to their power using the binomial theorem. It provides examples of expanding binomial expressions like (x + y)2, (x + y)3, and (x + y)4. It also notes that the sum of the exponents in each term equals the overall power, and the number of terms is always one more than the power. Finally, it provides the binomial theorem for expanding any binomial expression and finding a particular term.
Day 4 examples u7w14
This document discusses how to calculate arrangements when some items must be together or apart. It explains that when items need to be together, they should be counted as a single item to reduce the total items being arranged. Then the total number of arrangements is calculated by finding the total possible arrangements and subtracting the arrangements that do not satisfy the constraints of certain items being together or apart. Examples provided include arranging people in a row when some must or cannot sit together and arranging books on a shelf keeping books of each subject together.
Day 3 examples u7w14
Permutations refer to arrangements of objects in a definite order. Some key points:
- Permutations are represented by "nPn" where n is the total number of objects and r is the number being arranged.
- Permutations are used to calculate possibilities like license plates, phone numbers, and locker combinations.
- Restrictions like starting/ending conditions or requiring alternating arrangements reduce the number of possible permutations.
- Objects that are identical only count once toward the total number of permutations rather than being distinguishable.
Day 2 examples u7w14
Factorial notation represents the product of all positive integers less than or equal to the given number. For example, 5! = 5 x 4 x 3 x 2 x 1 and 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1. The document also provides examples of simplifying factorials without a calculator by using properties such as 5! + 4! = 6 x 4! and (k + 1)! + k! = (k + 2)k!.
Day 1 examples u7w14
This document provides examples and explanations of the fundamental counting principle and addition counting principle to solve combinatorics problems. It gives 8 examples of using the fundamental counting principle to count the number of possible outcomes of independent events. These include counting the number of volleyball shoe combinations, outfits that can be created from different clothing items, ways to select committees from groups of people, and 3-digit numbers with no repeating digits. It also provides 5 examples of using the addition counting principle to count outcomes when events are dependent, such as selecting a president and vice president of opposite sexes from a group.
Day 7 examples u6w14
This document discusses graphing composite functions. It provides examples of determining the composite functions f(g(x)) and g(f(x)) for various functions f(x) and g(x), sketching the graphs of the composite functions, and stating their domains. It also gives examples of determining possible functions f(x) and g(x) that satisfy given composite functions.
Day 5 examples u6w14
1. The document discusses composite functions, which involve combining two functions f(x) and g(x) where the output of one is used as the input of the other. It provides examples of evaluating composite functions using tables and graphs.
2. Key steps for evaluating composite functions are: 1) Substitute the inner function into the outer function and 2) Simplify the expression. Order matters as f(g(x)) and g(f(x)) may have different values.
3. Examples are worked through to find composite functions given basic functions like f(x) = x + 1 and g(x) = 2x as well as more complex rational functions.
Day 4 examples u6w14
The graph is a linear function with a domain of all real numbers and a range of real numbers greater than or equal to 3. The graph is a line with a y-intercept of 3 that increases at a rate of 1 as x increases.
Day 3 examples u6w14
Rational functions are functions of the form f(x) = p(x)/q(x) where p(x) and q(x) are polynomials. For example, comparing rational functions like 2x/(x^2 - 4) and (x-1)/(x+1). Horizontal asymptotes of rational functions occur when the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator.
Day 2 examples u6w14
This document discusses combining functions by graphing. When two functions f(x) and g(x) are combined, their graphs are overlayed on the same coordinate plane. The result is a new combined function where the output is determined by applying both functions f(x) and g(x) to the same input x.
Day 1 examples u6w14
This document discusses how to find the sum, difference, product, and quotient of functions. The sum of functions is found by adding the y-coordinates of each function. The difference is found by subtracting the y-coordinates. The product is represented as h(x) = f(x)g(x) and the quotient is represented as h(x) = f(x)/g(x). Examples are provided for adding and subtracting functions.
Mental math test outline
The document outlines a mental math test covering polynomials. It includes short answer questions testing long division, synthetic division, the remainder theorem, and finding the degree, leading coefficient, and y-intercept of polynomials. The test also covers matching graphs to polynomial equations and word problems involving fully factoring polynomials and two graphs. Multiple choice questions will require explaining solutions, while long answer questions involve fully factoring polynomials and word problems.
Day 8 examples u5w14
The document contains two polynomial word problems. The first asks to write a function V(x) to express the volume of a box with dimensions x, x+2, x+10 in terms of x, and find possible x values if the volume is 96 cm^3. The second problem describes a block of ice that is initially 3 ft by 4 ft by 5 ft, and asks to write a function to model reducing each dimension by the same amount to reach a volume of 24 ft^3, and determine how much to remove from each dimension.
Day 7 examples u5w14
The document provides 3 polynomial word problems: 1) finding the equation for a polynomial given its graph f(x) = -(x - 2)2(x + 1), 2) determining the polynomial P(x) when divided by (x - 3) with a quotient of 2x^2 + x - 6 and remainder of 4, and 3) finding the value of a if (x - 2) is a factor of ax^3 + 4x^2 + x - 2. It also gives a 4th problem of determining the value of k when 2x^3 + kx^2 - 3x + 2 is divided by x - 2 with a remainder of 4.
Day 5 examples u5w14
Polynomial functions are described by their degree and have certain characteristics. The graph of a polynomial is smooth and continuous without sharp corners. Odd degree polynomials rise on the left and fall on the right, while even degree polynomials rise on both sides. The number of x-intercepts and local maxima/minima are limited by the degree. Polynomials can be matched based on their degree, leading coefficient, even/odd nature, and number of x-intercepts and local extrema. The x-intercepts of a polynomial correspond to the roots of the equation, and a repeated root indicates a zero of higher multiplicity which affects the graph.
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Combs perms review pascals
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive function. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms for those who already suffer from conditions like anxiety and depression.
Day 8 examples u7w14
This document contains 5 problems involving finding terms in expansions of polynomials. The problems involve finding specific terms that contain a given power of x in expansions of polynomials such as (3x^4 - 1)^9, (-x^3 + 2)^6, (x + 1)^3x, (x + 1)^x, and determining the value of m if one term in the expansion of (2x - m)^7 is -262500x^2y^5.
Day 7 examples u7w14
The document discusses binomial expansion, which is the process of multiplying out terms with two variables according to their power using the binomial theorem. It provides examples of expanding binomial expressions like (x + y)2, (x + y)3, and (x + y)4. It also notes that the sum of the exponents in each term equals the overall power, and the number of terms is always one more than the power. Finally, it provides the binomial theorem for expanding any binomial expression and finding a particular term.
Day 4 examples u7w14
This document discusses how to calculate arrangements when some items must be together or apart. It explains that when items need to be together, they should be counted as a single item to reduce the total items being arranged. Then the total number of arrangements is calculated by finding the total possible arrangements and subtracting the arrangements that do not satisfy the constraints of certain items being together or apart. Examples provided include arranging people in a row when some must or cannot sit together and arranging books on a shelf keeping books of each subject together.
Day 3 examples u7w14
Permutations refer to arrangements of objects in a definite order. Some key points:
- Permutations are represented by "nPn" where n is the total number of objects and r is the number being arranged.
- Permutations are used to calculate possibilities like license plates, phone numbers, and locker combinations.
- Restrictions like starting/ending conditions or requiring alternating arrangements reduce the number of possible permutations.
- Objects that are identical only count once toward the total number of permutations rather than being distinguishable.
Day 2 examples u7w14
Factorial notation represents the product of all positive integers less than or equal to the given number. For example, 5! = 5 x 4 x 3 x 2 x 1 and 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1. The document also provides examples of simplifying factorials without a calculator by using properties such as 5! + 4! = 6 x 4! and (k + 1)! + k! = (k + 2)k!.
Day 1 examples u7w14
This document provides examples and explanations of the fundamental counting principle and addition counting principle to solve combinatorics problems. It gives 8 examples of using the fundamental counting principle to count the number of possible outcomes of independent events. These include counting the number of volleyball shoe combinations, outfits that can be created from different clothing items, ways to select committees from groups of people, and 3-digit numbers with no repeating digits. It also provides 5 examples of using the addition counting principle to count outcomes when events are dependent, such as selecting a president and vice president of opposite sexes from a group.
Day 7 examples u6w14
This document discusses graphing composite functions. It provides examples of determining the composite functions f(g(x)) and g(f(x)) for various functions f(x) and g(x), sketching the graphs of the composite functions, and stating their domains. It also gives examples of determining possible functions f(x) and g(x) that satisfy given composite functions.
Day 5 examples u6w14
1. The document discusses composite functions, which involve combining two functions f(x) and g(x) where the output of one is used as the input of the other. It provides examples of evaluating composite functions using tables and graphs.
2. Key steps for evaluating composite functions are: 1) Substitute the inner function into the outer function and 2) Simplify the expression. Order matters as f(g(x)) and g(f(x)) may have different values.
3. Examples are worked through to find composite functions given basic functions like f(x) = x + 1 and g(x) = 2x as well as more complex rational functions.
Day 4 examples u6w14
The graph is a linear function with a domain of all real numbers and a range of real numbers greater than or equal to 3. The graph is a line with a y-intercept of 3 that increases at a rate of 1 as x increases.
Day 3 examples u6w14
Rational functions are functions of the form f(x) = p(x)/q(x) where p(x) and q(x) are polynomials. For example, comparing rational functions like 2x/(x^2 - 4) and (x-1)/(x+1). Horizontal asymptotes of rational functions occur when the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator.
Day 2 examples u6w14
This document discusses combining functions by graphing. When two functions f(x) and g(x) are combined, their graphs are overlayed on the same coordinate plane. The result is a new combined function where the output is determined by applying both functions f(x) and g(x) to the same input x.
Day 1 examples u6w14
This document discusses how to find the sum, difference, product, and quotient of functions. The sum of functions is found by adding the y-coordinates of each function. The difference is found by subtracting the y-coordinates. The product is represented as h(x) = f(x)g(x) and the quotient is represented as h(x) = f(x)/g(x). Examples are provided for adding and subtracting functions.
Mental math test outline
The document outlines a mental math test covering polynomials. It includes short answer questions testing long division, synthetic division, the remainder theorem, and finding the degree, leading coefficient, and y-intercept of polynomials. The test also covers matching graphs to polynomial equations and word problems involving fully factoring polynomials and two graphs. Multiple choice questions will require explaining solutions, while long answer questions involve fully factoring polynomials and word problems.
Day 8 examples u5w14
The document contains two polynomial word problems. The first asks to write a function V(x) to express the volume of a box with dimensions x, x+2, x+10 in terms of x, and find possible x values if the volume is 96 cm^3. The second problem describes a block of ice that is initially 3 ft by 4 ft by 5 ft, and asks to write a function to model reducing each dimension by the same amount to reach a volume of 24 ft^3, and determine how much to remove from each dimension.
Day 7 examples u5w14
The document provides 3 polynomial word problems: 1) finding the equation for a polynomial given its graph f(x) = -(x - 2)2(x + 1), 2) determining the polynomial P(x) when divided by (x - 3) with a quotient of 2x^2 + x - 6 and remainder of 4, and 3) finding the value of a if (x - 2) is a factor of ax^3 + 4x^2 + x - 2. It also gives a 4th problem of determining the value of k when 2x^3 + kx^2 - 3x + 2 is divided by x - 2 with a remainder of 4.
Day 5 examples u5w14
Polynomial functions are described by their degree and have certain characteristics. The graph of a polynomial is smooth and continuous without sharp corners. Odd degree polynomials rise on the left and fall on the right, while even degree polynomials rise on both sides. The number of x-intercepts and local maxima/minima are limited by the degree. Polynomials can be matched based on their degree, leading coefficient, even/odd nature, and number of x-intercepts and local extrema. The x-intercepts of a polynomial correspond to the roots of the equation, and a repeated root indicates a zero of higher multiplicity which affects the graph.
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Generative AI Deep Dive: Advancing from Proof of Concept to Production
Presentation of the OECD Artificial Intelligence Review of Germany
Presentation of the OECD Artificial Intelligence Review of Germany
Unlock the Future of Search with MongoDB Atlas_ Vector Search Unleashed.pdf
Unlock the Future of Search with MongoDB Atlas_ Vector Search Unleashed.pdf
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024
Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdf
Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdf
UiPath Test Automation using UiPath Test Suite series, part 5
UiPath Test Automation using UiPath Test Suite series, part 5
Pushing the limits of ePRTC: 100ns holdover for 100 days
Pushing the limits of ePRTC: 100ns holdover for 100 days
“Building and Scaling AI Applications with the Nx AI Manager,” a Presentation...
“Building and Scaling AI Applications with the Nx AI Manager,” a Presentation...
Day 1 examples u4f13
- 1. Unit Circle / Circular Functions terminal arm/ side initial arm/ side Standard Position of Initial arm lays on the x-axis the origin
- 2. e) = 2 f) = -3.5
- 3. Acute angle formed between the terminal arm of an angle and the nearest x-axis. Reference Angle
- 4. Coterminal Angles angles which share the same terminal side/ arm ex. Give two coterminal angles for 30 There are infinitely many coterminal angles. ex. Determine the measures of angles that are coterminal with 85o for -400o≤ ≤400o