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The document provides examples of permutations problems with restrictions. It gives examples of finding arrangements of letters with conditions such as starting or ending with vowels, using only consonants, and alternating consonants and vowels. It also gives examples of counting arrangements of people sitting in chairs with the condition they must alternate, and counting multi-digit numbers with conditions such as being greater than 300 or being even.
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New day 4 examples
The document provides examples of counting permutations with restrictions. It discusses counting arrangements that must start or end with certain letters or digits, or that require certain items to be together or separate. The examples cover topics like counting letter arrangements with consonants and vowels alternating, counting numbers that are even or divisible by 5, and counting seating arrangements that require some people or books to be together or separate.
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.
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This document provides a spelling lesson reviewing words containing the "oi" sound and the "er" sound spelled as "er", "ir", and "ur". It includes a spelling race with 10 words, examples of each spelling, sentences to correct misspellings, filling sentences with provided words, and a challenge to create new sentences using words from the lesson.
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This document discusses verifying trigonometric identities and determining non-permissible values for identities. It first verifies two identities by showing that = and = 30° are solutions. It then identifies the non-permissible values of θ for the identity tan θ = 1/cot θ on the interval from 0 to 2π as being θ = nπ where n is any integer.
Day 2 examples u5w14
Synthetic division is a method for dividing a polynomial by a linear factor such as (x - a). The coefficients of the polynomial are written in a column. The divisor is written above the column. Working from right to left, each term of the divisor is multiplied by the coefficient above and subtracted from the number above. The result is the quotient and remainder of dividing the polynomial by the linear factor.
Day 2 examples 2
Synthetic division is a method for dividing a polynomial by a linear factor using long division. It allows one to divide a polynomial by (x - a) without performing the long division process step-by-step. One writes the coefficients of the polynomial being divided in a vertical arrangement and performs successive subtractions and multiplications to obtain the coefficients of the quotient and remainder polynomials.
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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.
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The document discusses angles and their multiples in degrees and radians. It provides examples of angle measures and their trigonometric functions. Key points covered include:
1) Examples of multiples of common angle measures like 60, 45, 30 degrees.
2) Use of trigonometric functions like cosine and sine to find coordinates on the unit circle for angles such as 180 degrees.
3) Conclusions about the general form of coordinates as functions of an angle measure using cosine and sine.
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New day 4 examples
The document provides examples of counting permutations with restrictions. It discusses counting arrangements that must start or end with certain letters or digits, or that require certain items to be together or separate. The examples cover topics like counting letter arrangements with consonants and vowels alternating, counting numbers that are even or divisible by 5, and counting seating arrangements that require some people or books to be together or separate.
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.
Spelling and grammar 8
This document provides a spelling lesson reviewing words containing the "oi" sound and the "er" sound spelled as "er", "ir", and "ur". It includes a spelling race with 10 words, examples of each spelling, sentences to correct misspellings, filling sentences with provided words, and a challenge to create new sentences using words from the lesson.
Day 11 examples u4w14
This document discusses verifying trigonometric identities and determining non-permissible values for identities. It first verifies two identities by showing that = and = 30° are solutions. It then identifies the non-permissible values of θ for the identity tan θ = 1/cot θ on the interval from 0 to 2π as being θ = nπ where n is any integer.
Day 2 examples u5w14
Synthetic division is a method for dividing a polynomial by a linear factor such as (x - a). The coefficients of the polynomial are written in a column. The divisor is written above the column. Working from right to left, each term of the divisor is multiplied by the coefficient above and subtracted from the number above. The result is the quotient and remainder of dividing the polynomial by the linear factor.
Day 2 examples 2
Synthetic division is a method for dividing a polynomial by a linear factor using long division. It allows one to divide a polynomial by (x - a) without performing the long division process step-by-step. One writes the coefficients of the polynomial being divided in a vertical arrangement and performs successive subtractions and multiplications to obtain the coefficients of the quotient and remainder polynomials.
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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.
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The document discusses angles and their multiples in degrees and radians. It provides examples of angle measures and their trigonometric functions. Key points covered include:
1) Examples of multiples of common angle measures like 60, 45, 30 degrees.
2) Use of trigonometric functions like cosine and sine to find coordinates on the unit circle for angles such as 180 degrees.
3) Conclusions about the general form of coordinates as functions of an angle measure using cosine and sine.
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This document discusses two methods for finding the volume of solids of revolution: the disk method, which revolves a function around the x-axis, and the washer method, which revolves a function around a horizontal line. It provides an example problem for each method, asking to find the volume generated by revolving a function between given x-values around either the x-axis or a horizontal line y=-1.
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This document discusses the concept of the definite integral in calculating distance traveled given a velocity-time graph. It explains that the definite integral provides an exact value for distance by calculating the area under the velocity-time curve between two time points, while upper and lower estimates can be found by using the greatest or least values in places where the graph is ambiguous. Students are assigned problems from their textbook involving calculating distance using integrals or upper and lower estimates from velocity-time graphs.
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The document contains links to various photos hosted on Flickr, Fotopedia, and other photo sharing sites. The photos seem to depict a variety of subjects including landscapes, portraits, and everyday scenes. There is no other text providing additional context for the collection of random photo links.
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To find an approximate value for √37, the document notes that the function f(x)=√x is concave up for positive values of x. It then states that a local linearization technique can be used to find an approximation, which involves determining the tangent line for f(x) at x=36 to estimate the value of f(37).
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1) The document discusses logarithmic properties and how to solve logarithmic equations and expressions. It provides examples of evaluating logarithmic expressions using log properties and rewriting expressions as single logarithms.
2) Logarithmic properties and theorems covered include the log property that log(MN) = logM + logN, and rewriting expressions with more than one logarithm as a single logarithm.
3) Examples show how to use these properties to evaluate expressions such as log6 8 + log6 9 - log6 2 and rewrite expressions like (x - 2) - 3logx as a single logarithm.
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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.
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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.
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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.
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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!.
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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.
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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.
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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.
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This document discusses two methods for finding the volume of solids of revolution: the disk method, which revolves a function around the x-axis, and the washer method, which revolves a function around a horizontal line. It provides an example problem for each method, asking to find the volume generated by revolving a function between given x-values around either the x-axis or a horizontal line y=-1.
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This document discusses the concept of the definite integral in calculating distance traveled given a velocity-time graph. It explains that the definite integral provides an exact value for distance by calculating the area under the velocity-time curve between two time points, while upper and lower estimates can be found by using the greatest or least values in places where the graph is ambiguous. Students are assigned problems from their textbook involving calculating distance using integrals or upper and lower estimates from velocity-time graphs.
Mystory
The document contains links to various photos hosted on Flickr, Fotopedia, and other photo sharing sites. The photos seem to depict a variety of subjects including landscapes, portraits, and everyday scenes. There is no other text providing additional context for the collection of random photo links.
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To find an approximate value for √37, the document notes that the function f(x)=√x is concave up for positive values of x. It then states that a local linearization technique can be used to find an approximation, which involves determining the tangent line for f(x) at x=36 to estimate the value of f(37).
Day 3 examples
1) The document discusses logarithmic properties and how to solve logarithmic equations and expressions. It provides examples of evaluating logarithmic expressions using log properties and rewriting expressions as single logarithms.
2) Logarithmic properties and theorems covered include the log property that log(MN) = logM + logN, and rewriting expressions with more than one logarithm as a single logarithm.
3) Examples show how to use these properties to evaluate expressions such as log6 8 + log6 9 - log6 2 and rewrite expressions like (x - 2) - 3logx as a single logarithm.
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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.
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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.
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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!.
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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.
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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.
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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.
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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.
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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.
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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.
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Day 4 examples
- 1. Perms with Restrictions Note: Must ALWAYS fill restrictions first!! Ex) Using the first 10 letters of the alphabet, how many different arrangements can you make if you must (a) start with a vowel (b) start and end with a vowel (c) Not start or end with a vowel
- 2. Ex) Using all the letters of the word BRAINS (no reps), how many 4-letter arrangements are possible (a) ending in "R" (b) with consonants only (c) with consonants and vowels alternating
- 3. Perms with Cases Ex) If there are six chairs in a row, how many ways can 3 boys and 3 girls sit if they must alternate?
- 4. Ex) How many different 3-digit numbers greater than 300 can be made using the digits 1, 2, 3, 4, 5, and 6? Ex) How many different four-digit numbers can be made that are even using the digits 0, 1, 2, 3, 4, 5, 6 ?