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Pre-Calc 30S May 7
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.
Day 3 examples
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.
Day 1 examples
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.
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.
Recommended
Pre-Calc 30S May 7
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.
Day 3 examples
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.
Day 1 examples
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.
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 4 examples
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.
Stress and breast feeding dr. shriniwas kashalikar
This document discusses merits that stress management techniques should possess. It states that stress management should be possible and feasible for most people, cause no financial loss or undesirable side effects, not be painful or a nuisance to others, involve intellectual surrender but not dependence, and be natural and conducive to growth. It provides the analogy that good stress management is like breastfeeding, which nourishes the infant for life without the infant consciously realizing it.
Love beyond success dr. shriniwas janardan kashalikar
The document discusses the state of despair that individuals may experience during spiritual evolution when they have moved past material pleasures but not yet experienced divine love. A person pursuing higher purposes often feels restless from a lack of fulfillment in material success or universal love. This leads to loneliness as others remain engrossed in worldly pursuits. With persistence in spiritual practice like Namasmaran, one can gain a revelation of the ever-present support of divine love, experiencing all people as loving. This love beyond success is a profound and precious feeling.
Stress god and we dr. shriniwas kashalikar
The document discusses the misinterpretation and misunderstanding of concepts like "I am Brahma" and God. It states that misunderstanding these concepts can lead to inaction, helplessness, and false beliefs. It argues that God is an internal experience of self-realization, not something to be believed in or have paranoid beliefs about. The document says that practicing NAMASMARAN, or remembrance of the divine name, can help people become self-sufficient by connecting to their true self and gaining a holistic perspective. NAMASMARAN also frees people from rigid beliefs by appreciating that needs vary between individuals and societies depending on life stage and circumstances.
G R O W T H & S U P E R L I V I N G D R
The document discusses two main points. First, people tend to see problems everywhere and come up with solutions, believing their solutions will create lasting change and harmony, but they fail to address the root issues. Second, people tend to exclude themselves and demand change from others while ignoring their own flaws. However, the author argues that transcending the subjective desire for perfection through meditation and focusing on our shared humanity can help us realize and manifest the potential for freedom that already exists. Practicing meditation collectively could help realize this freedom for all.
Day 5 examples
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 8 examples
The document contains three word problems involving related rates of change: 1) finding the rate of change of radius of a spherical balloon filling with gas, when the diameter is 18 inches. 2) finding the rate of change of circumference of circular waves expanding at 0.5 m/sec, when the radius is 4 meters. 3) involving a girl walking away from a pole and rates of change of the tip of her shadow and length of the shadow.
Day 5 examples
This document discusses graphing polynomials. Polynomials with zeros of even multiplicity will have a horizontal tangent line at that zero, while polynomials with zeros of odd multiplicity will cross the x-axis at that zero. For example, a polynomial with a zero of multiplicity 2 at x=1 will have a horizontal tangent line there, while a polynomial with a zero of multiplicity 1 at x=-2 will cross the x-axis at -2.
Stress and selfishness dr. shriniwas kashalikar
Stress arises from selfish desires and cravings that are never fully satisfied. As humans, we are able to make choices but this also leads to conflicts and dilemmas that produce stress. One way to help transition from selfishness to selflessness is through the regular practice and promotion of Namasmaran, chanting the name of the divine. By focusing on Namasmaran, which is not as strong a drive as physiological needs and desires, it can help burn away selfishness and allow selflessness and cosmic consciousness to grow in one's heart and in society.
Day 8 examples
This document discusses trigonometric functions such as sine, cosine, and tangent. It defines periodic functions as having repeating patterns, with the period being the length of one full pattern. For sine waves, the period is 2. The amplitude is the maximum vertical deviation from the central axis. Graphs of sine, cosine, and tangent are examined over various domains and their characteristics such as zeroes and asymptotes are explored.
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.
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!.
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Day 4 examples
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.
Stress and breast feeding dr. shriniwas kashalikar
This document discusses merits that stress management techniques should possess. It states that stress management should be possible and feasible for most people, cause no financial loss or undesirable side effects, not be painful or a nuisance to others, involve intellectual surrender but not dependence, and be natural and conducive to growth. It provides the analogy that good stress management is like breastfeeding, which nourishes the infant for life without the infant consciously realizing it.
Love beyond success dr. shriniwas janardan kashalikar
The document discusses the state of despair that individuals may experience during spiritual evolution when they have moved past material pleasures but not yet experienced divine love. A person pursuing higher purposes often feels restless from a lack of fulfillment in material success or universal love. This leads to loneliness as others remain engrossed in worldly pursuits. With persistence in spiritual practice like Namasmaran, one can gain a revelation of the ever-present support of divine love, experiencing all people as loving. This love beyond success is a profound and precious feeling.
Stress god and we dr. shriniwas kashalikar
The document discusses the misinterpretation and misunderstanding of concepts like "I am Brahma" and God. It states that misunderstanding these concepts can lead to inaction, helplessness, and false beliefs. It argues that God is an internal experience of self-realization, not something to be believed in or have paranoid beliefs about. The document says that practicing NAMASMARAN, or remembrance of the divine name, can help people become self-sufficient by connecting to their true self and gaining a holistic perspective. NAMASMARAN also frees people from rigid beliefs by appreciating that needs vary between individuals and societies depending on life stage and circumstances.
G R O W T H & S U P E R L I V I N G D R
The document discusses two main points. First, people tend to see problems everywhere and come up with solutions, believing their solutions will create lasting change and harmony, but they fail to address the root issues. Second, people tend to exclude themselves and demand change from others while ignoring their own flaws. However, the author argues that transcending the subjective desire for perfection through meditation and focusing on our shared humanity can help us realize and manifest the potential for freedom that already exists. Practicing meditation collectively could help realize this freedom for all.
Day 5 examples
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 8 examples
The document contains three word problems involving related rates of change: 1) finding the rate of change of radius of a spherical balloon filling with gas, when the diameter is 18 inches. 2) finding the rate of change of circumference of circular waves expanding at 0.5 m/sec, when the radius is 4 meters. 3) involving a girl walking away from a pole and rates of change of the tip of her shadow and length of the shadow.
Day 5 examples
This document discusses graphing polynomials. Polynomials with zeros of even multiplicity will have a horizontal tangent line at that zero, while polynomials with zeros of odd multiplicity will cross the x-axis at that zero. For example, a polynomial with a zero of multiplicity 2 at x=1 will have a horizontal tangent line there, while a polynomial with a zero of multiplicity 1 at x=-2 will cross the x-axis at -2.
Stress and selfishness dr. shriniwas kashalikar
Stress arises from selfish desires and cravings that are never fully satisfied. As humans, we are able to make choices but this also leads to conflicts and dilemmas that produce stress. One way to help transition from selfishness to selflessness is through the regular practice and promotion of Namasmaran, chanting the name of the divine. By focusing on Namasmaran, which is not as strong a drive as physiological needs and desires, it can help burn away selfishness and allow selflessness and cosmic consciousness to grow in one's heart and in society.
Day 8 examples
This document discusses trigonometric functions such as sine, cosine, and tangent. It defines periodic functions as having repeating patterns, with the period being the length of one full pattern. For sine waves, the period is 2. The amplitude is the maximum vertical deviation from the central axis. Graphs of sine, cosine, and tangent are examined over various domains and their characteristics such as zeroes and asymptotes are explored.
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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Stress and breast feeding dr. shriniwas kashalikar
Stress and breast feeding dr. shriniwas kashalikar
Love beyond success dr. shriniwas janardan kashalikar
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Sahastranetra A Bestseller On Vishnusahasranam Dr Shriniwas Kashalikar
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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.
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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.
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.
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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.
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.
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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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Pre-Calc 30S March 6
- 1. -6x