4.1.08 Pascals Triangle2
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Pascal's Triangle is an array of numbers arranged in triangular form where each number is derived from adding the two numbers directly above it. To generate the triangle, a chart is made where the terms are found by taking the binomial coefficients of n over r, and properties include that each row has one more term than the previous row and the first and last terms of each row are always 1.
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Variance and standard deviation are measures of spread used to describe the shape of distributions associated with the mean. While two distributions may have the same mean, their variance and standard deviation can show that they have very different shapes. To calculate variance and standard deviation, you first find the deviation of each value from the mean, square the deviations, and sum them. You then divide the sum by n-1 to get the variance, and take the square root of the variance to find the standard deviation.
Section 1-5
The document discusses the concept of limits in calculus, including the formal definition of a limit as the value a function approaches as the input gets arbitrarily close to a given number. It provides examples of evaluating limits graphically and algebraically, such as determining the limit of a function as x approaches 3 is 6, even though the function is undefined at x=3 due to a removable discontinuity. The document also discusses using graphs to identify different types of discontinuities and limits.
Section 1-4 -- Trapezoid Rule
The document discusses using the trapezoidal rule to evaluate definite integrals by approximating the area under a curve between bounds with trapezoids. It explains that the trapezoidal rule is a more exact method than counting squares. To use the rule, the integral bounds are divided into trapezoids, the function is evaluated at the endpoints, and the areas are calculated by multiplying the average base by the height and summing the results, with endpoints weighted by one half.
Lesson 1-4 -- Five Number Summary
The document discusses the five number summary, which includes the minimum, first quartile, median, third quartile, and maximum values. It provides an example of the heights of a basketball team, calculates their five number summary and interquartile range, and discusses how outliers can be identified using 1.5 times the interquartile range.
Section 1-3 -- Definite Integrals
The document discusses using integrals to calculate distance traveled based on a velocity-time graph. It explains that the area under the graph between two times represents the distance traveled during that interval. Specifically, it asks the reader to:
1) Find the velocity between 30-50 seconds from the given graph and explain why the distance can be found from the rectangular area.
2) Count the squares and find the distance traveled from 0-20 seconds by finding the area under the curved graph.
3) Estimate the definite integral of a function representing a football's cross-sectional area to find its total volume.
Lesson 1-3
The document discusses different measures of central tendency, including the mean and median. It introduces the concept of using sigma notation to simplify writing out long sums, such as when calculating the mean of a data set. It provides an example of entree prices from a restaurant and encourages writing out the formula to calculate the mean using sigma notation.
Lesson 1 2
A stemplot is a method to display data by splitting each number into a stem and leaf, with the ones digit as the leaf and other digits as the stem. It allows comparison of two data sets side by side. A dotplot displays the frequency of data values and is useful when the data is categorical. It may not be suitable to display continuous numeric data like golf scores.
5.12.08 Advanced Factoring1
Factoring by grouping allows factoring out the greatest common factor from some but not all terms of a polynomial, which can make the entire factorization easier. This technique can be used to factor difficult quadratic trinomials. Examples are provided to illustrate factoring by grouping.
5.6.08 Fundamental Theorem Of Algebra1
The fundamental theorem of algebra states that any polynomial of degree greater than 1 with complex coefficients has at least one complex zero. A related theorem is that a polynomial of degree n has at most n zeros. Another theorem is that a polynomial of degree greater than 1 with complex coefficients has exactly n complex zeros counting multiplicities.
4.14.08 Finding Polynomial Models1 2
The document discusses finding polynomial models to describe patterns like triangular numbers and tetrahedral numbers. It explains that a polynomial difference theorem can be used to find the polynomial formula for functions where the differences between successive values form a known pattern. Specifically, if the fourth differences are constant, the function is a fourth degree polynomial that can be determined by plugging known points into the general cubic polynomial formula.
4.9.08 Polynomial Models3
The document discusses polynomial models and volume equations. It provides an example of a box made from cardboard where squares are cut from the corners and the sides are folded up. An equation is written for the volume of the box in terms of x, which is a third degree polynomial. A polynomial's degree is the maximum number of variables being multiplied. The standard form of a polynomial of degree n is also discussed. The volume of the cardboard box is written in standard form. A polynomial with more than one variable, like (x+y)^4, has a degree of 4.
4.10.08 Graphs Of Polynomials
This document discusses key concepts related to graphs of polynomial functions, including:
- Extrema (maximum, minimum, relative maximum, relative minimum points) and where they occur on the sample graph
- Zeros (x-intercepts) and that the sample function has four zeros
- Increasing and decreasing intervals based on the slope, with the function increasing where driving uphill and decreasing where driving downhill
- Positive and negative intervals, referring to where the function is positive or negative based on the y-value.
4.3.08 Binomial Prob 1
The document discusses binomial probabilities and permutations for repeated independent trials with two possible outcomes. It asks what the probability is of getting 4 or more blues when spinning a spinner 6 times, and the probability of getting 3 or more reds in 6 spins. It introduces the binomial probability theorem for calculating these probabilities.
4.1.08 Pascals Triangle2
Pascal's Triangle is an array of numbers arranged in triangular form where each number is derived from adding the two numbers directly above it. To generate the triangle, a chart is made where the terms are found by taking the binomial coefficients of n over r, and properties include that each row has one more term than the previous row and the first and last terms of each row are always 1.
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- 1. Pascal’s Triangle An array of numbers derived from many different combinations . To generate this array, we make a chart: 6 5 4 3 2 1 0 6 5 4 3 2 1 0 n r
- 2. Pascal’s Triangle An array of numbers derived from many different combinations . To generate this array, we make a chart: 6 5 4 3 2 1 0 6 5 4 3 2 1 0 n r
- 3. Pascal’s Triangle We more commonly see the triangle written in its isosceles form: To find the r th term of the n th row, what would you do? . In other words, gives you the (r +1) st term in the n th row.
- 4. Some Properties of Pascal’s Triangle 1. 2. 3. 4. 5.