mathematics at a deeper level. Review of homogeneous equations The homogeneous constant coefficient linear equation an y (n) +· · ·+a1 y +a0 y = 0 has the characteristic polynomial an rn +· · ·+a1 r+a0 = 0. From the roots r1 ‚ . . . ‚ rn of the polynomial we can construct the solutions y1 ‚ . . . ‚ yn ‚ such as y1 = er1 x . We can also rewrite the equation in a weird-looking but useful way‚ using the symbol d D = dx . Examples: equation: y − 5y + 6y = 0. polynomial: r2 − 5r + 6 = 0. (factored):
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Diagonally Implicit Block Backward Differentiation Formulas for Solving Ordinary Differential Equations 1.0 Introduction In mathematics‚ if y is a function of x‚ then an equation that involves x‚ y and one or more derivatives of y with respect to x is called an ordinary differential equation (ODE). The ODEs which do not have additive solutions are non-linear‚ and finding the solutions is much more sophisticated because it is rarely possible to represent them by elementary function in close
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from now" is six times Miguel’s "age last year" or‚ in math: g + 3 = 6(m – 1) This gives me two equations with two variables: m + g = 68 g + 3 = 6(m – 1) Solving the first equation‚ I get m = 68 – g. (Note: It’s okay to solve for "g = 68 – m"‚ too. The problem will work out a bit differently in the middle‚ but the answer will be the same at the end.) I’ll plug "68 – g" into the second equation in place of "m": g + 3 = 6m – 6 g + 3 = 6(68 – g) – 6 g + 3 = 408 – 6g – 6 g + 3 = 402 – 6g
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Section 2.3 Linear Functions and Slopes 1 Section 2.3 Linear Functions and Slopes The Slope of a Line 2 Section 2.3 Linear Functions and Slopes Find the slope of the line that passes through (-2‚5) and (3‚-1) change in y 5 1 6 6 m or change in x 2 3 5 5 3 Section 2.3 Linear Functions and Slopes 4 Section 2.3 Linear Functions and Slopes Example Find the slope of the line passing through the pair of points (5‚-2) and (-1‚7). 5 Section 2.3 Linear Functions and Slopes First:
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_________________ Teacher: _______________ Reviewer: Quadratic Equations I. Multiple Choice: Choose the letter of the correct answer. Show your solution. 1. What are the values of x that satisfy the equation 3 – 27x2 = 0? A. x = [pic]3 B. x = [pic] C. x = [pic] D. x = [pic] 2. What are the solutions of the equation 6x2 + 9x – 15 = 0? A. 1‚ - 15 B. 1‚ [pic] C. – 1‚ - 5 D. 3‚ [pic] 3. For which equation is – 3 NOT a solution? A. x2 – 2x – 15 = 0 C. 2x2
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completed worksheet. Question 1: What trend does the graph illustrate? (2) 4. After the graph is complete‚ right-click on one of the data points on the graph. Select ‘Add trendline” 5. Choose “Linear” for the trend/regression type. Then at the bottom of the window‚ select “display equation on chart” and “Display R-squared value on chart”. Question 2:
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–––– Run ∆y y 2 - y1 ––– = ––––––– ∆x x 2 - x1 1- Solve for y to put the equation in slope intercept form. 2- Plot the y-intercept. 3- Using the slope as a fraction‚ rise y and run x to get second point. 4- Graph the line. Ex: 2x+3y=12 -2x -2x ––––––––– 3y=-2x+12 –– –––– 3 3 y= -2/3x+4 m= -2/3 b= 4 Horizontal and Vertical Lines: - A Horizontal Line has the form y=#. (In an equation of a horizontal line‚ there is no x) - The slope of a horizontal line is 0. Picture:
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Earliest Methods used to solve Quadratic Equation 1. Babylonian mathematics (also known as Assyro-Babylonian mathematics) was any mathematics developed or practiced by the people of Mesopotamia‚ from the days of the early Sumerians to the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited.[7] In respect of time they fall in two distinct groups: one from the Old Babylonian period (1830-1531 BC)‚ the other mainly Seleucid from the last three or four centuries BC
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Unit Test‚ Part 2: Linear Equations and Systems Answer the questions below. When you have finished‚ submit this assignment to your teacher by the due date for full credit. (10 points) 1. The table gives the population of a town for the years 2000–2009. a. Make a scatter plot of the data‚ draw a line of best fit‚ and then find the equation of the line of best fit. Show and explain your work. b. Describe what the slope of the line of best fit represents. c. Use the equation to predict the population
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Heat Equation from Partial Differential Equations An Introduction (Strauss) These notes were written based on a number of courses I taught over the years in the U.S.‚ Greece and the U.K. They form the core material for an undergraduate course on Markov chains in discrete time. There are‚ of course‚ dozens of good books on the topic. The only new thing here is that I give emphasis to probabilistic methods as soon as possible. Also‚ I introduce stationarity before even talking about state classification
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