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Published byGyles Webb Modified over 8 years ago
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Without using gsolv, equation solver, etc. Only calculator operations: x,+,-,^.
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-1.6 0.6 1.3
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Solve sinx = 1 – x
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x-5-4-3-20123456 y x-5-4-3-20123456 y Table Mode
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To “show that” 2 given x-values lead to a f(x) sign change between them Apply the sign change rule to approximate the roots of f(x)=0 Apply the sign change rule to approximate the roots of f(x)=g(x) Rearrange an equation to make x the subject Topic ObjectivesSkills Practice Clear explanation of a “show that” question Calculator skills (evaluating functions) Checking skills Algebra (rearrange an equation) Discussion/teamwork
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By now we know that it is possible to find the solutions to equations such as... sinx = log(x) by looking at the graphs y=sinx and y=log(x) and seeing where they cross. Equations and Graphs
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Puzzle Triangle In table groups Equation solver and G-Solv
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Notice that where the graphs cross the upper graph before the root (solution) becomes the lower afterward (and vice versa for the lower). Change in Biggest
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Let us call the two functions being equated f(x) and g(x)... The equation to solve is f(x) = g(x) If we input two values for x into f(x) and g(x) and which is the bigger function changes for the different x values there is a solution to the equation between those values. Numerical Test
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Instead of working with f(x)=g(x) we could work with f(x)- g(x) = 0 the solutions of the equation are now where y=0 We can now identify the existence of a root between two x values if the result of this difference for the x values gives different signs (+ and -). A simpler way to define this... Sign Change Rule
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1.Change f(x)=g(x) into f(x)-g(x)=0 (if needed) 2.Substitute lower bound (given) value for x 3.Substitute upper bound (given) value for x 4.There should be a sign change (if not something must have gone wrong; they won’t tell you numbers should indicate a root if they don’t) 5.Make reference to the fact this sign change indicates the presence of a root between the x values. Steps in the sign test
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Which of these equations have a root between x=3 and x=4? A: log 10 x = 4 – x B: sinx = 3 - 2x C: 2 x = 5x-3 D: cosx = 6 – 2x Answers on whiteboards =0x = 3x = 4Root? logx + x – 4 -0.522880.60206 Yes sinx + 2x – 3 3.141124.243198 No 2 x – 5x + 3 -4 No cosx + 2x - 6 -0.989991.346356 Yes
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Which of these equations have a root between x=1 and x=2? A: sinx = 3 – x B: log(2x) + 1 = x C: 3 x – 2 = 0 D: cos2x = 1 – x Answers on whiteboards =0x = 1x = 2Root? sinx + x – 3 -1.15853-0.0907 No log(2x) + 1 – x 0.30103-0.39794 Yes 3 x – 2 17 No cos2x - 1 + x -0.416150.346356 Yes
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Which of these equations have a root between x=1 and x=2? A: x 2 = 5cosx B: 2 x = 4sinx C: 1 / cosx = 4cosx D: 3log(5sinx) = 4 – 2x Answers on whiteboards =0x = 1x = 2Root? x 2 - 5cosx -1.701516.080734 Yes 2 x - 4sinx -1.365880.36281 Yes 1 / cosx - 4cosx -0.31039-0.73841 YES!!! 3log(5sinx) - 4 + 2x -0.127971.973028 Yes
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There are some times when the sign change occurs but there is no root. 1) The graphs must be continuous (no breaks) in the region. E.g. Between x=-1 and x=1 Limitations
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There are some times when a root exists but there is no sign change… 2) There may be more than one root in the region E.gBetween x=0 and x=1 Limitations
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There are some times when a root exists but there is no sign change… 3) There may be a repeated root in the region E.gBetween x=1 and x=2 Limitations
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Sketch y=sinx and y= 1 – x on the same axes. Using the sketch explain why there is just one root for the equation sinx = 1 - x The root of the equation is show that lies between 0 and 1. Final Check
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Show that f(x)=0 has a solution in the interval 0<x<1 Show that the larger root, p, is such that 4<p<5 Sign Change Rule
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Rearranging to make x the subject
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