2 interactive labs · AS & A Level Mathematics 9709
Locating a Root by a Change of Sign
3.6 Numerical solution of equations (also 2.6) · Search consecutive integers, tenths and hundredths for a change of sign, predict what the test proves, and meet the cases where it fails.
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About this lab
Search consecutive integers, tenths and hundredths for a change of sign, predict what the test proves, and meet the cases where it fails.
You will be able to
Locate a root of f(x) = 0 between consecutive integers by a change of sign, and write the justification a mark scheme awards.
Show that a root is correct to 1 or 2 decimal places by a change of sign between x − 0.05 and x + 0.05, or x − 0.005 and x + 0.005.
Explain why a change of sign across an asymptote is not a root, and why no change of sign does not rule out a root.
Relate the roots of f(x) = 0 to the intersections of y = g(x) and y = h(x) when the equation is written as g(x) = h(x).
Iterating to Find a Root
3.6 Numerical solution of equations (also 2.6) · Rearrange an equation to x = F(x), predict what the iteration does, step it on a cobweb and a table, then prove the accuracy by a change of sign.
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About this lab
Rearrange an equation to x = F(x), predict what the iteration does, step it on a cobweb and a table, then prove the accuracy by a change of sign.
You will be able to
Explain how a rearrangement x = F(x) relates to the equation being solved, and use the notation x₁, x₂, … for the iterates.
Use a given iteration xₙ₊₁ = F(xₙ) to find a root to a stated number of decimal places, carrying unrounded values.
Justify the accuracy of a root by a change of sign of f at x − 0.0005 and x + 0.0005 (3 d.p.), not by two iterates agreeing.
Recognise from the iterates that an iteration may fail to converge, or may converge to a different root from the one wanted.
