Solving Equations (Edexcel A-Level Mathematics): Quizzes

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Change of Sign Failure
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What is change of sign failure?

When method detects no root but one exists

Why does f(x)=(x−2)2f(x) = (x - 2)^2 cause change of sign failure at x=2x = 2?

Function touches x-axis without crossing

What happens at a repeated root?

No sign change occurs

Why do discontinuous functions cause change of sign failure?

Function may jump over the root

Which method can help when change of sign fails?

Newton-Raphson method

If f(1)<0f(1) < 0 and f(2)<0f(2) < 0 with two roots in [1,2][1,2], what occurs?

No change of sign

For f(x)=1xf(x) = \frac{1}{x}, why does sign change occur without a root?

Discontinuity at y-axis

When does change of sign method work?

When function is continuous in interval

If f(−1)<0f(-1) < 0 and f(1)>0f(1) > 0 for f(x)=1xf(x) = \frac{1}{x}, is there a root?

No, due to discontinuity

If f(1)>0f(1) > 0 and f(2)>0f(2) > 0 with a root in [1,2][1,2], why does the method fail?

Lack of sign change in interval

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