Sign Change with Intersection Points

Questions that ask to show two curves intersect (equivalently, that their difference has a root) in a given interval by demonstrating a sign change, typically requiring rearrangement into the form f(x) - g(x) = 0 or analysis of where curves meet.

1 questions

OCR C3 Q9
9.
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The diagram shows the curve with equation \(y = 2 x - 3 \ln ( 2 x + 5 )\) and the normal to the curve at the point \(P ( - 2 , - 4 )\).
  1. Find an equation for the normal to the curve at \(P\). The normal to the curve at \(P\) intersects the curve again at the point \(Q\) with \(x\)-coordinate \(q\).
  2. Show that \(1 < q < 2\).
  3. Show that \(q\) is a solution of the equation $$x = \frac { 12 } { 7 } \ln ( 2 x + 5 ) - 2 .$$
  4. Use an iterative process based on the equation above with a starting value of 1.5 to find the value of \(q\) to 3 significant figures and justify the accuracy of your answer.