Question 1 Foundation
is continuous, and . What can you conclude?
Answer. A root lies between 2 and 3.
A sign change across a continuous function guarantees a crossing. Both conditions are needed.
Roots of Equations
Original questions written at exam standard. Work each one before you open the solution.
Difficulty
Question 2 Foundation
For , evaluate .
Answer.
. Positive, so the root lies below 1.5.
Question 3 Standard
State the Newton–Raphson formula.
Answer.
Geometrically it follows the tangent at the current estimate down to the -axis.
Question 4 Standard
Starting from an interval of width 1, how many bisections are needed to reduce it below 0.1?
Answer.
The widths are . The fourth is the first below 0.1.
Question 5 Stretch
Give a rearrangement of suitable for fixed-point iteration, and justify it.
Answer. , since near the root.
From , take the cube root. The derivative is well under 1 near , so the iteration converges.
Question 6 Stretch
Why can Newton–Raphson fail even when a root exists nearby?
Answer. If is zero or near zero, the tangent is almost horizontal and the next estimate is thrown far away.
It may then converge to a different root, or cycle without settling. A sketch helps choose a starting value in a region where the gradient is steep.