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A second set of iteration questions for GCSE Maths. They are different from the ones on the topic page, so work through both. Try each question, then open the answer to check your working.
Use x = square root of (7 - x), starting at 2. Do 3 steps and round to 1 decimal place.
Answer: 2.2
f(x) = x^2 + x - 2. Check the sign of f(1) and f(2). Is there a sign change between them?
Answer: no, one end is already a root
Iterate x = square root of (8 - x) from a start at 3. Do 3 steps, then give 1 decimal place.
Answer: 2.4
Use x = square root of (10 - x), starting at 3, for 3 steps and round to 1 decimal place.
Answer: 2.7
Iterate x = square root of (13 - x), starting at 3. Do 3 steps and give 1 decimal place.
Answer: 3.1
Do 3 steps of x = square root of (11 - x), starting at 3, and round to 1 decimal place.
Answer: 2.9
A longer question with marks, in the style of a written paper. This is a Yumio practice question, not a past paper question.
For f(x) = x^2 + x - 11, work out the sign of f(2) and of f(3), and state the conclusion.
Answer: a root lies between 2 and 3
Short one-step questions. Pick an option before you open the answer.
What does a sign change of f between 1 and 2 prove?
A root lies between 1 and 2. It does not give the root to 1 decimal place by itself. A root lies between 1 and 2. It does not give the root to 1 decimal place by itself.
f(1) = -3 and f(2) = 1 for x^2 + x - 5. What is the conclusion?
A root lies between 1 and 2. A root lies between 1 and 2.
Why is f(1) = 0 not a sign change between 1 and 2?
The root is at the endpoint 1. The two values are not opposite signs. The root is at the endpoint 1. The two values are not opposite signs.
f(1) = -6 and f(2) = -2. Is that a sign change?
No. Both values are negative. No. Both values are negative.
Where is the sign change for x^2 + x - 8?
Between 2 and 3, because f(2) = -2 and f(3) = 4. Between 2 and 3, because f(2) = -2 and f(3) = 4.
What is the first iteration of √(5 - x) starting at 2?
√3, which is about 1.732. You then feed that value back in. √3, which is about 1.732. You then feed that value back in.
Why not round during the iteration?
Each next input has to be the full output. Rounding early moves the later steps. Each next input has to be the full output. Rounding early moves the later steps.
1.7867 to 1 decimal place is what?
1.8 (rounds up from 1.7867). 1.8 (rounds up from 1.7867).
2.366 to 1 decimal place is what?
2.4 (rounds up from 2.366). 2.4 (rounds up from 2.366).
Which rearrangement is being iterated by x = √(5 - x)?
x^2 = 5 - x, or x^2 + x - 5 = 0. x^2 = 5 - x, or x^2 + x - 5 = 0.
f(3) = -3 and f(4) = 5 for x^2 + x - 15. Where is the root?
Between 3 and 4. Between 3 and 4.
What must you include besides the two function values?
The conclusion: a root lies between those inputs, or that the signs do not change. The conclusion: a root lies between those inputs, or that the signs do not change.
Start 3, formula √(10 - x), after 3 steps to 1 decimal place?
2.7 (rounded to 1 decimal place). 2.7 (rounded to 1 decimal place).
Can both f(1) and f(2) be negative and a root still exist?
Yes. It may lie in the next interval, such as between 2 and 3. Yes. It may lie in the next interval, such as between 2 and 3.
How many steps do you do if the question says 3 steps?
Three substitutions. The starting value is not one of the three outputs. Three substitutions. The starting value is not one of the three outputs.
AQA-style and Edexcel-style questions. Yumio is not affiliated with, endorsed by or connected to AQA, Pearson Edexcel or any other exam board.