Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Derive Newton's iteration and check a concrete root approximation.
- Justify the conclusion "f(8/3)=1/9, smaller than f(3)=2" using the stated assumptions.
Before you start
Derivatives and tangent lines.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Use Newton's method once from x₀=3 to approximate √7.
Why this math matters
Derive Newton's iteration and check a concrete root approximation. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The starting value is positive.
- Arithmetic in the worked step is exact.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Linearize a nonlinear root problem and inspect the update
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Use Newton's method once from x₀=3 to approximate √7.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
f(x)=x²−7 and f′(x)=2x
The desired positive root solves f(x)=0.
Work through the mathematics
x₁=x₀−f(x₀)/f′(x₀)=3−2/6=8/3
The tangent's zero supplies the next iterate.
Check the conclusion
f(8/3)=1/9, smaller than f(3)=2
The residual is reduced; local quadratic convergence requires a simple root and a sufficiently suitable starting point.
The result
f(8/3)=1/9, smaller than f(3)=2
The residual is reduced; local quadratic convergence requires a simple root and a sufficiently suitable starting point.
Common mistakes to catch
- Newton iteration is not globally guaranteed for every function and starting point.
- A small step and a small residual are different diagnostics.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Write the general square-root update.
Show a hint
Simplify x−(x²−7)/(2x).
Reveal answer and explanation
xₙ₊₁=(xₙ+7/xₙ)/2
The tangent formula becomes an arithmetic average.
Practice 2
Can Newton's method always start at zero here?
Show a hint
Inspect f′(0).
Reveal answer and explanation
No
The derivative is zero, so the displayed update divides by zero.
Take the idea with you
Use a bracket or other safeguard when an unrestricted tangent step leaves a useful region.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Guarantee a root bracket after a fixed number of steps
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