How do you choose enough Taylor terms for a tolerance?
You will be able to: Select a degree using a verified error bound and distinguish degree from term count.
How do you choose enough Taylor terms for a tolerance?
A calculator needs enough terms to meet a requested accuracy. Adding terms without a stopping rule is less useful than checking a bound after each candidate degree.
A useful starting point: How can a derivative bound guarantee Taylor accuracy? →
Words and symbols before equations
- Degree n
- The highest power allowed in Pₙ; some coefficients may be zero.
- Nonzero term count
- How many nonzero summands are actually used.
- Tolerance
- The requested maximum absolute error.
- Uniform bound
- A bound valid over a whole stated input interval.
What this picture assumes
For eˣ, every derivative is eᵗ. M=e^max(a,x) bounds the next derivative over the whole interval between a and x. The exact function value is used only to compare actual error with the guarantee.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- At x=0.5, P3=1.645833, eˣ=1.648721, actual error=0.002888. M=1.648721 on the full interval; bound=0.004294. Increasing degree and moving nearer the center change the approximation; the theorem supplies the guarantee.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
For e⁰·⁵ at center 0, a simple bound is M=2 because eᵗ≤e⁰·⁵<2 on [0,0.5]. Test 2(0.5)ⁿ⁺¹/(n+1)! against the tolerance.
For tolerance 0.001, n=3 gives 2(0.5)⁴/24≈0.005208, too large. n=4 gives 2(0.5)⁵/120≈0.000521, sufficient.
An alternating Taylor series may admit a smaller next-term bound if its evaluated magnitudes decrease to zero. Degree and nonzero term count differ: x−x³/6 has degree 3 but two nonzero terms.
A worked example, step by step
Use the alternating bound to estimate sin(0.2) with x−x³/6 and bound its error.
- At x=0.2 the sine series alternates with decreasing magnitudes.
- The approximation is 0.2−0.2³/6≈0.198666667.
- The next nonzero magnitude is 0.2⁵/5!=0.0000026667.
- The error is at most that value; this cubic polynomial has two nonzero terms.
A bound larger than the tolerance does not prove the approximation is inaccurate; it means this bound has not guaranteed enough accuracy.
If an error bound is 0.01, does that disprove accuracy within 0.001?
Compare with an explanation
No. The actual error may be smaller; this bound simply does not guarantee the tighter tolerance.
Predict. Change one thing. Explain.
Set the center to 0 and x to 0.5. Raise n until the Lagrange bound is below 0.001. Compare the guarantee with the actual error and explain their distinction.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
At x=0.5, P3=1.645833, eˣ=1.648721, actual error=0.002888. M=1.648721 on the full interval; bound=0.004294. Increasing degree and moving nearer the center change the approximation; the theorem supplies the guarantee.
For eˣ, every derivative is eᵗ. M=e^max(a,x) bounds the next derivative over the whole interval between a and x. The exact function value is used only to compare actual error with the guarantee.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the term formula, partial sums, test conditions, remainder bound or interval of convergence. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionUsing M=2 on [0,0.5], show that P₄ for eˣ guarantees error below 0.001 at x=0.5. State P₄.
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Compare with the answer and four-point rubric
- 1 point: P₄=1+x+x²/2+x³/6+x⁴/24.
- 1 point: The fifth derivative is eᵗ≤2 on the interval.
- 1 point: The bound is 2(0.5)⁵/5!=1/1920.
- 1 point: 1/1920≈0.000521<0.001, so degree 4 is sufficient.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1What does an insufficiently small bound mean?
This bound cannot certify the requested tolerance.
RECALL 2When can a next-term bound be used?
When the evaluated series satisfies alternating-error hypotheses.
RECALL 3Is degree the same as term count?
No; zero coefficients can create gaps.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do you choose enough Taylor terms for a tolerance?
- Choose n so the verified bound meets the tolerance.
- Check alternating conditions before using the next-term bound.
- Degree is not always the number of nonzero terms minus one.
Remember: A bound larger than the tolerance does not prove the approximation is inaccurate; it means this bound has not guaranteed enough accuracy.
Conditions: For eˣ, every derivative is eᵗ. M=e^max(a,x) bounds the next derivative over the whole interval between a and x. The exact function value is used only to compare actual error with the guarantee.
Refresh Kid · AP Calculus BC Unit 10 · Objectives LIM-8.C · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 10.12, LIM-8.C. CED effective Fall 2020 and Fall 2026 clarifications checked September 17, 2026. This unit covers BC topics 10.1–10.15, Infinite Sequences and Series. Required convergence methods are the nth-term, integral, comparison, alternating and ratio tests; the root test is not assigned as required AP content.
Finite plots illustrate terms and partial sums but do not establish infinite convergence. Positive-series comparisons retain their hypotheses and inequality directions. Alternating bounds use the next omitted magnitude. Taylor bounds use the next derivative over the full interval. Power-series endpoints are checked separately, including after differentiation and integration. Taylor representation requires a remainder tending to zero.
All focused explanations, examples, practice and models are original Refresh Kid work. OpenStax was consulted for mathematical cross-checking. Khan Academy’s destination was checked, but JavaScript lesson content was not fully readable by the research tool. Organic Chemistry Tutor video titles, creator and destinations were checked; full videos were not reviewed. No provider questions, diagrams or scripts were copied. Resources are optional, and no paid resource is needed. Refresh Kid is not affiliated with or endorsed by these providers.
GitHub’s 3D website collection informed optional camera and spatial inspection. The original geometric slab model uses self-hosted Three.js with its MIT license. A unit cube receives slabs of widths 1/2, 1/4, 1/8 and so on, each spanning unit height and depth. The complete 2D strip and numerical volume readout remain available. No autoplay or WebGL is required to learn the mathematics.
Independent teacher review and observation of students remain pending. Technical checks do not certify mathematical accuracy, accessibility or learning effectiveness. This is a review edition.
Optional official resource: Released AP Calculus BC questions and scoring guides. The archive spans multiple units; no entire exam question is assigned here.
Learn → Explore → Practice → Review is informed by the IES learning guide; this implementation has not been evaluated with learners.
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