Learn with Amar
Teaching video
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Browse grades and teaching videos01 · Read and understand
What you will learn
- Build a quadratic Taylor polynomial.
- Evaluate its numerical prediction.
- Bound the omitted remainder.
Before you start
Powers, factorials, exponential derivatives, and absolute error.
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 a quadratic Taylor polynomial centered at zero to approximate e^0.1. Can you guarantee the absolute error is less than 0.000185?
Why this math matters
A local polynomial can replace a complicated function in a calculator, simulation, or analysis. The important question is not only whether the approximation looks reasonable, but how much error it permits. Taylor's theorem connects that error to a bound on a higher derivative over the interval being used.
Set up the model
A useful answer starts with clear assumptions:
- The function is eˣ, whose derivatives of every order are eˣ.
- The center is zero and the target input is 0.1.
- Use the verified inequality e^0.1 < 1.106 to bound derivatives on [0, 0.1].
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.
How close is a short polynomial to an exponential?
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Use a quadratic Taylor polynomial centered at zero to approximate e^0.1. Can you guarantee the absolute error is less than 0.000185?
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 from derivative values at zero
P₂(x) = 1 + x + x²/2
The function and its first two derivatives equal one at zero. The factorial denominator 2! makes the polynomial's second derivative match as well.
Compute the approximation
P₂(0.1) = 1 + 0.1 + 0.005 = 1.105
This is a polynomial evaluation, not yet a statement of accuracy. Keep its approximation sign when comparing with the exponential.
Bound the third-order remainder
|R₂(0.1)| ≤ 1.106(0.1)³/6 ≈ 0.000184334 < 0.000185
The third derivative is eˣ and never exceeds the chosen bound on this interval. Taylor's remainder formula therefore establishes the requested guarantee.
The result
e^0.1 is approximately 1.105, with absolute error below 0.000185.
The actual value is approximately 1.105170918, consistent with the bound. A bound need not equal the actual error. Moving farther from the center can require more terms or a different derivative bound.
Common mistakes to catch
- Using only the derivative at zero does not bound the derivative throughout the interval.
- An infinite Taylor series represents its function only where its remainder tends to zero.
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
Use P₂ to approximate e^0.2.
Show a hint
Substitute 0.2 into 1 + x + x²/2.
Reveal answer and explanation
1.22
1 + 0.2 + 0.04/2 = 1.22. The earlier error bound for 0.1 does not automatically apply.
Practice 2
Approximate sin(0.1) by x − x³/6 in radians. Bound the error using the next alternating term.
Show a hint
For these decreasing alternating terms, use 0.1⁵/120.
Reveal answer and explanation
Approximately 0.0998333333; error at most 0.00000008334
The next omitted term is about 8.333 × 10⁻⁸, and the alternating-series remainder bound applies at this input.
Take the idea with you
Report an approximation together with its center, order, input interval, and an error argument when accuracy matters.
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.
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