How do successive polynomial terms affect the local shape of the Maclaurin approximation of $e^x$?

Conditions

• The function is $f(x) = e^x$.
• The expansion is centered at $x=0$.
• Terms are added sequentially from degree 2 to 4.

Reasoning, step by step

1. Start with the linear approximation $S_1(x) = 1+x$.
2. Add the quadratic term $x^2/2!$ to match the second derivative $f''(0)=1$.
3. Add the cubic term $x^3/3!$ to match the third derivative $f'''(0)=1$.
4. Add the quartic term $x^4/4!$ to match the fourth derivative $f^{(4)}(0)=1$.
5. Observe that each addition refines the fit near $x=0$ by capturing more curvature information.

Example

The script states: 'The quadratic, cubic and quartic terms successively match more derivatives at zero. Their local shapes improve even though their global shapes differ from the exponential.'

Common misconceptions

• Assuming that adding more terms always improves the approximation globally.
• Believing that the polynomial will eventually match the exponential function exactly at all points.

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Connected concepts

Explore next

• Why do global shapes differ from the exponential?
• Does the error decrease at every point after adding a term?
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