What a Taylor polynomial is approximating

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2D Graphing

The graph shows the main idea directly: near the center, the polynomial hugs the original function closely; farther away, the match degrades. CAS then explains where the coefficients came from.

The local-fit idea

A Taylor polynomial centered at $a$ is built so that the polynomial and the original function agree at $x = a$ , and their first several derivatives agree there too.

That is why the approximation is best near the center. The polynomial is engineered to match the local behavior of the function, not to imitate the whole graph perfectly on every interval.

What the graph is showing

For $sin (x)$ near $0$ , the degree-5 Taylor polynomial $x - x^{3} /6 + x^{5} /120$ tracks the original curve extremely well around the origin.

As you move farther from $0$ , the two graphs separate because the higher-order curvature of $sin (x)$ is no longer captured by only five degrees of polynomial data.

What changes when you change the center or the degree

Changing the center changes which part of the function you are approximating well. A polynomial centered at $a = 0$ is optimized near $0$ , not near $x = 3$ .

Increasing the degree adds more derivative information and usually improves the fit over a wider neighborhood around the center. But the idea is still local approximation, not global equality.

Common Pitfall

A Taylor polynomial is not supposed to match the original function everywhere. If the graphs drift apart far from the center, that does not mean the polynomial is wrong.

Try a Variation

Compare the degree-3 and degree-5 Taylor polynomials for $sin (x)$ . On what interval around $0$ does the extra degree produce a visibly better fit?

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Worked ExampleCalculusMath Workspace (CAS)

Find a Taylor polynomial and estimate the error

This example builds the degree-5 Taylor polynomial of $sin (x)$ centred at $0$ , evaluates it at $x = 0.5$ , and confirms that the actual error stays inside the bound given by Taylor's theorem.

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ProofCalculusProof Builder

Proof of the chain rule

The chain rule says the derivative of $f (g (x))$ is $f^{'} (g (x)) \cdot g^{'} (x)$ . The proof uses a continuous extension of the difference quotient to handle all cases cleanly.

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ExplanationCalculus2D Graphing

What a derivative means geometrically

The derivative is the slope of the tangent line, but that phrase only becomes useful when you compare the curve to nearby secant lines and local linear approximations.

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