Where is the local approximation of $e^x$ most visible in the animation?

Conditions

• The animation includes a zoom or close-up feature.
• The focus is on the neighborhood of the expansion point $x=0$.

Reasoning, step by step

1. Identify the expansion point at $x=0$.
2. Observe the behavior of the curves in the wide view, where they may diverge.
3. Switch to the close-up view centered at $x=0$.
4. Note that in this small neighborhood, the difference between $e^x$ and $S_n(x)$ is visually minimal for higher $n$.
5. Conclude that the close-up reveals the effectiveness of the local Taylor approximation.

Example

The script states: 'The close-up makes the local approximation visible.'

Common misconceptions

• Looking for the best approximation far from $x=0$.
• Assuming the global view shows the accuracy of the derivative matching.

Watch the explanation

Connected concepts

Explore next

• Why does the approximation fail far from $x=0$?
• What does 'local' mean in the context of Taylor series?
• How does the degree of the polynomial affect the width of the accurate region?