Let's solve each part step by step.

Part 1: Approximating $f(x) = \sin(x) \approx x$ near zero

For small values of $x$, the approximation $\sin(x) \approx x$ is often used. This approximation is the first term in the Taylor series expansion of $\sin(x)$, which is given by: $\sin(x) = x - \frac{x^3}{6} + \frac{x^5}{120} - \cdots$ Using only the first term gives us the approximation $\sin(x) \approx x$. To find the maximum error using $n = 2$, we’ll need to consider up to the next term, which includes the $x^3$ term.

Part 1a: Estimate $f(0.1)$ and find the maximum error

1. Estimate: $f(0.1) \approx 0.1$

2. Maximum Error Calculation: The error in the approximation can be estimated using the next term in the Taylor series: $E(x) \approx \frac{|x^3|}{6}$ For $x = 0.1$: $E(0.1) \approx \frac{|(0.1)^3|}{6} = \frac{0.001}{6} \approx 0.0001667$ Rounding to two decimal places, the maximum error is approximately $0.00017$.

So, for $f(0.1)$:

• Approximation: $f(0.1) \approx 0.1$
• Maximum Error: $0.00017$

Part 1b: Estimate $f(0.6)$ and find the maximum error

1. Estimate: $f(0.6) \approx 0.6$

2. Maximum Error Calculation: Using the same error term as above, for $x = 0.6$: $E(0.6) \approx \frac{|(0.6)^3|}{6} = \frac{0.216}{6} = 0.036$

So, for $f(0.6)$:

• Approximation: $f(0.6) \approx 0.6$
• Maximum Error: $0.036$

Summary

1. For $f(0.1)$:

   • Approximation: $f(0.1) \approx 0.1$
   • Maximum Error: $0.00017$

2. For $f(0.6)$:

   • Approximation: $f(0.6) \approx 0.6$
   • Maximum Error: $0.036$

Would you like a detailed breakdown of the Taylor series or any clarification on the error estimation?

Here are five follow-up questions that may further your understanding:

1. What is the Taylor series expansion for $\sin(x)$ up to the fourth term?
2. How does the error in the Taylor series approximation change with increasing $x$ values?
3. What would the approximation be for $f(x) = \sin(x)$ at $x = 0.01$ and its associated error?
4. Can we improve the approximation accuracy by including higher-order terms in the Taylor series?
5. How would the error behave if we chose a different function, such as $f(x) = \cos(x)$, near zero?

Tip: When using Taylor series approximations, including more terms from the series generally decreases the error, especially for larger values of $x$ close to the center of expansion (in this case, zero).