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Let's work through the problem step by step.

### Given:
\[ f(x) = \frac{x^2}{x^2 + 3} \]

### (a) Find the first Taylor polynomial \( T_1(x) \) at \( x = 0 \) for \( f(x) \).

To find the first Taylor polynomial \( T_1(x) \), we need to find the value of \( f(0) \) and the first derivative \( f'(0) \).

1. **Compute \( f(0) \):**

   \[
   f(0) = \frac{0^2}{0^2 + 3} = 0
   \]

2. **Compute \( f'(x) \):**

   We use the quotient rule to differentiate \( f(x) \).

   If \( f(x) = \frac{g(x)}{h(x)} \), then
   \[
   f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}
   \]
   
   Here, \( g(x) = x^2 \) and \( h(x) = x^2 + 3 \).
   
   - \( g'(x) = 2x \)
   - \( h'(x) = 2x \)
   
   Then:
   \[
   f'(x) = \frac{(2x)(x^2 + 3) - (x^2)(2x)}{(x^2 + 3)^2}
   \]
   Simplifying this:
   \[
   f'(x) = \frac{2x(x^2 + 3 - x^2)}{(x^2 + 3)^2} = \frac{6x}{(x^2 + 3)^2}
   \]

3. **Evaluate \( f'(0) \):**

   \[
   f'(0) = \frac{6 \times 0}{(0^2 + 3)^2} = 0
   \]

The first Taylor polynomial \( T_1(x) \) at \( x = 0 \) is:
\[
T_1(x) = f(0) + f'(0)x = 0 + 0 \cdot x = 0
\]

### (b) Use the Taylor polynomial to approximate \( f(1.05) \).

Since the first Taylor polynomial \( T_1(x) = 0 \), the approximation is:
\[
f(1.05) \approx T_1(1.05) = 0
\]

### (c) Calculate the error in the Taylor Series approximation \( |f(1.05) - T_1(1.05)| \).

To find the error, we calculate the exact value of \( f(1.05) \):
\[
f(1.05) = \frac{(1.05)^2}{(1.05)^2 + 3}
\]
I'll compute this numerically.

Let's calculate that value.The exact value of \( f(1.05) \) is approximately \( 0.2687 \).

### Error Calculation
Since the Taylor polynomial approximation \( T_1(1.05) = 0 \), the error is:
\[
\text{Error} = |f(1.05) - T_1(1.05)| = |0.2687 - 0| = 0.2687
\]

### Summary:
1. The first Taylor polynomial \( T_1(x) \) at \( x = 0 \) is \( 0 \).
2. The approximation of \( f(1.05) \) using \( T_1(x) \) is \( 0 \).
3. The error in the Taylor Series approximation is approximately \( 0.2687 \).

Would you like more details or have any other questions?

Here are 5 related questions to explore:
1. How would the Taylor polynomial change if we considered more terms in the expansion?
2. What is the second derivative \( f''(x) \) at \( x = 0 \)?
3. How does the error change if you use the second Taylor polynomial instead of the first?
4. What is the interval of convergence for the Taylor series of \( f(x) \)?
5. How can we use Taylor series to approximate functions near points other than \( x = 0 \)?

**Tip:** When working with Taylor series, increasing the number of terms generally improves the accuracy of approximations.

Explore how to approximate f(x) using Taylor polynomials. Learn the steps involved in finding the first Taylor polynomial, estimating f(1.05), and calculating the error in the Taylor Series approximation. Suitable for advanced undergraduate students studying calculus and series approximations.

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