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Let's analyze the problem step by step.

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

### Part (a): Find the first Taylor polynomial \( P_1(x) \) of \( f(x) \) at \( x = 0 \).

The Taylor series expansion of a function \( f(x) \) around \( x = a \) is given by:
\[ f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \dots \]

Since we are looking for the first Taylor polynomial \( P_1(x) \), we only need the first two terms of the series:
\[ P_1(x) = f(0) + f'(0)x \]

1. **Find \( f(0) \):**
   \[ f(0) = \frac{0}{2(0) + 3} = 0 \]

2. **Find \( f'(x) \):**
   Use the quotient rule:
   \[
   f'(x) = \frac{(2x + 3)(1) - x(2)}{(2x + 3)^2} = \frac{2x + 3 - 2x}{(2x + 3)^2} = \frac{3}{(2x + 3)^2}
   \]
   Now evaluate \( f'(0) \):
   \[
   f'(0) = \frac{3}{(2(0) + 3)^2} = \frac{3}{9} = \frac{1}{3}
   \]

Thus, the first Taylor polynomial is:
\[ P_1(x) = 0 + \frac{1}{3}x = \frac{1}{3}x \]

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

Now, approximate \( f(1.05) \) using \( P_1(x) \):
\[ P_1(1.05) = \frac{1}{3} \times 1.05 = \frac{1.05}{3} \approx 0.35 \]

### Part (c): Calculate the error in the Taylor series approximation, \( |f(1.05) - P_1(1.05)| \), to 4 decimal places.

1. **Calculate \( f(1.05) \):**
   \[
   f(1.05) = \frac{1.05}{2(1.05) + 3} = \frac{1.05}{2.1 + 3} = \frac{1.05}{5.1} \approx 0.2059
   \]

2. **Calculate the error:**
   \[
   \text{Error} = |f(1.05) - P_1(1.05)| = |0.2059 - 0.35| = 0.1441
   \]

So, the error is approximately **0.1441**.

Would you like any further details or explanations on these calculations?

Here are five related questions:
1. What is the second-degree Taylor polynomial \( P_2(x) \) for \( f(x) \) at \( x = 0 \)?
2. How does the error change if you use the second-degree polynomial instead of the first-degree polynomial?
3. What is the radius of convergence for the Taylor series of \( f(x) \) around \( x = 0 \)?
4. How would the Taylor polynomial change if you expanded around \( x = 1 \) instead of \( x = 0 \)?
5. How does the Taylor series help in approximating functions that are difficult to evaluate directly?

**Tip:** Remember that the error in a Taylor series approximation depends on the number of terms used. The more terms you include, the smaller the error, especially near the point of expansion.

Explore the calculation of Taylor series approximation for the function f(x) = x/(2x+3). Learn how to find the first Taylor polynomial, approximate values using the polynomial, and calculate errors in approximation. Suitable for advanced undergraduate students studying calculus and Taylor series.

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