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.