From the uploaded image, the task involves the Taylor polynomial of order 2 and the corresponding remainder term for the function $f(x) = \ln(x+1)$ around the expansion point $a = 0$. Let me break down and explain each part of the problem.

a) Construct the Taylor polynomial of second order

The Taylor polynomial of $f(x) = \ln(x+1)$ at $a = 0$ is given by:

$T_2(x) = f(0) + f'(0)x + \frac{f''(0)}{2}x^2$

Step 1: Compute derivatives of $f(x)$

1. $f(x) = \ln(x+1)$
   $f'(x) = \frac{1}{x+1}$
2. $f''(x) = -\frac{1}{(x+1)^2}$

Step 2: Evaluate at $x = 0$

• $f(0) = \ln(0+1) = 0$
• $f'(0) = \frac{1}{0+1} = 1$
• $f''(0) = -\frac{1}{(0+1)^2} = -1$

Step 3: Substitute into the Taylor polynomial formula

$T_2(x) = 0 + 1\cdot x + \frac{-1}{2}x^2 = x - \frac{x^2}{2}$

Thus, the second-order Taylor polynomial is: $T_2(x) = x - \frac{x^2}{2}$

b) Corresponding remainder term

The remainder term (Lagrange form) for $f(x)$ is: $R_3(x) = \frac{f^{(3)}(c)}{3!}x^3$ where $c \in [0, x]$ and $f^{(3)}(x)$ is the third derivative.

Step 1: Compute $f^{(3)}(x)$

$f^{(3)}(x) = \frac{2}{(x+1)^3}$

Step 2: Express $R_3(x)$

$R_3(x) = \frac{2}{3! (c+1)^3}x^3 = \frac{1}{3(c+1)^3}x^3, \quad c \in [0, x]$

c) Approximation at $x = 1.5$ and absolute error

Step 1: Compute $T_2(1.5)$

Using $T_2(x) = x - \frac{x^2}{2}$: $T_2(1.5) = 1.5 - \frac{(1.5)^2}{2} = 1.5 - \frac{2.25}{2} = 1.5 - 1.125 = 0.375$

Step 2: Compute absolute error

The absolute error is: $\text{Error} = |f(1.5) - T_2(1.5)|$

Evaluate $f(1.5) = \ln(1.5+1) = \ln(2.5) \approx 0.9163$ (rounded to 4 decimals).

Thus: $\text{Error} = |0.9163 - 0.375| = 0.5413$

Final Answers:

1. Taylor polynomial: $T_2(x) = x - \frac{x^2}{2}$

2. Remainder term: $R_3(x) = \frac{1}{3(c+1)^3}x^3$

3. Approximation at $x = 1.5$: $T_2(1.5) = 0.375$

4. Absolute error: $\text{Error} = 0.5413$

Do you need more detailed calculations or additional explanations? Here are some related questions:

1. How is the Lagrange remainder term used to estimate the error?
2. Why does the approximation improve as the degree of the Taylor polynomial increases?
3. Can this method be extended to functions with discontinuities?
4. What is the significance of rounding in numerical approximations?
5. How can one verify that $c \in [0, x]$ satisfies the conditions for the remainder term?

Tip: Always compute derivatives carefully and keep track of negative signs when constructing Taylor series expansions!