To find the Taylor polynomial $T_2(x)$ of the function $f(x) = e^{\sin x}$ centered at $a = \frac{\pi}{2}$ and then compute the error $|f(x) - T_2(x)|$ at $x = 1.3$, we can follow these steps:

1. Calculate the derivatives of $f(x)$:

Given $f(x) = e^{\sin x}$, we need the first and second derivatives:

• First derivative: $f'(x) = e^{\sin x} \cdot \cos x$

• Second derivative: $f''(x) = e^{\sin x} \cdot (\cos^2 x - \sin x)$

2. Evaluate the function and its derivatives at $a = \frac{\pi}{2}$:

• $f\left(\frac{\pi}{2}\right) = e^{\sin\left(\frac{\pi}{2}\right)} = e^1 = e$
• $f'\left(\frac{\pi}{2}\right) = e^{\sin\left(\frac{\pi}{2}\right)} \cdot \cos\left(\frac{\pi}{2}\right) = e \cdot 0 = 0$
• $f''\left(\frac{\pi}{2}\right) = e^{\sin\left(\frac{\pi}{2}\right)} \cdot \left(\cos^2\left(\frac{\pi}{2}\right) - \sin\left(\frac{\pi}{2}\right)\right) = e \cdot (0 - 1) = -e$

3. Write the Taylor polynomial $T_2(x)$ at $a = \frac{\pi}{2}$:

The second-degree Taylor polynomial is given by: $T_2(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2}(x - a)^2$ Substituting the values, we get: $T_2(x) = e + 0 \cdot (x - \frac{\pi}{2}) + \frac{-e}{2}(x - \frac{\pi}{2})^2 = e - \frac{e}{2}(x - \frac{\pi}{2})^2$

4. Compute $T_2(1.3)$:

First, compute $x - \frac{\pi}{2}$ at $x = 1.3$: $1.3 - \frac{\pi}{2} \approx 1.3 - 1.570796 = -0.270796$ Now, compute $T_2(1.3)$: $T_2(1.3) = e - \frac{e}{2}(-0.270796)^2$ $T_2(1.3) \approx e - \frac{e}{2} \times 0.073331 \approx e(1 - 0.036665) \approx e \times 0.963335 \approx 2.620557$

5. Compute the actual function value $f(1.3)$:

$f(1.3) = e^{\sin(1.3)} \approx e^{0.963558} \approx 2.621273$

6. Calculate the error $|f(1.3) - T_2(1.3)|$:

$|f(1.3) - T_2(1.3)| \approx |2.621273 - 2.620557| \approx 0.000716$

Final Answers:

• The Taylor polynomial $T_2(x)$ is approximately $T_2(x) = e - \frac{e}{2}(x - \frac{\pi}{2})^2$.
• $T_2(1.3) \approx 2.620557$
• The error $|f(1.3) - T_2(1.3)|$ is approximately $0.000716$.

All values are rounded to 6 decimal places.

Would you like further details or have any questions? Here are some related questions you might be interested in:

1. How does the error bound for Taylor polynomials relate to the remainder term?
2. What happens to the Taylor series if we include more terms?
3. How does the choice of $a$ (expansion point) affect the accuracy of the Taylor polynomial?
4. Can you explain the geometric interpretation of Taylor polynomials?
5. How does the error behave as $x$ moves farther from $a$?

Tip: When calculating Taylor series, always check the convergence interval to ensure that your approximation is valid for the given $x$.