Questions: Compute (T2(x)) at (x=0.7) for (y=e^x) and use a calculator to compute the error (lefte^x-T2(x)right) at (x=1.4). [ T2(x)=square mathrmm ] [ lefte^x-T2(x)right=square ]

Transcript text: Compute $T_{2}(x)$ at $x=0.7$ for $y=e^{x}$ and use a calculator to compute the error $\left|e^{x}-T_{2}(x)\right|$ at $x=1.4$. \[ \begin{array}{l} T_{2}(x)=\square \mathrm{m} \\ \left|e^{x}-T_{2}(x)\right|=\square \\ \hline \end{array} \]

Solution

Solution Steps

To solve this problem, we need to compute the second-degree Taylor polynomial $ T_2(x) $ for the function $ y = e^x $ at a given point. The Taylor polynomial is given by:

\[ T_2(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2}(x-a)^2 \]

where $ f(x) = e^x $, and $ a $ is the point around which we are expanding the polynomial. For this problem, we assume $ a = 0 $ (Maclaurin series). After computing $ T_2(x) $ at $ x = 0.7 $, we calculate the error at $ x = 1.4 $ using the formula:

\[ \text{Error} = \left| e^{1.4} - T_2(1.4) \right| \]

Step 1: Compute $ T_2(0.7) $

To compute the second-degree Taylor polynomial $ T_2(x) $ for $ y = e^x $ at $ x = 0.7 $, we use the formula:

\[ T_2(x) = f(0) + f'(0)(x - 0) + \frac{f''(0)}{2}(x - 0)^2 \]

Calculating the values: