Let's break down the problem and approach it step by step:

Given Information:

• The function is $f(x) = e^{\frac{x}{500}}$.
• The linear approximation is $y = 1 + \frac{e^2 - 0.1}{100}$ for $0 < x < 100$.
• This approximation agrees with $f(x)$ at $x = 0$ and $x = 100$.

We need to calculate the percentage error in the approximation when $x = 50$.

Steps:

1. Evaluate the original function $f(x)$ at $x = 50$: $f(50) = e^{\frac{50}{500}} = e^{0.1}$

2. Evaluate the linear approximation at $x = 50$: The linear approximation is $y = 1 + \frac{e^2 - 0.1}{100}$. However, we want to find the value of $y$ at $x = 50$.

   Since we don't have $x$ explicitly in the approximation formula, we'll assume it is meant to be a constant linear approximation valid for the interval $0 < x < 100$. So, we will treat the approximation formula as fixed and use it directly at $x = 50$.

   $y_{\text{approx}} = 1 + \frac{e^2 - 0.1}{100}$

3. Calculate the percentage error: The formula for percentage error is: $\text{Percentage Error} = \frac{|f(50) - y_{\text{approx}}|}{|f(50)|} \times 100$

Let's perform the calculations to find the error.

Calculations:

1. $f(50) = e^{0.1} \approx 1.1052$.

2. $y_{\text{approx}} = 1 + \frac{e^2 - 0.1}{100}$. First, calculate $e^2$: $e^2 \approx 7.3891$ Now substitute: $y_{\text{approx}} = 1 + \frac{7.3891 - 0.1}{100} = 1 + \frac{7.2891}{100} \approx 1 + 0.072891 = 1.072891$

3. Now, calculate the percentage error: $\text{Percentage Error} = \frac{|1.1052 - 1.072891|}{|1.1052|} \times 100 \approx \frac{0.032309}{1.1052} \times 100 \approx 2.92\%$

Conclusion:

The percentage error in the approximation at $x = 50$ is approximately 2.92%.

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Would you like further clarification on any of the steps or any other related questions?

Here are 5 related questions you might find interesting:

1. How would the error change if the approximation was made at $x = 0$ instead of $x = 50$?
2. What is the formula for the linear approximation of $f(x) = e^{x/500}$ at $x = 0$?
3. How would the percentage error behave if we used a second-order Taylor expansion instead of a linear approximation?
4. What is the general method for finding the linear approximation of a function at a point?
5. How can the error in the linear approximation be minimized for functions like $f(x) = e^{x/500}$?

Tip: To improve the accuracy of approximations, using higher-order Taylor expansions (quadratic or cubic approximations) can help reduce the error significantly!