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Approximate to four decimal places using the linearization L(x) of f(x) = at a = 100 and use a calculator to compute 104 the percentage error. (Use decimal notation. Give your answer to two decimal places.) The percentage error 2 %

Approximate to four decimal places using the linearization L(x) of f(x) = at a = 100 and use a calculator to compute 104 the percentage error. (Use decimal notation. Give your answer to two decimal places.) The percentage error 2 %

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Exercise: Approximating \(\frac{1}{104}\) Using Linearization and Calculating Percentage Error Approximate \(\frac{1}{104}\) to four decimal places using the linearization \(L(x)\) of \(f(x) = \frac{1}{x}\) at \(a = 100\) and use a calculator to compute the percentage error. _(Use decimal notation. Give your answer to two decimal places.)_ #### Solution 1. **Finding the Linearization**: - Function \( f(x) = \frac{1}{x} \) - Point of approximation \( a = 100 \) - Linearization formula: \( L(x) = f(a) + f'(a)(x - a) \) 2. **Calculations**: - \( f(a) = \frac{1}{100} = 0.01 \) - Derivative \( f'(x) = -\frac{1}{x^2} \) - At \( a = 100 \), \( f'(a) = -\frac{1}{100^2} = -0.0001 \) - \( L(x) = 0.01 + (-0.0001)(x - 100) \) 3. **Approximating \(\frac{1}{104}\)**: - \( x = 104 \) - \( L(104) = 0.01 + (-0.0001)(104 - 100) = 0.01 - 0.0004 = 0.0096 \) 4. **Actual Value and Percentage Error**: - Actual value \( \frac{1}{104} \approx 0.0096 \) - Percentage error = \(\left| \frac{\text{approximate value} - \text{actual value}}{\text{actual value}} \right| \times 100 \% \) Finally, complete the form below: ``` The percentage error ≈ ``` #### Explanation for the Graphs/Diagrams (if necessary): - No graphs or diagrams are included in this question. The problem involves mathematical functions and their linear approximations. This exercise helps students understand how linearization can be used to approximate function values and the concept of percentage error in approximations.
Transcribed Image Text:### Exercise: Approximating \(\frac{1}{104}\) Using Linearization and Calculating Percentage Error Approximate \(\frac{1}{104}\) to four decimal places using the linearization \(L(x)\) of \(f(x) = \frac{1}{x}\) at \(a = 100\) and use a calculator to compute the percentage error. _(Use decimal notation. Give your answer to two decimal places.)_ #### Solution 1. **Finding the Linearization**: - Function \( f(x) = \frac{1}{x} \) - Point of approximation \( a = 100 \) - Linearization formula: \( L(x) = f(a) + f'(a)(x - a) \) 2. **Calculations**: - \( f(a) = \frac{1}{100} = 0.01 \) - Derivative \( f'(x) = -\frac{1}{x^2} \) - At \( a = 100 \), \( f'(a) = -\frac{1}{100^2} = -0.0001 \) - \( L(x) = 0.01 + (-0.0001)(x - 100) \) 3. **Approximating \(\frac{1}{104}\)**: - \( x = 104 \) - \( L(104) = 0.01 + (-0.0001)(104 - 100) = 0.01 - 0.0004 = 0.0096 \) 4. **Actual Value and Percentage Error**: - Actual value \( \frac{1}{104} \approx 0.0096 \) - Percentage error = \(\left| \frac{\text{approximate value} - \text{actual value}}{\text{actual value}} \right| \times 100 \% \) Finally, complete the form below: ``` The percentage error ≈ ``` #### Explanation for the Graphs/Diagrams (if necessary): - No graphs or diagrams are included in this question. The problem involves mathematical functions and their linear approximations. This exercise helps students understand how linearization can be used to approximate function values and the concept of percentage error in approximations.
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