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1 Use linear approximation, i.e. the tangent line, to approximate as follows: Let f(x) == 1 1.002 and find the equation of the tangent line to f(x) at a "nice" point near 1.002. Then use this to 1 approximate 1.002

1 Use linear approximation, i.e. the tangent line, to approximate as follows: Let f(x) == 1 1.002 and find the equation of the tangent line to f(x) at a "nice" point near 1.002. Then use this to 1 approximate 1.002

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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**Linear Approximation to Estimate \(\frac{1}{1.002}\)** To approximate \(\frac{1}{1.002}\) using linear approximation (i.e., the tangent line), follow these steps: 1. **Define the Function:** Let \( f(x) = \frac{1}{x} \). 2. **Select a "Nice" Point:** Choose a point near 1.002 where calculations are simpler. This point will help to find the tangent line, which is the basis for the approximation. 3. **Find the Tangent Line Equation:** Compute the derivative of \( f(x) \) to determine the slope of the tangent line at the chosen point. Use the derivative to form the equation of the tangent line. 4. **Use the Tangent Line for Approximation:** Once the tangent line equation is established, use it to approximate \(\frac{1}{1.002}\). This method leverages the concept that the tangent to a function at a point is a good local approximation of the function itself.
Transcribed Image Text:**Linear Approximation to Estimate \(\frac{1}{1.002}\)** To approximate \(\frac{1}{1.002}\) using linear approximation (i.e., the tangent line), follow these steps: 1. **Define the Function:** Let \( f(x) = \frac{1}{x} \). 2. **Select a "Nice" Point:** Choose a point near 1.002 where calculations are simpler. This point will help to find the tangent line, which is the basis for the approximation. 3. **Find the Tangent Line Equation:** Compute the derivative of \( f(x) \) to determine the slope of the tangent line at the chosen point. Use the derivative to form the equation of the tangent line. 4. **Use the Tangent Line for Approximation:** Once the tangent line equation is established, use it to approximate \(\frac{1}{1.002}\). This method leverages the concept that the tangent to a function at a point is a good local approximation of the function itself.
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