Math Problem Statement
Solution
To solve this problem, let’s proceed through the steps required:
Step 1: Find the Linear Approximation of at
The linear approximation at is given by: Here, , and we’re approximating near .
-
Calculate :
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Find : Using the power rule,
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Calculate :
Thus, the linear approximation at is:
Step 2: Use the Linear Approximation for Values Close to
Now, using , calculate the linear approximation values for each in the table. Here’s how it works for the first value, :
- For :
You can fill in similar calculations for each value of in the table. The linear approximation column should match the values in your image.
Step 3: Calculate the Exact Values
Calculate for each in the table by substituting into . Here’s an example:
- For :
Calculate similarly for the other values of , matching the values in the table.
Step 4: Calculate the Percent Error
The percent error formula is:
Using this formula, calculate the percent error for each value.
- For :
Summary of Steps for Completion:
- Step 1: Complete the linear approximation column using .
- Step 2: Fill in the exact values by computing for each .
- Step 3: Calculate the percent error for each pair of values using the formula above.
Would you like further clarification on any of these steps or assistance with the calculations?
Related Questions for Practice:
- How would the linear approximation change if the function were different, like ?
- How does the error behave if you expand around a different point, such as instead of ?
- Can you derive a
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Math Problem Analysis
Mathematical Concepts
Calculus
Linear Approximation
Error Analysis
Percent Error
Formulas
Linear approximation formula: L(x) = f(a) + f'(a)(x - a)
Percent error formula: 100 * |approximation - exact| / |exact|
Theorems
Linear approximation theorem
Suitable Grade Level
College level (Calculus I)
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