Math Problem Statement

Suppose f(x) = x^(3/4) is to be approximated near x = 1. Find the linear approximation of f at x = 1. Then, complete the following table, showing the errors in the various approximations. Use a calculator to obtain the exact values. The percent error is calculated using the formula: 100 * |approximation - exact| / |exact|. Comment on the behavior of the errors as x approaches 1.

Solution

To solve this problem, let’s proceed through the steps required:

Step 1: Find the Linear Approximation of f(x)=x34f(x) = \sqrt[4]{x^3} at x=1x = 1

The linear approximation L(x)L(x) at x=ax = a is given by: L(x)=f(a)+f(a)(xa)L(x) = f(a) + f'(a)(x - a) Here, f(x)=x3/4f(x) = x^{3/4}, and we’re approximating near x=1x = 1.

  1. Calculate f(1)f(1): f(1)=13/4=1f(1) = 1^{3/4} = 1

  2. Find f(x)f'(x): Using the power rule, f(x)=34x(3/41)=34x1/4=34x4f'(x) = \frac{3}{4}x^{(3/4 - 1)} = \frac{3}{4}x^{-1/4} = \frac{3}{4\sqrt[4]{x}}

  3. Calculate f(1)f'(1): f(1)=3414=34=0.75f'(1) = \frac{3}{4\sqrt[4]{1}} = \frac{3}{4} = 0.75

Thus, the linear approximation L(x)L(x) at x=1x = 1 is: L(x)=1+0.75(x1)L(x) = 1 + 0.75(x - 1)

Step 2: Use the Linear Approximation for Values Close to x=1x = 1

Now, using L(x)=1+0.75(x1)L(x) = 1 + 0.75(x - 1), calculate the linear approximation values for each xx in the table. Here’s how it works for the first value, x=1.1x = 1.1:

  1. For x=1.1x = 1.1: L(1.1)=1+0.75(1.11)=1+0.75×0.1=1+0.075=1.075L(1.1) = 1 + 0.75(1.1 - 1) = 1 + 0.75 \times 0.1 = 1 + 0.075 = 1.075

You can fill in similar calculations for each value of xx in the table. The linear approximation column should match the values in your image.

Step 3: Calculate the Exact Values

Calculate f(x)f(x) for each xx in the table by substituting into f(x)=x3/4f(x) = x^{3/4}. Here’s an example:

  1. For x=1.1x = 1.1: f(1.1)=(1.1)3/41.0746f(1.1) = (1.1)^{3/4} \approx 1.0746

Calculate similarly for the other values of xx, matching the values in the table.

Step 4: Calculate the Percent Error

The percent error formula is: Percent Error=ApproximationExactExact×100\text{Percent Error} = \left| \frac{\text{Approximation} - \text{Exact}}{\text{Exact}} \right| \times 100

Using this formula, calculate the percent error for each xx value.

  1. For x=1.1x = 1.1: Percent Error=1.0751.07461.0746×1000.0372%\text{Percent Error} = \left| \frac{1.075 - 1.0746}{1.0746} \right| \times 100 \approx 0.0372\%

Summary of Steps for Completion:

  • Step 1: Complete the linear approximation column using L(x)=1+0.75(x1)L(x) = 1 + 0.75(x - 1).
  • Step 2: Fill in the exact values by computing f(x)=x3/4f(x) = x^{3/4} for each xx.
  • 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:

  1. How would the linear approximation change if the function were different, like f(x)=x2/3f(x) = x^{2/3}?
  2. How does the error behave if you expand around a different point, such as x=2x = 2 instead of x=1x = 1?
  3. 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)