https://file.aiquickdraw.com/mathbot/file/1720957169008w7oyx1tu.jpg

explain each and every step of this equation

Sure, let's break down the steps of the linear interpolation shown in the image.

The interpolation formula being used is:
\[ f(x) = f(x_1) + \frac{(x - x_1)}{(x_2 - x_1)} \cdot (f(x_2) - f(x_1)) \]

We are given:
\[ x_1 = 0.40, \quad x_2 = 0.45 \]
\[ f_s(0.40) = 0.0860190, \quad f_s(0.45) = 0.1123850 \]
\[ f_T(0.40) = 0.3935501, \quad f_T(0.45) = 0.4786301 \]

We want to interpolate to find:
\[ f_s(0.444) \text{ and } f_T(0.444) \]

### Interpolation for \( f_s(0.444) \):

1. **Write the formula:**
   \[ f_s(0.444) = f_s(0.45) + \frac{(0.444 - 0.45)}{(0.40 - 0.45)} \cdot (f_s(0.40) - f_s(0.45)) \]

2. **Substitute the known values:**
   \[ f_s(0.444) = 0.1123850 + \frac{(0.444 - 0.45)}{(0.40 - 0.45)} \cdot (0.0860190 - 0.1123850) \]

3. **Simplify the fraction:**
   \[ \frac{(0.444 - 0.45)}{(0.40 - 0.45)} = \frac{-0.006}{-0.05} = 0.12 \]

4. **Calculate the difference in function values:**
   \[ 0.0860190 - 0.1123850 = -0.026366 \]

5. **Multiply the fraction by the difference:**
   \[ 0.12 \cdot -0.026366 = -0.003164 \]

6. **Add this result to \( f_s(0.45) \):**
   \[ 0.1123850 + (-0.003164) = 0.109221 \]

So, 
\[ f_s(0.444) \approx 0.109221 \]

### Interpolation for \( f_T(0.444) \):

1. **Write the formula:**
   \[ f_T(0.444) = f_T(0.45) + \frac{(0.444 - 0.45)}{(0.40 - 0.45)} \cdot (f_T(0.40) - f_T(0.45)) \]

2. **Substitute the known values:**
   \[ f_T(0.444) = 0.4786301 + \frac{(0.444 - 0.45)}{(0.40 - 0.45)} \cdot (0.3935501 - 0.4786301) \]

3. **Simplify the fraction:**
   \[ \frac{(0.444 - 0.45)}{(0.40 - 0.45)} = 0.12 \]

4. **Calculate the difference in function values:**
   \[ 0.3935501 - 0.4786301 = -0.08508 \]

5. **Multiply the fraction by the difference:**
   \[ 0.12 \cdot -0.08508 = -0.01021 \]

6. **Add this result to \( f_T(0.45) \):**
   \[ 0.4786301 + (-0.01021) = 0.46842 \]

So, 
\[ f_T(0.444) \approx 0.46842 \]

**Summary:**
- \( f_s(0.444) \approx 0.109221 \)
- \( f_T(0.444) \approx 0.46842 \)

Would you like any further details or have any questions?

### Here are 5 questions you might consider asking:
1. How is linear interpolation different from polynomial interpolation?
2. Can you explain other methods of interpolation like cubic spline interpolation?
3. What are some real-world applications of linear interpolation?
4. How do you handle interpolation when you have more than two data points?
5. Can you show me an example of extrapolation using similar data points?

### Tip:
When performing interpolation, ensure that the x-values are ordered correctly to avoid negative intervals, which can lead to incorrect results.

Explore linear interpolation step-by-step with examples. Learn how to interpolate values between given data points using the linear interpolation formula. Detailed explanations and calculations provided.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation