Use the finite-difference formulae from page 133 of the notes (print version) to find approximate values of the derivatives  f′j
  in the following table.

 

Use the centred-difference approximation wherever possible. Enter your answers to two decimal places accuracy.

It seems like you're referring to a problem involving finite-difference formulas to approximate derivatives, particularly using centered-difference approximations. However, since the specific table or set of data is not included in your message, I can walk you through the general process and give an example of how to apply these formulas.

### Finite-Difference Approximation Formulae:

1. **Forward Difference Approximation**:
   \[
   f'_{j} \approx \frac{f(x_{j+1}) - f(x_j)}{h}
   \]
   where \(h\) is the step size.

2. **Backward Difference Approximation**:
   \[
   f'_{j} \approx \frac{f(x_j) - f(x_{j-1})}{h}
   \]

3. **Centered Difference Approximation**:
   \[
   f'_{j} \approx \frac{f(x_{j+1}) - f(x_{j-1})}{2h}
   \]

The centered-difference formula is more accurate and is used when possible (i.e., when both forward and backward data points are available). For boundary points, you'd need to use forward or backward differences.

### General Steps for Approximating the Derivatives:
1. **Determine the step size** \( h \), which is the difference between consecutive \( x_j \) values (assuming uniform spacing).
2. **Apply the centered-difference formula** for interior points (i.e., not at the boundaries).
3. **Use forward or backward difference** for boundary points.
4. **Round answers to two decimal places** as requested.

### Example (if we had a table):

Assume you are given the following points in a table:

| \( x_j \)   | \( f(x_j) \) |
|-------------|--------------|
| 1.0         | 2.718        |
| 1.1         | 3.004        |
| 1.2         | 3.320        |
| 1.3         | 3.669        |
| 1.4         | 4.055        |

Let \( h = 0.1 \).

- **At \( x_2 = 1.1 \)**, you can use the centered-difference approximation:
  \[
  f'_2 \approx \frac{f(1.2) - f(1.0)}{2h} = \frac{3.320 - 2.718}{2(0.1)} = 3.01
  \]

- **At the boundary \( x_0 = 1.0 \)**, you can use the forward-difference approximation:
  \[
  f'_0 \approx \frac{f(1.1) - f(1.0)}{h} = \frac{3.004 - 2.718}{0.1} = 2.86
  \]

Without the actual table you need to work from, this is the general method.

If you can provide the table of \( x_j \) and \( f(x_j) \) values, I can compute the actual derivatives for you.

Would you like to upload the table or clarify the specific data so we can work on this further?

Here are some follow-up questions that relate to this:
1. What is the exact table of values you're working with?
2. Are the points equally spaced, and if so, what is the step size \( h \)?
3. Do you need to approximate derivatives for boundary points or just interior ones?
4. Do you require higher-order derivatives as well, or just the first derivative?
5. Would you like help in understanding the errors associated with different finite-difference formulas?

**Tip:** Centered-difference approximations tend to be more accurate because they minimize the error by averaging forward and backward estimates.

Use the finite-difference formulae from page 133 of the notes (print version) to find approximate values of the derivatives f′j in the following table. Use the centred-difference approximation wherever possible. Enter your answers to two decimal places accuracy.

Learn how to use finite difference methods, including forward, backward, and centered-difference approximations, to find approximate values of derivatives. This guide covers key numerical analysis concepts and provides step-by-step instructions for applying these formulas, with examples rounding answers to two decimal places.

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