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The problem involves using the modified Euler's method to approximate the solution to the differential equation:

\[
\dot{x} = t x^2
\]
with the initial condition \(x(0) = 1\), over one time step \(\Delta t\).

### Modified Euler's Method
Modified Euler's method is a second-order numerical method that improves on the basic Euler's method by averaging the slope at the beginning and at the end of the interval. The steps for applying modified Euler's method are:

1. **First step (Euler step):** Use the differential equation to approximate the value at the next time step using the slope at the beginning of the interval.
   \[
   x_1 = x_0 + \Delta t \cdot f(t_0, x_0)
   \]
   where \(f(t, x) = t x^2\).

2. **Second step (correction step):** Correct the estimate using the slope at the midpoint:
   \[
   x(\Delta t) = x_0 + \frac{\Delta t}{2} \left( f(t_0, x_0) + f(t_1, x_1) \right)
   \]

### Step-by-step solution:

1. **Initial condition**: \(x(0) = 1\).
   
2. **Euler step**:
   \[
   f(t_0, x_0) = t_0 x_0^2 = 0 \cdot 1^2 = 0
   \]
   So the Euler estimate for \(x_1\) becomes:
   \[
   x_1 = x_0 + \Delta t \cdot 0 = 1
   \]

3. **Correction step**: Now, we calculate the correction using the value of \(f(t_1, x_1)\):
   \[
   f(t_1, x_1) = \Delta t \cdot x_1^2 = \Delta t \cdot 1^2 = \Delta t
   \]
   Therefore, the corrected value of \(x(\Delta t)\) becomes:
   \[
   x(\Delta t) = 1 + \frac{\Delta t}{2} (0 + \Delta t) = 1 + \frac{\Delta t^2}{2}
   \]

### Answer:
The correct answer is the last option:
\[
x(\Delta t) = 1 + \frac{1}{2} (\Delta t)^2
\]

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Would you like more details on how Modified Euler's method works or any of the steps? Here are 5 related questions to deepen understanding:

1. How does Modified Euler's method differ from the standard Euler's method?
2. What are the limitations of Modified Euler's method for solving differential equations?
3. How would the solution change if the function was non-linear or had higher-order terms?
4. Can Modified Euler's method be applied to systems of differential equations? How?
5. How does Modified Euler's method compare to the Runge-Kutta methods?

**Tip**: When solving numerically, ensure that the step size \(\Delta t\) is small enough for accurate results but not so small that the computation becomes inefficient.

The modified Euler's method over one time step applied to \( \dot{x} = t x^2 \) with \( x(0) = 1 \) results in...

Understand the application of the modified Euler's method to approximate the solution of the differential equation dx/dt = t*x^2, with the initial condition x(0) = 1, over one time step. Step-by-step solution provided, including key formulas for solving numerically.

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