Find the approximate change in the value of `(1)/(x^(2)).` when x changes from `x=2` to `x=2.002`.

Answer

To find the approximate change in the value of \( \frac{1}{x^2} \) when \( x \) changes from \( 2 \) to \( 2.002 \), we will use the concept of derivatives.

### Step-by-Step Solution:

1. **Define the Function**:
Let \( y = \frac{1}{x^2} \).

2. **Find the Derivative**:
We need to find the derivative \( \frac{dy}{dx} \).
\[ y = x^{-2} \]
Using the power rule of differentiation:
\[ \frac{dy}{dx} = -2x^{-3} = -\frac{2}{x^3} \]

3. **Evaluate the Derivative at \( x = 2 \)**:
Substitute \( x = 2 \) into the derivative:
\[ \frac{dy}{dx} \bigg|_{x=2} = -\frac{2}{2^3} = -\frac{2}{8} = -\frac{1}{4} \]

4. **Determine \( dx \)**:
The change in \( x \) is given by:
\[ dx = 2.002 - 2 = 0.002 \]

5. **Calculate \( dy \)**:
Now, we can find the approximate change in \( y \) using the formula:
\[ dy \approx \frac{dy}{dx} \cdot dx \]
Substitute the values:
\[ dy \approx -\frac{1}{4} \cdot 0.002 = -0.0005 \]

6. **Conclusion**:
The approximate change in the value of \( \frac{1}{x^2} \) when \( x \) changes from \( 2 \) to \( 2.002 \) is \( -0.0005 \).