Find the approximate value of `(1. 999)^6` .

Answer

To find the approximate value of \( (1.999)^6 \), we can use the concept of derivatives and linear approximation. Here’s a step-by-step solution:

### Step 1: Define the function
Let \( y = f(x) = x^6 \). We want to find \( f(1.999) \).

### Step 2: Choose a point close to 1.999
We can choose \( x = 2 \) because it is close to 1.999 and the calculation of \( 2^6 \) is straightforward.

### Step 3: Calculate \( f(2) \)
Calculate \( f(2) \):
\[ f(2) = 2^6 = 64 \]

### Step 4: Determine \( \Delta x \)
Now, we find \( \Delta x \):
\[ \Delta x = 1.999 - 2 = -0.001 \]

### Step 5: Find the derivative \( f'(x) \)
Next, we calculate the derivative \( f'(x) \):
\[ f'(x) = 6x^5 \]

### Step 6: Evaluate the derivative at \( x = 2 \)
Now, evaluate \( f'(2) \):
\[ f'(2) = 6 \cdot 2^5 = 6 \cdot 32 = 192 \]

### Step 7: Use linear approximation
Using the linear approximation formula:
\[ f(1.999) \approx f(2) + f'(2) \cdot \Delta x \]
Substituting the values we have:
\[ f(1.999) \approx 64 + 192 \cdot (-0.001) \]
\[ f(1.999) \approx 64 - 0.192 \]
\[ f(1.999) \approx 63.808 \]

### Final Result
Thus, the approximate value of \( (1.999)^6 \) is:
\[ \boxed{63.808} \]