The approximate value of `(1.0002)^3000`, is

Options

A. `1.2`
B. `1.4`
C. `1.6`
D. `1.8`

Explanation

To find the approximate value of \( (1.0002)^{3000} \), we can use the concept of differentials. Here’s a step-by-step solution:

### Step 1: Define the function
Let \( y = x^{3000} \). We want to evaluate \( y \) at \( x = 1.0002 \).

### Step 2: Identify the small change
We can express \( x \) as:
\[ x = 1 + dx \]
where \( dx = 0.0002 \). Thus, we have:
\[ x = 1 + 0.0002 \]

### Step 3: Calculate \( y \) at \( x = 1 \)
Now, calculate \( y \) when \( x = 1 \):
\[ y = 1^{3000} = 1 \]

### Step 4: Differentiate the function
Next, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = 3000x^{2999} \]

### Step 5: Evaluate the derivative at \( x = 1 \)
Now, substitute \( x = 1 \) into the derivative:
\[ \frac{dy}{dx} \bigg|_{x=1} = 3000 \cdot 1^{2999} = 3000 \]

### Step 6: Calculate \( dy \)
Using the differential \( dy \):
\[ dy = \frac{dy}{dx} \cdot dx = 3000 \cdot 0.0002 \]
Calculating this gives:
\[ dy = 3000 \cdot 0.0002 = 0.6 \]

### Step 7: Find \( y + dy \)
Now, we can find \( y + dy \):
\[ y + dy = 1 + 0.6 = 1.6 \]

### Conclusion
Thus, the approximate value of \( (1.0002)^{3000} \) is:
\[ \boxed{1.6} \]

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