Using differentiation, approximate value of `f(x)=x^(2)-2x+1` at x=2.99 is . . .

Options

A. 3.96
B. 9.96
C. 4.98
D. 5.98

Explanation

To approximate the value of the function \( f(x) = x^2 - 2x + 1 \) at \( x = 2.99 \) using differentiation, we can follow these steps:

### Step 1: Identify the function and the point of approximation
We have the function:
\[ f(x) = x^2 - 2x + 1 \]
and we want to approximate it at \( x = 2.99 \).

### Step 2: Calculate the derivative of the function
The first step in using differentiation for approximation is to find the derivative of the function:
\[ f'(x) = \frac{d}{dx}(x^2 - 2x + 1) = 2x - 2 \]

### Step 3: Choose a point close to 2.99
Since \( 2.99 \) is very close to \( 3 \), we will use \( x = 3 \) as our point of approximation.

### Step 4: Calculate \( f(3) \)
Now we calculate the value of the function at \( x = 3 \):
\[ f(3) = 3^2 - 2 \cdot 3 + 1 = 9 - 6 + 1 = 4 \]

### Step 5: Calculate \( f'(3) \)
Next, we calculate the derivative at \( x = 3 \):
\[ f'(3) = 2 \cdot 3 - 2 = 6 - 2 = 4 \]

### Step 6: Determine the change in \( x \)
The change in \( x \) from \( 3 \) to \( 2.99 \) is:
\[ \Delta x = 2.99 - 3 = -0.01 \]

### Step 7: Use the linear approximation formula
The linear approximation formula is:
\[ f(x + \Delta x) \approx f(x) + f'(x) \cdot \Delta x \]
Substituting our values:
\[ f(2.99) \approx f(3) + f'(3) \cdot (-0.01) \]
\[ f(2.99) \approx 4 + 4 \cdot (-0.01) \]
\[ f(2.99) \approx 4 - 0.04 = 3.96 \]

### Final Result
Thus, the approximate value of \( f(2.99) \) is:
\[ \boxed{3.96} \]