If `f(x)=3x^(2)+4x-1`, then find the approximate value of `f(3.1)`.

Answer

To find the approximate value of \( f(3.1) \) for the function \( f(x) = 3x^2 + 4x - 1 \), we can use the concept of linear approximation. Here’s a step-by-step solution:

### Step 1: Identify the function and the point of approximation
We have:
\[ f(x) = 3x^2 + 4x - 1 \]
We want to approximate \( f(3.1) \). We can express this as:
\[ f(3.1) \approx f(3) + f'(3) \cdot (3.1 - 3) \]
where \( (3.1 - 3) = 0.1 \).

### Step 2: Calculate \( f(3) \)
First, we need to find \( f(3) \):
\[ f(3) = 3(3^2) + 4(3) - 1 \]
Calculating this step-by-step:
\[ = 3(9) + 12 - 1 \]
\[ = 27 + 12 - 1 \]
\[ = 39 - 1 = 38 \]

### Step 3: Calculate the derivative \( f'(x) \)
Next, we need to find the derivative \( f'(x) \):
\[ f'(x) = \frac{d}{dx}(3x^2 + 4x - 1) = 6x + 4 \]

### Step 4: Calculate \( f'(3) \)
Now, we evaluate the derivative at \( x = 3 \):
\[ f'(3) = 6(3) + 4 \]
Calculating this:
\[ = 18 + 4 = 22 \]

### Step 5: Calculate \( f(3.1) \) using linear approximation
Now we can substitute back into our linear approximation formula:
\[ f(3.1) \approx f(3) + f'(3) \cdot (0.1) \]
\[ = 38 + 22 \cdot 0.1 \]
Calculating this:
\[ = 38 + 2.2 = 40.2 \]

### Final Answer
Thus, the approximate value of \( f(3.1) \) is:
\[ \boxed{40.2} \]