Find the approximate value of `f(3.02)`, where `f(x)=3x^2+5x+3`.

Options

A. `45.46`
B. `37.46`
C. `27.56`
D. `39.40`

Explanation

To find the approximate value of \( f(3.02) \) where \( f(x) = 3x^2 + 5x + 3 \), we can use the concept of derivatives. Here’s a step-by-step solution:

### Step 1: Identify the function and the point of approximation
Let \( f(x) = 3x^2 + 5x + 3 \). We want to approximate \( f(3.02) \).

### Step 2: Choose a point close to 3.02
We will take \( x = 3 \) as our point of approximation.

### Step 3: Calculate \( \Delta x \)
Since we are approximating \( f(3.02) \), we calculate:
\[ \Delta x = 3.02 - 3 = 0.02 \]

### Step 4: Calculate \( f(3) \)
Now, we need to find \( f(3) \):
\[ f(3) = 3(3^2) + 5(3) + 3 \]
Calculating this:
\[ = 3(9) + 15 + 3 = 27 + 15 + 3 = 45 \]

### Step 5: Find the derivative \( f'(x) \)
Next, we find the derivative of \( f(x) \):
\[ f'(x) = \frac{d}{dx}(3x^2 + 5x + 3) = 6x + 5 \]

### Step 6: Evaluate the derivative at \( x = 3 \)
Now, we evaluate \( f'(3) \):
\[ f'(3) = 6(3) + 5 = 18 + 5 = 23 \]

### Step 7: Calculate \( \Delta y \)
Using the derivative, we can find the change in \( y \) (denoted as \( \Delta y \)):
\[ \Delta y = f'(3) \cdot \Delta x = 23 \cdot 0.02 = 0.46 \]

### Step 8: Approximate \( f(3.02) \)
Finally, we can approximate \( f(3.02) \):
\[ f(3.02) \approx f(3) + \Delta y = 45 + 0.46 = 45.46 \]

### Conclusion
Thus, the approximate value of \( f(3.02) \) is:
\[ \boxed{45.46} \]