If `f(x)=3x^(2)+5x+2`, then the approximate value `f(5.02)` is

Options

A. 47.66
B. 57.66
C. 67.66
D. 102.76

Answer

To find the approximate value of \( f(5.02) \) for the function \( f(x) = 3x^2 + 5x + 2 \), we can use the concept of derivatives and linear approximation. Here’s the step-by-step solution:

### Step 1: Identify the function and the point of approximation
Given:
\[ f(x) = 3x^2 + 5x + 2 \]
We want to approximate \( f(5.02) \). We will let \( x = 5 \) and \( \Delta x = 0.02 \).

### Step 2: Calculate the derivative of the function
We need to find the derivative \( f'(x) \):
\[ f'(x) = \frac{d}{dx}(3x^2 + 5x + 2) = 6x + 5 \]

### Step 3: Evaluate the derivative at the point \( x = 5 \)
Now, we evaluate the derivative at \( x = 5 \):
\[ f'(5) = 6(5) + 5 = 30 + 5 = 35 \]

### Step 4: Use the linear approximation formula
The linear approximation formula is:
\[ f(x + \Delta x) \approx f(x) + f'(x) \Delta x \]
Substituting \( x = 5 \) and \( \Delta x = 0.02 \):
\[ f(5.02) \approx f(5) + f'(5) \cdot 0.02 \]

### Step 5: Calculate \( f(5) \)
Now, we need to calculate \( f(5) \):
\[ f(5) = 3(5^2) + 5(5) + 2 = 3(25) + 25 + 2 = 75 + 25 + 2 = 102 \]

### Step 6: Substitute values into the linear approximation formula
Now we can substitute \( f(5) \) and \( f'(5) \) into the approximation:
\[ f(5.02) \approx 102 + 35 \cdot 0.02 \]
Calculating \( 35 \cdot 0.02 \):
\[ 35 \cdot 0.02 = 0.7 \]
Thus,
\[ f(5.02) \approx 102 + 0.7 = 102.7 \]

### Final Answer
The approximate value of \( f(5.02) \) is:
\[ \boxed{102.7} \]