If `f(x) = 3x^(2) + 15x+5`, then the approximate value of `f(3.02)` is :

Options

A. `47.66`
B. `57.66`
C. `67.66`
D. `77.66`

Explanation

To find the approximate value of \( f(3.02) \) for the function \( f(x) = 3x^2 + 15x + 5 \), we can use the concept of derivatives. Here’s a step-by-step solution:

### Step 1: Calculate \( f(3) \)
First, we need to find the value of the function at \( x = 3 \).

\[ f(3) = 3(3^2) + 15(3) + 5 \]
Calculating each term:
\[ = 3(9) + 45 + 5 \]
\[ = 27 + 45 + 5 = 77 \]

### Step 2: Calculate \( f'(x) \)
Next, we find the derivative of \( f(x) \).

\[ f'(x) = \frac{d}{dx}(3x^2 + 15x + 5) \]
Using the power rule:
\[ f'(x) = 6x + 15 \]

### Step 3: Calculate \( f'(3) \)
Now, we evaluate the derivative at \( x = 3 \).

\[ f'(3) = 6(3) + 15 \]
Calculating:
\[ = 18 + 15 = 33 \]

### Step 4: Use the linear approximation formula
We can approximate \( f(3.02) \) using the formula:

\[ f(x + h) \approx f(x) + h \cdot f'(x) \]

Here, \( x = 3 \) and \( h = 0.02 \).

Substituting the values we found:
\[ f(3.02) \approx f(3) + 0.02 \cdot f'(3) \]
\[ = 77 + 0.02 \cdot 33 \]

### Step 5: Calculate the final approximation
Now, calculate \( 0.02 \cdot 33 \):
\[ 0.02 \cdot 33 = 0.66 \]
Adding this to \( f(3) \):
\[ f(3.02) \approx 77 + 0.66 = 77.66 \]

### Final Answer
Thus, the approximate value of \( f(3.02) \) is \( \boxed{77.66} \).

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