Find the approximate value of :
`f(3.02)," where "f(x)=3x^(2)+15x+3`

Answer

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

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

\[ f(3) = 3(3^2) + 15(3) + 3 \]
Calculating this step-by-step:
- \( 3^2 = 9 \)
- \( 3 \times 9 = 27 \)
- \( 15 \times 3 = 45 \)

Now, adding these values together:
\[ f(3) = 27 + 45 + 3 = 75 \]

### Step 2: Find \( \frac{dy}{dx} \)
Next, we need to differentiate \( f(x) \) with respect to \( x \) to find \( \frac{dy}{dx} \).

\[ f'(x) = \frac{d}{dx}(3x^2 + 15x + 3) \]
Using the power rule:
- The derivative of \( 3x^2 \) is \( 6x \)
- The derivative of \( 15x \) is \( 15 \)
- The derivative of a constant (3) is \( 0 \)

Thus,
\[ f'(x) = 6x + 15 \]

### Step 3: Calculate \( \frac{dy}{dx} \) at \( x = 3 \)
Now, we will evaluate \( f'(x) \) at \( x = 3 \):

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

### Step 4: Calculate \( dy \)
We will now find \( dy \) using the formula \( dy = f'(x) \cdot dx \), where \( dx = 0.02 \) (since \( 3.02 = 3 + 0.02 \)).

\[ dy = f'(3) \cdot dx = 33 \cdot 0.02 \]
Calculating this:
\[ dy = 0.66 \]

### Step 5: Find \( f(3.02) \)
Finally, we can find \( f(3.02) \) using the approximation:

\[ f(3.02) \approx f(3) + dy \]
Substituting the values we found:
\[ f(3.02) \approx 75 + 0.66 = 75.66 \]

Thus, the approximate value of \( f(3.02) \) is \( \boxed{75.66} \).

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