Using differentiation, find the approximate value of `f(3.01)`, where `f(x)=4x^(2)+5x+2.`

Answer

To find the approximate value of \( f(3.01) \) for the function \( f(x) = 4x^2 + 5x + 2 \) using differentiation, we can follow these steps:

### Step 1: Identify the function and the point of interest
We have the function:
\[ f(x) = 4x^2 + 5x + 2 \]
We want to find \( f(3.01) \). We will use \( x = 3 \) as our base point and \( dx = 0.01 \).

### Step 2: Calculate \( f(3) \)
First, we need to find the value of the function at \( x = 3 \):
\[ f(3) = 4(3)^2 + 5(3) + 2 \]
Calculating this step-by-step:
\[ = 4(9) + 15 + 2 \]
\[ = 36 + 15 + 2 \]
\[ = 53 \]
So, \( f(3) = 53 \).

### Step 3: Differentiate the function
Next, we differentiate \( f(x) \) to find \( f'(x) \):
\[ f'(x) = \frac{d}{dx}(4x^2 + 5x + 2) \]
Calculating the derivative:
\[ = 8x + 5 \]

### Step 4: Evaluate the derivative at \( x = 3 \)
Now, we substitute \( x = 3 \) into the derivative:
\[ f'(3) = 8(3) + 5 \]
Calculating this:
\[ = 24 + 5 = 29 \]

### Step 5: Calculate \( dy \)
Now, we can find \( dy \) using the formula:
\[ dy = f'(x) \cdot dx \]
Substituting \( f'(3) \) and \( dx = 0.01 \):
\[ dy = 29 \cdot 0.01 = 0.29 \]

### Step 6: Approximate \( f(3.01) \)
Finally, we can approximate \( f(3.01) \) using:
\[ f(3.01) \approx f(3) + dy \]
Substituting the values we found:
\[ f(3.01) \approx 53 + 0.29 = 53.29 \]

### Final Answer
Thus, the approximate value of \( f(3.01) \) is:
\[ \boxed{53.29} \]