Find the approximate value of f(2.01) where `f(x) =x^(3)-4x+7`.

Answer

To find the approximate value of \( f(2.01) \) where \( f(x) = x^3 - 4x + 7 \), we can use the concept of derivatives for approximation. However, since the video solution provided calculates the value directly, we will follow that method step by step.

### Step 1: Substitute \( x = 2.01 \) into the function
We start by calculating \( f(2.01) \):
\[ f(2.01) = (2.01)^3 - 4(2.01) + 7 \]

### Step 2: Calculate \( (2.01)^3 \)
To find \( (2.01)^3 \):
\[ (2.01)^3 = 2.01 \times 2.01 \times 2.01 \]
Calculating \( 2.01 \times 2.01 \):
\[ 2.01 \times 2.01 = 4.0401 \]
Now multiplying by \( 2.01 \):
\[ 4.0401 \times 2.01 \approx 8.120601 \]
For approximation, we can round this to \( 8.12 \).

### Step 3: Calculate \( -4(2.01) \)
Now we calculate \( -4(2.01) \):
\[ -4(2.01) = -8.04 \]

### Step 4: Combine the results
Now we combine all parts:
\[ f(2.01) = 8.12 - 8.04 + 7 \]
Calculating this step by step:
\[ 8.12 - 8.04 = 0.08 \]
Adding \( 7 \):
\[ 0.08 + 7 = 7.08 \]

### Final Answer
Thus, the approximate value of \( f(2.01) \) is:
\[ \boxed{7.08} \]