The approximate value of ` f(X) = x^(3) +5x^(2)-7x+9` at x= 1.1 is

Options

A. `8.6`
B. `8.5`
C. `8.4`
D. `8.3`

Answer

To find the approximate value of the function \( f(x) = x^3 + 5x^2 - 7x + 9 \) at \( x = 1.1 \), we can use the method of linear approximation. Here’s the step-by-step solution:

### Step 1: Identify the function and the point of approximation
We have:
\[ f(x) = x^3 + 5x^2 - 7x + 9 \]
We want to approximate \( f(1.1) \).

### Step 2: Choose a point close to 1.1
We can choose \( x = 1 \) as a point close to \( 1.1 \).

### Step 3: Calculate \( f(1) \)
Now, we calculate \( f(1) \):
\[ f(1) = 1^3 + 5(1^2) - 7(1) + 9 \]
\[ = 1 + 5 - 7 + 9 \]
\[ = 8 \]

### Step 4: Calculate the derivative \( f'(x) \)
Next, we find the derivative \( f'(x) \):
\[ f'(x) = \frac{d}{dx}(x^3 + 5x^2 - 7x + 9) \]
\[ = 3x^2 + 10x - 7 \]

### Step 5: Evaluate the derivative at \( x = 1 \)
Now, we evaluate \( f'(1) \):
\[ f'(1) = 3(1^2) + 10(1) - 7 \]
\[ = 3 + 10 - 7 \]
\[ = 6 \]

### Step 6: Calculate \( \Delta x \)
We have:
\[ \Delta x = 1.1 - 1 = 0.1 \]

### Step 7: Calculate \( \Delta y \)
Using the linear approximation formula:
\[ \Delta y = f'(1) \cdot \Delta x \]
\[ \Delta y = 6 \cdot 0.1 = 0.6 \]

### Step 8: Calculate the approximate value of \( f(1.1) \)
Now we can approximate \( f(1.1) \):
\[ f(1.1) \approx f(1) + \Delta y \]
\[ f(1.1) \approx 8 + 0.6 = 8.6 \]

### Conclusion
The approximate value of \( f(1.1) \) is \( 8.6 \).

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