Question

If \( f(x) = x^2 \), find \( \frac{f(1.1) - f(1)}{1.1 - 1} \).

Hint:

The expression \( \frac{f(1.1) - f(1)}{1.1 - 1} \) is used to approximate the derivative of the function at \( x = 1 \).

Answer 1

Step 1: Define the function.
We are given the function: \[ f(x) = x^2 \]
Step 2: Calculate \( f(1.1) \) and \( f(1) \).
We have: \[ f(1.1) = (1.1)^2 = 1.21, \quad f(1) = (1)^2 = 1 \]
Step 3: Compute the expression.
Now, substitute into the given expression: \[ \frac{f(1.1) - f(1)}{1.1 - 1} = \frac{1.21 - 1}{0.1} = \frac{0.21}{0.1} = 2.1 \]
Step 4: Conclusion.
The value of the expression is: \[ \boxed{2.1} \]