Let's work through the problem step-by-step.

Given:

$f(x) = 2x^2$

We need to simplify the expression for: $\frac{f(1 + \Delta x) - f(1)}{\Delta x}$

Part (A): Simplify $\frac{f(1 + \Delta x) - f(1)}{\Delta x}$

1. First, calculate $f(1 + \Delta x)$: $f(1 + \Delta x) = 2(1 + \Delta x)^2$ Expand $(1 + \Delta x)^2$: $(1 + \Delta x)^2 = 1 + 2\Delta x + (\Delta x)^2$ Now substitute this back into the expression for $f(1 + \Delta x)$: $f(1 + \Delta x) = 2(1 + 2\Delta x + (\Delta x)^2) = 2 + 4\Delta x + 2(\Delta x)^2$

2. Now calculate $f(1)$: $f(1) = 2(1)^2 = 2$

3. Substitute both $f(1 + \Delta x)$ and $f(1)$ into the given difference quotient: $\frac{f(1 + \Delta x) - f(1)}{\Delta x} = \frac{(2 + 4\Delta x + 2(\Delta x)^2) - 2}{\Delta x}$ Simplify the numerator: $\frac{(2 + 4\Delta x + 2(\Delta x)^2) - 2}{\Delta x} = \frac{4\Delta x + 2(\Delta x)^2}{\Delta x}$ Factor out $\Delta x$ from the numerator: $= \frac{\Delta x(4 + 2\Delta x)}{\Delta x}$ Cancel the $\Delta x$ terms: $= 4 + 2\Delta x$

Thus, the simplified expression is: $\frac{f(1 + \Delta x) - f(1)}{\Delta x} = 4 + 2\Delta x$

Part (B): What does the quantity approach as $\Delta x \to 0$?

As $\Delta x \to 0$, the term $2\Delta x$ in the expression $4 + 2\Delta x$ approaches 0. Therefore, the expression approaches: $4$

This means that the limit of the difference quotient as $\Delta x \to 0$ is 4. This is also the derivative of the function $f(x) = 2x^2$ evaluated at $x = 1$.

Would you like further details or explanations on any step?

Here are 5 questions to expand this topic:

1. How do we use the limit definition of the derivative in calculus?
2. Why does canceling $\Delta x$ in the difference quotient work when simplifying?
3. What is the significance of the result in part (B) for the derivative at $x = 1$?
4. How can you generalize the derivative for $f(x) = 2x^2$ at any point $x$?
5. What are some real-world applications of finding derivatives like this?

Tip:

The difference quotient is the foundation for understanding derivatives. Mastering this concept will make calculus much more intuitive!