https://file.aiquickdraw.com/mathbot/file/1725698560008ccib2c1k.jpg

The expression shown in the image appears to be part of a limit problem, specifically related to the definition of a derivative. Here's the mathematical expression in the image:

\[
\lim_{\Delta x \to 0} \frac{(x + \Delta x)^2 - 7(x + \Delta x) + 2 - (x^2 - 7x + 2)}{\Delta x}
\]

This looks like the difference quotient used in finding the derivative of a quadratic function:

\[
f(x) = x^2 - 7x + 2
\]

The formula for the derivative using the difference quotient is:

\[
f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}
\]

In this case, \( f(x) = x^2 - 7x + 2 \), so you're calculating:

\[
f(x + \Delta x) = (x + \Delta x)^2 - 7(x + \Delta x) + 2
\]

The next step would be expanding and simplifying the numerator, then dividing by \( \Delta x \), and finally taking the limit as \( \Delta x \to 0 \).

Would you like me to go through the full solution step by step?

Here are 5 related questions to explore:
1. What is the purpose of the difference quotient in calculus?
2. How do you expand binomials like \( (x + \Delta x)^2 \)?
3. Why does \( \Delta x \) approach 0 in the definition of a derivative?
4. How do you simplify expressions with higher-order terms of \( \Delta x \)?
5. Can we use this method to find the derivative of any polynomial?

**Tip:** Always check your algebraic expansions and cancellations carefully when working with limits and derivatives.

lim(Δx→0) [(x + Δx)^2 - 7(x + Δx) + 2 - (x^2 - 7x + 2)] / Δx

Learn how to calculate the derivative of a quadratic function using the difference quotient method. This detailed explanation walks through the process of expanding and simplifying the expression for f(x) = x^2 - 7x + 2, applying limits, and finding the derivative step by step. Suitable for calculus students in Grades 11-12.

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