https://file.aiquickdraw.com/mathbot/file/1725694772318ux9to80r.jpg

The given image depicts the difference quotient expression used in calculus, which is a core step in deriving the derivative of a function using limits. Let's break it down:

\[
\lim_{\Delta x \to 0} \frac{(5 + \Delta x)^2 - 3(2 + \Delta x) + 9 - (5^2 - 3(2) + 9)}{\Delta x}
\]

To clarify, this limit formula is designed to compute the derivative of a given function \( f(x) \) at a specific point by using the difference quotient.

Here's a step-by-step breakdown of how to simplify and solve this expression:

### Step 1: Expand the terms inside the numerator
1. Expand \( (5 + \Delta x)^2 \) using the binomial formula:
   \[
   (5 + \Delta x)^2 = 25 + 10\Delta x + (\Delta x)^2
   \]

2. Distribute \( -3 \) over \( (2 + \Delta x) \):
   \[
   -3(2 + \Delta x) = -6 - 3\Delta x
   \]

### Step 2: Substitute expanded forms back into the difference quotient
\[
\frac{(25 + 10\Delta x + (\Delta x)^2) - (6 + 3\Delta x) + 9 - (25 - 6 + 9)}{\Delta x}
\]

### Step 3: Simplify the constant terms
The expression \( 25 - 6 + 9 \) results in:
\[
25 - 6 + 9 = 28
\]

Thus, the numerator becomes:
\[
(25 + 10\Delta x + (\Delta x)^2) - (6 + 3\Delta x) + 9 - 28
\]

### Step 4: Combine like terms
Simplifying the constants and combining terms involving \( \Delta x \):
\[
(25 - 6 + 9 - 28) + (10\Delta x - 3\Delta x) + (\Delta x)^2 = 0 + 7\Delta x + (\Delta x)^2
\]

So, the expression becomes:
\[
\frac{7\Delta x + (\Delta x)^2}{\Delta x}
\]

### Step 5: Simplify the expression by canceling \( \Delta x \)
Factor \( \Delta x \) from the numerator:
\[
\frac{\Delta x(7 + \Delta x)}{\Delta x}
\]

Cancel out the \( \Delta x \):
\[
7 + \Delta x
\]

### Step 6: Take the limit as \( \Delta x \to 0 \)
As \( \Delta x \to 0 \), the expression becomes:
\[
7
\]

Thus, the derivative (or the value of the limit) is \( 7 \).

### Final Answer:
\[
\lim_{\Delta x \to 0} \frac{(5 + \Delta x)^2 - 3(2 + \Delta x) + 9 - (25 - 6 + 9)}{\Delta x} = 7
\]

Would you like more details on any specific steps or have any other questions?

Here are some related questions for further practice:
1. How do you find the derivative of a quadratic function in general?
2. Can the difference quotient method be applied to any function?
3. How would you apply this method to the function \( f(x) = 3x^2 - 5x + 2 \)?
4. What does the derivative represent geometrically?
5. How do you interpret the limit in terms of instantaneous rate of change?

**Tip:** Always simplify and group like terms before applying the limit to make the calculations smoother.

The limit of the difference quotient expression to find the derivative of a function as \( \Delta x \to 0 \): \( \lim_{\Delta x \to 0} \frac{(5 + \Delta x)^2 - 3(2 + \Delta x) + 9 - (25 - 6 + 9)}{\Delta x} \)

Learn how to calculate the derivative of a function using the difference quotient and limits. This step-by-step solution covers key calculus concepts including the expansion of binomials, simplifying expressions, and applying the limit as \( \Delta x \to 0 \). Ideal for high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation