could you explain why u derived the slope and explain what it does

Certainly! Let's break down why the slope (and hence its derivative) is important in this problem and how it relates to the tangent line.

### Understanding the Slope and its Role

#### 1. **What is Slope?**
The slope of a line is a measure of how steep the line is. Mathematically, for a straight line, the slope is defined as the "rise over run" – that is, how much the line goes up (or down) for a certain horizontal distance. If you have a line described by \( y = mx + c \), the slope is \( m \).

#### 2. **Tangent Line and the Curve**
In the context of a curve (a function that isn't necessarily a straight line), the slope isn't constant. It changes at different points on the curve. The **tangent line** at a specific point on the curve is the straight line that just touches the curve at that point and has the same slope as the curve at that point.

#### 3. **Why Derive the Slope of the Curve?**
The slope of the curve at a particular point can be found by taking the derivative of the curve's equation with respect to \( x \). The derivative, \( \frac{dy}{dx} \), gives us a formula that tells us the slope of the curve at any point \( x \).

When you're asked to find where a tangent line touches the curve, you're essentially being asked to find the point where the slope of the curve equals the slope of the tangent line.

### Application in This Problem

#### Given:
- Curve: \( y = \frac{2x^3}{3} - \frac{3x^2}{2} - 2 \)
- Tangent Line: \( 18x - 2y - 49 = 0 \), rearranged to \( y = 9x - \frac{49}{2} \), showing a slope \( m = 9 \).

#### Why We Derive the Slope:
To find the point \( P \) where the tangent touches the curve, we need the slope of the curve at \( P \) to match the slope of the tangent. 

1. **Derivative of the Curve**:
   \[
   \frac{dy}{dx} = 2x^2 - 3x
   \]
   This gives us the slope of the curve at any point \( x \).

2. **Set the Derivative Equal to the Tangent's Slope**:
   To find where the curve's slope equals the tangent's slope, we set \( \frac{dy}{dx} = 9 \).

   \[
   2x^2 - 3x = 9
   \]

   Solving this equation helps us find the specific \( x \)-coordinates where the curve's slope matches the slope of the tangent line.

### What Does the Slope Tell Us?

- **At a specific point on a curve**: The slope tells us the direction and steepness of the curve at that point. It is the slope of the tangent line to the curve at that point.
- **Matching Slope**: If a line is tangent to a curve at a point, the line must have the same slope as the curve at that point. This condition is why we set the derivative of the curve equal to the slope of the tangent line to find the correct point of tangency.

### Conclusion
By deriving the slope of the curve, we can determine where the curve and the tangent line have the same slope, which tells us exactly where the tangent line touches the curve. This point of intersection is critical to solving problems like the one you've posted.

Would you like to explore any specific part of this explanation further?

Here are 5 related questions that might interest you:
1. What is the geometric meaning of the derivative in a real-world context?
2. How would you find the equation of a tangent line if the point on the curve is given?
3. Can a tangent line intersect a curve at more than one point?
4. What happens to the slope of the curve as \( x \) approaches infinity?
5. How does the concavity of the curve affect the tangent line?

**Tip:** Understanding the derivative as the slope of the curve at a point can help you visualize how functions change and predict their behavior without graphing them.

Explore the concept of slope and tangent lines in calculus with detailed explanations and examples. Learn how to find the slope of a curve at a point, derive tangent lines, and apply these concepts to solve problems. Suitable for advanced high school students and calculus enthusiasts.

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