Find the slope of the line tangent to f(x) at x=3. The graph of f(x) is shown below. 

Move the point on the curve to x=3. Then plot two points on the tangent line. Finally, calculate the slope of the tangent line at x=3.




To find the slope of the tangent line to the graph of a function \( f(x) \) at a specific point, follow these steps:

1. **Identify the point on the curve at \( x = 3 \)**:  
   You need to first determine the point where \( x = 3 \) on the graph of \( f(x) \). This gives you the coordinate \( (3, f(3)) \).

2. **Draw the tangent line at \( x = 3 \)**:  
   A tangent line is a straight line that touches the curve at just one point and has the same slope as the curve at that point. Once you identify the point at \( x = 3 \), draw a tangent line.

3. **Choose two points on the tangent line**:  
   Pick two points on the tangent line. Let’s say these points are \( (x_1, y_1) \) and \( (x_2, y_2) \).

4. **Calculate the slope**:  
   The slope of the tangent line is given by the formula for the slope between two points:
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1}
   \]
   Using the coordinates of the two points you selected on the tangent line, plug them into this formula to calculate the slope.

Once you provide an image of the graph or the coordinates of the points, I can help you calculate the exact slope of the tangent line at \( x = 3 \).

Would you like to upload an image of the graph, or do you have specific coordinates for the points?

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Here are 5 follow-up questions for better understanding:

1. How is the slope of a curve at a point related to the derivative of the function?
2. Can the slope of a tangent line be negative? Why or why not?
3. What does it mean for a function to have a horizontal tangent line at a point?
4. How does the concept of limits relate to finding the slope of a tangent line?
5. Why are tangent lines important in understanding the behavior of functions?

**Tip**: When selecting points to calculate slope, choose points that are easy to read on the graph to avoid calculation errors.

Find the slope of the line tangent to f(x) at x=3. The graph of f(x) is shown below. Move the point on the curve to x=3. Then plot two points on the tangent line. Finally, calculate the slope of the tangent line at x=3.

This problem helps students find the slope of a tangent line to the graph of f(x) at x=3. It covers key calculus concepts such as the slope formula and the definition of the derivative, providing a step-by-step solution for advanced high school or college students.

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