https://file.aiquickdraw.com/mathbot/file/172653846339624ampvti.jpg

The problem asks to find the slope of the tangent line to the function \( f(x) \) at \( x = 3 \), based on the graph provided.

### Step-by-step solution:

1. **Identify the point of tangency**:
   The point where \( x = 3 \) is marked on the curve. From the graph, you can visually estimate the corresponding \( y \)-coordinate.

2. **Draw the tangent line**:
   A tangent line is a straight line that touches the curve at exactly one point. By placing a point at \( x = 3 \) on the graph and visualizing the tangent, we can observe the line’s steepness or direction.

3. **Calculate the slope of the tangent line**:
   The slope of a tangent line is equivalent to the derivative of the function at that point, \( f'(3) \). From the graph, you can estimate two points on the tangent line (e.g., \( (3, y_1) \) and another point \( (x_2, y_2) \) on the line).

   The slope \( m \) is calculated using:
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1}
   \]
   where \( (x_1, y_1) \) and \( (x_2, y_2) \) are points on the tangent line.

   From the graph, it appears that the slope is negative (the tangent line slopes downward as \( x \) increases). Using visual estimation, you can approximate the values and calculate the slope.

Would you like more details or further clarification on any steps?

### Expanding concepts:
1. How do we calculate the slope of a line if given two points?
2. What is the significance of a tangent line in calculus?
3. How do you find the derivative of a function at a specific point?
4. Can the slope of the tangent line be different from the slope of the secant line?
5. What are the applications of tangent lines in real-world problems?

### Tip:
Remember that the slope of the tangent line can be approximated graphically, but for an exact value, you would typically need the function \( f(x) \) and its derivative.

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.

Learn how to find the slope of the tangent line to the function f(x) at x = 3 using a graphical method. This problem covers key calculus concepts such as the tangent line and slope calculation, and is suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation