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To estimate the gradient of the curve at \( x = 5 \) using the tangent line, we need to determine the slope of the tangent line drawn at that point. 

Here's a step-by-step breakdown of how to find this slope:

1. **Identify Points on the Tangent Line:** Locate two points on the tangent line that are as precise as possible. Using the graph, find two points with clear coordinates. Let's denote these points as \( (x_1, y_1) \) and \( (x_2, y_2) \).

2. **Calculate the Gradient (Slope):** The gradient of the tangent line is given by the formula:
   \[
   \text{Gradient} = \frac{y_2 - y_1}{x_2 - x_1}
   \]
   Substitute the coordinates of the two points into this formula to get an estimate for the gradient.

3. **Example Calculation:**
   From the graph, it appears that:
   - A possible point close to \( x = 3 \) on the tangent line is around \( (3, 2) \).
   - Another point on the tangent line around \( x = 7 \) is approximately \( (7, 8) \).

   Using these points:
   \[
   \text{Gradient} = \frac{8 - 2}{7 - 3} = \frac{6}{4} = 1.5
   \]

So, an estimate for the gradient of the curve at \( x = 5 \) is approximately \( 1.5 \).

Would you like more details on this method or have any other questions?

Here are five related questions you may find useful:
1. How can the accuracy of estimating the gradient from a graph be improved?
2. What is the significance of the gradient at a specific point on a curve?
3. How does this graphical method compare to using calculus for finding the gradient?
4. What other methods can be used to find tangents if the function is unknown?
5. How does the tangent line represent the instantaneous rate of change?

**Tip:** When estimating slopes from graphs, use points that are far apart along the line to reduce error in your gradient calculation.

The diagram shows the graph of y = f(x) and the tangent drawn to the graph at x = 5. Use the tangent to work out an estimate for the gradient of the curve at x = 5.

Estimate the gradient of a curve at x = 5 using the tangent line. This solution covers key calculus concepts like derivatives and tangent slopes. Ideal for students in Grades 10-12.

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