https://file.aiquickdraw.com/mathbot/file/1726090928688qthzlz10.jpg

Let's solve the given problem by examining the graph and the lines A, B, and C.

**Part (a): Which of the lines A, B, or C shows the best estimate for the tangent to the curve at \(x = 4\)?**

Looking at the graph, we need to determine which line is closest to being a tangent at the point where \(x = 4\) on the curve. A tangent line should touch the curve at \(x = 4\) and have the same slope as the curve at that point.

- **Line A** (magenta): This line intersects the curve but does not touch it smoothly at \(x = 4\). It is steeper than the curve's slope at that point.
- **Line B** (blue): This line is above the curve and does not match the slope at \(x = 4\); it is less steep than the curve's actual slope.
- **Line C** (orange): This line touches the curve at \(x = 4\) and appears to follow the curve's slope exactly.

Therefore, **Line C** is the best estimate for the tangent to the curve at \(x = 4\).

**Part (b): Estimate the gradient of the curve at \(x = 4\) using your answer to part (a).**

Since Line C is the best estimate of the tangent line at \(x = 4\), we can calculate its gradient to estimate the gradient of the curve at that point.

To find the gradient (slope) of Line C, we use the formula:

\[
\text{Gradient} = \frac{\text{Change in } y}{\text{Change in } x} = \frac{y_2 - y_1}{x_2 - x_1}
\]

From the graph, we can pick two points on Line C. For example:
- Point 1: \((4, 7)\)
- Point 2: \((8, 0)\)

Now, calculate the gradient:

\[
\text{Gradient} = \frac{0 - 7}{8 - 4} = \frac{-7}{4} = -1.75
\]

Therefore, the estimated gradient of the curve at \(x = 4\) is \(-1.75\).

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

Here are some related questions that might be useful:
1. What is the importance of finding the gradient of a curve at a specific point?
2. How do you differentiate between tangent and secant lines?
3. How can we improve our estimate of the gradient using calculus?
4. What are the implications of a negative gradient on a curve?
5. How can tangents be used in real-world applications?

**Tip:** When finding a tangent line's gradient, ensure you select points that lie exactly on the line for accurate calculations!

a) Which of the lines A, B or C shows the best estimate for the tangent to the curve at x = 4?
b) Use your answer to part a) to estimate the gradient of the curve at x = 4.

Understand how to estimate the tangent line to a curve at x = 4 and calculate the gradient using a graphical approach. This problem explores key concepts in graph interpretation, including the properties of tangent lines and slope calculations. Suitable for students in Grades 9-10.

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