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 $-1.75$.

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!