Let's break down the problem in the image you've shared:

Part (a) - Best Estimate for Tangent to the Curve at $x = 5$

To determine which of the lines A, B, or C best represents the tangent to the curve at $x = 5$, we need to recall that the tangent line touches the curve exactly at the point of tangency, without crossing it. The slope of the tangent line will represent the instantaneous rate of change (or gradient) at that point.

• Line A appears to have a steep negative slope.
• Line B appears to have a positive slope but isn't as steep as line A.
• Line C appears to have a gentle slope and may intersect the curve at the point $x = 5$.

Based on the graph, line B seems to be the best estimate for the tangent line because it appears to touch the curve without crossing it at $x = 5$, matching the expected behavior of a tangent.

Part (b) - Estimating the Gradient of the Curve at $x = 5$

The gradient of the tangent line at $x = 5$ is the slope of the line you identified in part (a). From the graph, line B is the best estimate for the tangent. To estimate the gradient, we need to calculate the slope of line B.

To find the slope of a line, we use the formula:

$\text{Slope} = \frac{\Delta y}{\Delta x} = \frac{\text{change in } y}{\text{change in } x}$

From the graph, you can select two points on line B and calculate the slope between them. For example, you might choose points where the line crosses the gridlines.

Would you like to proceed with specific points to calculate the slope of line B? Feel free to ask for more clarification or details!