To estimate the gradient of the curve at point $A$ using the tangent, we will calculate the gradient of the tangent line drawn at point $A$. The gradient of a straight line is given by the formula:

$\text{Gradient (m)} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$

From the graph:

• The tangent line passes through points approximately at $(5, 2)$ and $(14, 9)$.

Now, I'll calculate the gradient:

$\text{Gradient} = \frac{9 - 2}{14 - 5} = \frac{7}{9} \approx 0.78$

So, the estimated gradient of the curve at point $A$ is 0.8 to 1 decimal place.

Would you like further clarification or additional details?

Here are 5 related questions to extend your understanding:

1. How does the tangent line relate to the derivative of the function at a point?
2. What does a positive gradient indicate about the curve's behavior at point A?
3. How would the gradient change if the curve were steeper?
4. Can the gradient of the tangent line be negative? What would that signify?
5. How do we estimate gradients at points where no tangent is provided?

Tip: The steeper the tangent line, the higher the gradient. If a tangent line is horizontal, the gradient is zero.