Question 1: Calculate the gradient of the curve y = x^2 at the point where x = 2 using a small interval of 0.1 around x = 2.

Answer: 3.2

Question 2: Estimate the gradient at x = 1.5 on the curve y = 2x + 1 by drawing a tangent on the grid.

Answer: 2

Question 3: Using data points (1, 3), (2, 5), and (3, 7), estimate the gradient of the curve at x = 2.

Answer: 2

Question 4: On the graph provided, draw a tangent to the curve y = x^3 at x = 1 and estimate its gradient.

Answer: 3

Question 5: Explain why the gradient estimate between x = 1.9 and x = 2.1 for y = x^2 is close to the derivative at x = 2.

Answer: Because the points are close together, the average gradient approximates the instantaneous gradient at x=2.

Question 6: Estimate the gradient of y = √x at x = 4 by calculating the difference in y over the difference in x for points x=3.9 and x=4.1.

Answer: 0.5

Question 7: A car's speed is recorded at two points: 60 km/h at 2 seconds and 70 km/h at 4 seconds. Estimate the average acceleration between these times.

Answer: 5 km/h per second

Question 8: Construct a tangent to the curve y = x^3 at x = 2 on your grid and estimate its gradient.

Answer: 12

Question 9: Identify and correct the mistake in estimating the gradient of y = x^2 at x = 3 using points x=2.9 and x=3.1.

Answer: Using points too far apart can lead to inaccurate estimates; smaller intervals give better approximations.

Question 10: For the curve y = 1/x, estimate the gradient at x = 1 by calculating the difference quotient between x=0.9 and x=1.1.

Answer: -1

Question 11: Compare the gradient estimates obtained at x=2 for y = x^2 using intervals of 0.2 and 0.05. Which estimate is likely more accurate and why?

Answer: The estimate with interval 0.05 is more accurate because it approximates the derivative more closely.