Question 1: Plot the graph of y=2x on the grid. Using a tangent drawn at the point where x=1, estimate the gradient of the curve at that point by calculating the slope of the tangent.

Answer: 2

Question 2: On the curve y=x^2, draw a tangent at the point where y=4. Estimate the gradient at that point by calculating the slope of the tangent.

Answer: 4

Question 3: Using the curve y=3x+1, draw a tangent at x=2. Estimate the gradient of the tangent and compare it to the actual gradient of the line.

Answer: 3

Question 4: Construct a triangle with sides 3cm, 4cm, and 5cm on the grid. Use this to estimate the gradient of a tangent to the curve y=√x at x=4 by drawing a suitable tangent.

Answer: 0.5

Question 5: On the graph of y=log(x), draw a tangent at x=1. Estimate the gradient of this tangent by calculating the slope between two points close to x=1.

Answer: 0

Question 6: The curve y=sin(x) is plotted. At x=π/2, draw a tangent and estimate the gradient using the points at x=π/2±0.1. Show your working.

Answer: 0

Question 7: A curve y=f(x) is given by y=x^3 - 3x. At x=2, estimate the gradient of the tangent by drawing and calculating between two points near x=2.

Answer: 9

Question 8: Explain the potential mistake if a student estimates the gradient of a curve by choosing two points that are not close together. How does this affect accuracy?

Answer: Using points that are far apart can give a less accurate estimate of the tangent's gradient because the slope between distant points may not reflect the instantaneous rate of change at a point.

Question 9: The curve y=1/x is plotted. Draw a tangent at x=1 and estimate its gradient using points at x=0.8 and x=1.2. Show your working.

Answer: -1

Question 10: Challenge: For the curve y=x^4, estimate the gradient at x=1 by drawing a tangent and calculating the slope between points at x=0.9 and x=1.1. How does this compare to the actual derivative at x=1?

Answer: 4