Question 1: Construct the tangent to the curve y = x^2 at the point where x = 2. Show all working.

Answer: Tangent line passing through (2,4) with slope 4, y - 4 = 4(x - 2)

Question 2: Plot the graph of y = 3x + 1 and draw the tangent to the curve at x = 1.

Answer: Line y = 4x - 2 passing through (1,4) with slope 3

Question 3: Explain why a tangent line at a point on a curve touches but does not cross the curve at that point.

Answer: Because the tangent has the same slope as the curve at that point, it only touches and does not cross there.

Question 4: Construct the tangent to y = x^3 at x = 1. Show your steps.

Answer: Slope = 3x^2 at x=1, so slope=3, tangent y - 1=3(x - 1)

Question 1: Given y = 2x^2 + 1, construct the tangent line at x = 2. Justify your construction process.

Answer: Calculate slope = 8, point (2, 9). Draw the tangent through (2,9) with slope 8.

Question 2: Describe how the accuracy of your tangent line drawing can be checked using the derivative of the curve.

Answer: Compare the slope of the tangent line with the derivative at the point; they should be equal.

Question 3: Construct the tangent to y = x^2 - 4x + 3 at its minimum point. Show all steps and reasoning.

Answer: Find vertex at x=2, slope=0, tangent y -1=0(x-2) or y=1.

Question 1: A roller coaster follows a curve described by y = -x^2 + 4x. Construct the tangent at x=1 and explain its significance in the context of the roller coaster's slope.

Answer: Calculate slope = -2x + 4 at x=1, which is 2. Draw the tangent at (1,3) with slope 2.

Question 1: Construct the tangent to the curve y=sin(x) at x=π/2, using a calculator for the slope. Show your steps.

Answer: Derivative y'=cos(x), at π/2=0, so tangent y=0(x-π/2)+1

Question 2: Explain common misconceptions students have when constructing tangents to curves and how to avoid them.

Answer: Misconception: drawing lines without checking slope; solution: verify slope from derivatives.

Question 1: A student drew a tangent to y = x^2 at x=3, but the line passes through (3,9) with slope 5. Identify the mistake and correct the construction.

Answer: Mistake: slope should be 2*3=6. Correct line: y - 9=6(x - 3).

Question 2: A student attempts to draw a tangent to y=ln(x) at x=1, but draws a horizontal line. What is the error and how can it be fixed?

Answer: Error: slope of tangent at x=1 is 1, not 0. Fix: draw line with slope 1 passing through (1,0).