Worked examples: SACE Stage 2 Specialist Mathematics Formula Sheet

Selected worked examples

1. Vectors · projection component

   Formula or reference: The scalar component of u along v is (u · v)/|v|.

   Units and conditions: v is nonzero. A negative component points opposite v.

   Worked example: For u = (4,3,1) and v = (0,2,0), the component along v is 6 ÷ 2 = 3.

2. Calculus · derivative of an inverse tangent

   Formula or reference: d(arctan x)/dx = 1/(1 + x²).

   Units and conditions: The result is defined for every real x, with angles in radians.

   Worked example: For y = arctan(3x), the gradient is 3/(1 + 9x²), so y′(1) = 0.30.