pure

• the formula for the length of an arc using radians is angle x radius
• the formula for finding the area of a sector using radians is 1/2 (radius)^2 (angle)
• general formula for arithmetic sequences: a + (n - 1) d
• for the binomial series convergence is only guaranteed for |x| < 1
• • any triangle with a hypotenuse along the diameter of a circle has a right angle
• • a chord that is perpendicular to a radius of a circle is bisected by the radius
• • if a radius and a tangent meet at the same point on the circle the angle between them is a right angle
• • if a point is concave d^2y/dx^2 < 0
• • if a point is convex d^2y/dx^2 > 0
• • if a point is a point of inflection d^2y/dx^2 = 0
• method of separation for solving differential equations

  1. move all of the ‘y’ variables to one side, all the ‘x’ variables to the other
  2. integrate both sides
  3. add a ‘+c’ to one side
  4. manipulate into ‘y=’ form
• formula for a geometric sequence: ar^(n-1)
• sum of the first n terms of a geometric series: a(1 - r^n)/(1 - r)
• when implicitly differentiating, differentiate the y term with respect to y, and multiply by dy/dx
• the derivative of a^x = a^x lna
• formula for parametric integration (integrating y): ∫ y dx/dt dt
• in a proof question, you must:

  • state assumptions and information you are using
  • show every step clearly
  • cover all possible cases (e.g. odd and even numbers)
  • end with a statement
• to prove something by exhaustion you have to break the question up into smaller cases, for example even and odd numbers
• to prove something by counter-example you have to find an example that does not work for the given statement
• to prove the sum to n of an arithmetic series, list the terms in their normal order, then again in reverse, if you add these together each pair of terms add to the first term (a) + the last term (L), then divide by 2, leaving you with n/2(a + L)
• cos(2x) = 1 - 2sin^2(x)
• cos(2x) = 2cos^2(x) - 1
• derivative of cos^2(x):

  -2 sin(X) cos(x)