Grade 12

Mathematics, five units

The full five unit course: trigonometric identities and equations, the calculus of rational, root, exponential and trigonometric functions, vectors in space, and probability as far as the normal distribution.

8 lessons

  1. 1How do you prove a trigonometric identity?Work on one side only and transform it until it becomes the other. Do not move terms across, because that assumes the equality you are trying to establish. The method that almost always works is to write everything in terms of sine and cosine and then simplify using the fundamental identity.50m
  2. 2How do you solve a trigonometric equation over a full period?Find one solution with the inverse function, then use the symmetry of the function to find the second one in the period, and finally add the period to generate the rest. The number of solutions per period is fixed: two for sine and cosine, one for tangent.50m
  3. 3How do you find the asymptotes of a rational function?A vertical asymptote sits where the denominator is zero and the numerator is not — solve the denominator for zero and check each root. A horizontal asymptote is found by comparing the degrees of numerator and denominator: equal degrees give the ratio of the leading coefficients, a smaller numerator gives zero, and a larger numerator gives none.50m
  4. 4How do you differentiate a root and an exponential function?Rewrite a root as a fractional power and the ordinary power rule applies unchanged — a square root is a power of one half. The exponential function with base e is its own derivative, which is what makes e the natural base. When there is an expression inside either of them, finish with the chain rule.50m
  5. 5What is the natural logarithm and how do you differentiate it?The natural logarithm answers the question: to what power must e be raised to give this number. It is the inverse of the exponential function, its derivative is one over x, and its domain is the positive numbers only — which is a restriction that has to be stated in every answer, not assumed.50m
  6. 6What is a definite integral and how do you calculate it?Integration is differentiation run backwards: you look for the function whose derivative is the one given. A definite integral is that antiderivative evaluated at the upper limit minus its value at the lower limit, and the number it produces is the signed area between the graph and the horizontal axis.50m
  7. 7How do you find the length of a vector and the angle between two vectors?The length of a vector in space is the square root of the sum of the squares of its three components. The angle between two vectors comes from the scalar product: multiply component by component, add, and divide by the product of the lengths — the result is the cosine of the angle. A scalar product of zero means the vectors are perpendicular.50m
  8. 8What is the normal distribution and how do you standardise a measurement?The normal distribution is a symmetric bell curve centred on the mean, with its width set by the standard deviation. Standardising turns a measurement into a z-score: subtract the mean and divide by the standard deviation. The score says how many standard deviations the measurement is from the mean, which is what lets you read a probability from the table.50m