GR 12: CALCULUS

In this PROGRAM, you will: • Factorise third-degree polynomials. • Apply the remainder and factor theorems to polynomials of degree at most 3 (no proofs required) • Understanding the limit concept intuitively, in the context of approximating the rate of change or gradient of a function at a point. • Use limits to define the derivative of a function f at any point 𝑥 • Generalise to find the derivative of f at any point x in the domain of f i.e., define the derivative function f '(x) of the function f (x) • Understand intuitively that f '(a) is the gradient of the tangent to the graph of f at the point with x -coordinate. • Use the definition (first principle), determine the derivative, f '(x) where a, b and c are constants. • Use the differentiation rules to determine derivatives. • Determine equations of tangents to graphs of functions. • Introduce the second derivative and how it determines the concavity of a function • Sketch graphs of cubic polynomial functions using differentiation to determine the coordinates of stationary points, and points of inflection (where concavity changes). Also, determine the x -intercepts of the graph using the factor theorem and other techniques. • Solve practical problems concerning optimisation and rate of change, including calculus of motion.