IBDP Mathematics Analysis & Approaches HL Chapter 9 Notes

Differential Calculus 1


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All DP mathematics courses serve to accommodate the diverse range of needs, interests and abilities of students, and to fulfill the requirements of various university and career aspirations. The aims of these courses are to enable students to: develop mathematical knowledge, concepts, principles, logical, critical and creative thinking, employ and refine their powers of abstraction and generalization. Students are also encouraged to appreciate the international dimensions of mathematics and the multiplicity of its cultural and historical perspectives.

To begin with, the basic definition of limits is if f(x) is really close to some Real number A for all of the values x close to A, it can be said that f(x) has a limit of A when x approaches the real number A from either side. This will be written as lim x->a f(x) = A. However, x never equals a as we try to find the behaviour when x is only close to A. Alternatively, when x=0, we would be ignoring the definition of a limit, but will be known as a missing point. However, we don’t need to graph all the functions as they can be found algebraically. In addition, limits exist when x approaches A from only one side. In this case, it can be said that f(x) either converges or diverges to A. When it diverges, lim x->a f(x) does not exist. Knowing that there are infinite values before the value A, limits can be used to define the extreme behaviour when x approaches A. It is when x -> infinity from either side. Furthermore, you will also study the concept of continuity and discontinuities of a curve with the existence of f(A) and lim x->a f(x). In.
Differential calculus is a branch of mathematics that helps us delve deeper into certain fields such as engineering, science and finance. It deals with the rates of change of any 2 particular variables with different units. This change can either be constant, average or instantaneous. When the equation is graphed, the gradient of the graph is known to be the rate of change. For an average rate of change, the gradient of the chord joining two points is found. However, for the instantaneous rate of change, the gradient of the tangent at one particular point is found. The concept of gradient of tangent is further explored by investigating the behaviour when taking the limit h->0. In order to find the gradients of certain equations, a gradient function of y=f(x) is known as the derivative function which is denoted by f’(x). Furthermore, the concept of differentiability and continuity from the last chapter is carried onto differentiation. In order to confirm if either exists, tests need to be performed.