IBDP Mathematics Applications & Interpretations SL Chapter 12 Notes

Analyzing rates of change: Differential calculus

STUDY NOTES FOR MATHEMATICS – Chapter 12 – Analyzing rates of change: Differential calculus

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.

There are many properties when it comes to defining a curve. The simplest properties will further be elaborated in this chapter. To begin with, there are tangents which are the best approximated line for one point in the curve. Additionally, there are normals which are perpendicular to this tangent at the point of contact. A curve can also be defined as increasing (positive gradient) or decreasing (negative gradient) over particular intervals. Digging deeper, we will also be studying about the points that turn the graph from positive to negative, also known as stationary inflection points or stationary points. In order to determine these points, the use of sign diagrams come in handy. Upon studying the properties, the shape of the curves will also be learned in terms of how concave it is. By understanding the properties of the individual derivative curves, we can further establish a relationship between all of them. Finally, the concept of L’Hopital’s rule needs to be understood in case of indeterminate forms of limits.

Differentiation has a wide range of applications in the real world to show one variable changes relative to another. Firstly, the concept of rate of change is understood using the derivative of a function. After learning the rate of change, the idea of optimization can be understood. Optimisation is the process of finding a minimum or maximum value of a particular function. The applications of this can be used in areas where you need to optimise the material usage to make a product for instance. Essentially, the minima or maxima is not obtained after finding the first derivative. Sometimes, the functions need to be repeatedly differentiated in order to find the optimised value. To determine if you can find this value, various tests can be performed. This will be further explained in detail. The end of this chapter deals with related rates. This involves differentiating two variables with respect to a common variable, time, t. This chapter majorly includes the application of the differentiation concepts in real world problems.