STUDY NOTES FOR MATHEMATICS – CHAPTER 10 – Differential Calculus 2
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IBDP Mathematics is one of the Group 5 courses of the IB curriculum. This course is targeted towards students wishing to pursue studies in mathematics at university or subjects that have a large mathematical content; it is for those who enjoy developing mathematical arguments, problem solving and exploring real and abstract applications.
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.
This chapter includes the different rules that need to be followed in order to differentiate function in an easier manner. These rules allow us to skip the first principles and differentiate the function directly. We will be learning three basic rules which are the chain rule, product rule and the quotient rule. The product rule is used when two functions are multiplied and the quotient rule is used when the function is in the form of division. Furthermore, we will also understand that there are general equations that can be used in order to find the derivatives of exponential, logarithmic, trigonometric and inverse trigonometric functions without having to do elongated steps. After learning about first derivatives, the understanding of finding second derivatives is simpler to learn. Second derivatives are derivatives of the first derivative. Therefore, higher derivatives will be the derivative of the previous derivative. Sometimes, there are functions which are harder to differentiate given the complexity of the function. In this instance, implicit differentiation will be used.
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.
- Chapter 1 Algebra and Function Basics
- Chapter 2 Functions Equations and Inequalities
- Chapter 3 Sequences and Series
- Chapter 4 Exponential and Logarithmic Functions
- Chapter 5 Trigonometric Functions and Equations
- Chapter 6 Geometry and Trigonometry
- Chapter 7 Statistics
- Chapter 8 Probability
- Chapter 9 Differential Calculus 1
- Chapter 10 Differential Calculus 2
- Chapter 11 Integral Calculus
- Chapter 12 Probability Distributions