IBDP Mathematics Analysis & Approaches HL Chapter 11 Notes
STUDY NOTES FOR MATHEMATICS – CHAPTER 11 – Integral Calculus
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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.
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.
Kinematics is the study of motion. In this chapter, we learn about kinematics, taking into account direction as well as distance and speed. This will lead us to functions for displacement, velocity and acceleration, and how they are linked by calculus. Velocity of an object is its rate of change of displacement. The velocity function of the object at time t is v(t)=s’(t). It is the gradient of the tangent to the function s(t) at any given time. Whereas acceleration of an object is the rate of change of velocity. In order to find velocity from displacement, you differentiate once. To find acceleration from displacement, you differentiate twice. The opposite would require integration. Doing this helps express velocity and acceleration in terms of displacement. The speed of an object at any instant is the magnitude of the object’s velocity.
- 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