## IBDP Mathematics Analysis & Approaches SL Chapter 13 Notes

More calculus

STUDY NOTES FOR MATHEMATICS – CHAPTER 13 – More 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.
We explore definite integrals and how they are used to calculate areas. When calculating definite integrals we can omit the constant of integration c as this will always cancel out in the subtraction process. Some definite integrals are difficult or impossible to evaluate analytically, hence we use technology to solve them. We can evaluate definite integrals by substitution, by making sure the endpoints are converted to the new variable. The chapter further discusses how to calculate the area under the curve. If f(x) is positive and continuous on the interval a ≤ x ≤ b, then the area bounded by y= f(x), the x axis, and the vertical lines x=a and x=b. Whereas, if f(x) is negative and continuous on the interval a ≤ x ≤ b, then the area bounded by y= f(x), the x axis, and the vertical lines x=a and x=b, we calculate the area above the curve. Solids of revolution is the solid formed when the shaded area of a graph is revolved about the x axis through 360 degrees. Lastly, the chapter involves problem solving using integration. 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.

In this chapter, we explore what it really means for the relationships between two variables to be called a function. We will then explore properties of functions which will help us work with and understand them. A relation between variables x and y is any set of points in the (x,y) plane. We say that the points connect the two variables. The chapter later on discusses what a domain and range of a function is and how to calculate them. The domain of a relation is the set of values which the variable on the horizontal axis can take. The variable is usually x. The range of a relation is the set of values which the variable on the vertical axis can take. The variable is usually y. The domain and range of relation can be described using set notation, interval notation, or a number line graph. Finally, we graph functions to model real world situations.