Calculus Unit 1 Chapter 1 Lecture Notes PDF

Lecture 1 Class: FY BSc Subject : Calculus Subject Code: PUSASQF1.1 Chapter: Unit 1 – Chapter 1 Chapter Name: Pre-Calculus and Graphs 1 Syllabus Objectives Unit 1 Pre-Calculus and Graphs Limits and C...

Lecture 1 Class: FY BSc Subject : Calculus Subject Code: PUSASQF1.1 Chapter: Unit 1 – Chapter 1 Chapter Name: Pre-Calculus and Graphs 1 Syllabus Objectives Unit 1 Pre-Calculus and Graphs Limits and Continuity Unit 2 Differentiation Application of Derivatives Unit 3 Definite and Indefinite Integrals Unit 4 Differential Equations Taylor and Maclaurin series 2 Paper Pattern External Examination break up – 60 marks Question No. Particulars Total Marks Question type Q.1 10 Multiple choice 15 marks Concept checking questions of 1.5 mark questions each Q.2 3 questions of 5 marks 15 marks Knowledge Application each oriented Q.3 3 questions of 5 marks 15 marks Knowledge Application each oriented Option in Q4 (Part A or Part B) Q.4 Part A One question or can be 15 marks Higher order divided into multiple qts Application oriented Q.4 Part B One question or can be 15 marks Higher order divided into multiple qts Application oriented 3 Study Pattern Concept checkers – After each chapter Quiz – Before lectures Class task – Once in a while PPT – Read for revision Reference notes/ Class notes – Regular reading, understanding concepts Practice questions Assignment* - For mandatory practice (at least 2) Project/ Case study Mock exam paper – Additional practice under time constraint and exam difficulty * Considered in internal marking 4 Today’s Agenda 1. Calculus 2. Functions 3. Types of Functions 4. Transformations of Functions 5 1 Calculus What is calculus? What do you think is the scope of calculus? Why do we study calculus? Watch Video – The Essence of Calculus https://www.youtube.com/watch?v=WUvTyaaNkzM 6 2 Functions What are functions? Or simply state, what do you understand by functions? Do they have a practical use? 7 Motivation Toward the end of the twentieth century, the values of stocks of internet and technology companies rose dramatically. As a result, the Standard and Poor’s stock market average rose as well. Figure below tracks the value of that initial investment of just under $100 over the 40 years. It shows that an investment that was worth less than $500 until about 1995 skyrocketed up to about $1,100 by the beginning of 2000. That five-year period became known as the “dot-com bubble” because so many internet startups were formed. As bubbles tend to do, though, the dot-com bubble eventually burst. Many companies grew too fast and then suddenly went out of business. The result caused the sharp decline represented on the graph beginning at the end of 2000. Notice, as we consider this example, that there is a definite relationship between the year and stock market average. For any year we choose, we can determine the corresponding value of the stock market average. In this chapter, we will explore these kinds of relationships and their properties. 8 2 Functions Functions are used all the time in mathematics to describe relationships between two variables. The concept of function is one of the most important in mathematics. It is explained as follows: Given two sets A and B, a set with elements that are ordered pairs (x, y), where x is an element of A and y is an element of B, is a relation from A to B. A relation from A to B defines a relationship between those two sets. A function is a special type of relation in which each element of the first set is related to exactly one element of the second set. The element of the first set is called the input; the element of the second set is called the output. A function f consists of a set of inputs, a set of outputs, and a rule for assigning each input to exactly one output. Input Function Output 9 2 Functions & their graphs Let two sets X and Y be given. If for every element x in the set X there is exactly one element (an image) y=f(x) in the set Y, then it is said that the function f is defined on the set X. If we consider the number sets X⊂R, Y⊂R (where R is the set of real numbers), then the function y=f(x) can be represented as a graph in a Cartesian coordinate system Oxy. The input values make up the domain, and the output values make up the range. 10 Question Which table, Table 6 or Table 8, represents a function (if any)? 11 Solution Table 6 defines a function. Each input value corresponds to exactly one output value. Table 8 does not define a function because the input value of 5 corresponds to two different output values. 12 Functions Watch Video – Function, Domain and Range https://www.youtube.com/watch?v=vO5qqfsWzhc 13 3 Types of Functions What are the different types of functions that you know? 14 3 Types of Functions There are a wide range of functions we could think off: Even functions Odd functions Linear functions Polynomial functions Exponential functions Trigonometric functions Inverse functions Cubic functions Quadratic functions Logarithmic functions Identity functions 15 3 Types of Functions What is the difference between an even and an odd function? 16 3.1 Even & Odd Function If f(x) = f(−x) for all x in the domain of f, then f is an even function. An even function is symmetric about the y-axis. If f(−x) = −f(x) for all x in the domain of f, then f is an odd function. An odd function is symmetric about the origin. Even function has symmetry about the y-axis. Odd function has symmetry about the origin. 17 3.1 Even & Odd Functions Ex 1: Determine algebraically whether f (x) = –3x2 + 4 is even, odd, or neither Solution: I'll plug –x in for x, and simplify: f (–x) = –3(–x)2 + 4 = –3(x2) + 4 = –3x2 + 4 Since, f(x) = f(−x) f (x) is even function. 18 3.1 Even & Odd Functions Ex 2: Determine algebraically whether f (x) = 2x3 – 4x is even, odd, or neither. Solution: I'll plug –x in for x, and simplify: f (–x) = 2(–x)3 – 4(–x) = 2(–x3) + 4x = –2x3 + 4x f(-x) = -(2x3 – 4x) f(-x) = - f(x) f(x) is an odd function. 19 3.1 Even & Odd Functions Ex 3: Determine algebraically whether f (x) = 2x3 – 3x2 – 4x + 4 is even, odd, or neither. Solution: I'll plug –x in for x, and simplify: f (–x) = 2(–x)3 – 3(–x)2 – 4(–x) + 4 = 2(–x3) – 3(x2) + 4x + 4 = –2x3 – 3x2 + 4x + 4 f (x) is neither even nor odd. 20 3.2 Periodic Function Define a periodic function as f(x+kT)=f(x) , where k is an integer, T is the period of the function Periodic functions are functions that behave in a cyclic (repetitive) manner over a specified interval (called a period). The graph repeats itself over and over as it is traced from left to right. 21 3.2 Types & uses of Periodic Function 1. Euler's Formula: 2. Jaccobi Elliptic Functions: These functions are commonly used in the description of the motion of a pendulum. 3. Fourier Series : The Fourier series has applications in the representation of heatwaves , vibration analysis, quantum mechanics, electrical engineering, signal processing, image processing. 22 3.3 Inverse Function Given a function y=f(x). To find its inverse function of it, it is necessary solve the equation y=f(x) for x and then switch the variables x and y. The inverse function is often denoted as y= 𝑓 −1 (x). The graphs of the original and inverse functions are symmetric about the line y=x. 23 3.3 Inverse Function Ex 1: Find the inverse of h(x)=x3+2 Solution: 24 3.4 Composite Function Suppose that a function y=f(u) depends on an intermediate variable u, which in turn is a function of the independent variable x: u=g(x). In this case, the relationship between y and x represents a “function of a function” or a composite function, which can be written as y=f(g(x)). The two-layer composite functions can be easily generalized to an arbitrary number of “layers”. 25 3.4 Composite Function Ex 1: Find g [ f(x)] given that, f (x) = 2x + 3 and g (x) = –x2 + 5 Solution: Replace x in g(x) = –x2 + 5 with 2x + 3 g [f (x)] = – (2x + 3)2 + 5 = – (4x2 + 12x + 9) + 5 = –4x2 – 12x – 9 + 5 = –4x2 – 12x – 4 26 3.5 Linear Function Linear functions have the form y = ax + b, where a and b are constants. Here the number a is called the slope of the straight line. It is equal to the tangent of the angle between the straight line and the positive direction of the x- axis: a=tanα. The number b is the y-intercept. 27 3.5 Linear Function Ex 1: Find the linear function that has two points (-1, 15) and (2, 27) on it. Solution: The given points are (x₁, y₁) = (-1, 15) and (x₂, y₂) = (2, 27). Find the slope of the function using the slope formula : m = (y₂ - y₁) / (x₂ - x₁) = (27 - 15) / (2 - (-1)) = 12/3 = 4. Find the equation of linear function using the point slope form. y - y₁ = m (x - x₁) y - 15 = 4 (x - (-1)) y - 15 = 4 (x + 1) y - 15 = 4x + 4 y = 4x + 19 Therefore, the equation of the linear function is, f(x) = 4x + 19. 28 3.6 Cubic Function The simplest cubic function is given by y = x 3 , x∈R. In general, a cubic function is described by the formula y = ax 3 + bx 2 + cx + d, x∈R, where a, b, c, d are real numbers (a≠0). The graph of a cubic function is called a cubic parabola. When a>0, the cubic function is increasing, and when a0, a≠1, y = 𝑒 𝑥 when a = e ≈ 2.71828182846… An exponential function increases when a>1 and decreases when 01 and decreases if 0