Calculus-1-EMath-1101-3a-week5.pdf

EMath 1101 BSSE - 1 Week 3a LA Holleza NOTES BY: Sasa ○ The area A of a circle as a...

EMath 1101 BSSE - 1 Week 3a LA Holleza NOTES BY: Sasa ○ The area A of a circle as a function of its radius r is OUTLINE 2 𝐴 = Π𝑟 I. Function and Function Notation II. Domain ○ Here, r is the independent III. Range variable and A is the dependent IV. Function Notation variable. V. Explicit Form A. Note VI. Difference Quotient Domain VII. Domain and Range of Function VIII. Finding the Domain and Range of I. The domain of a function f is the set X, Function which consists of all the input values for IX. Piecewise Function X. Graph of a Function which the function is defined. XI. Vertical Line Test II. Relation to 𝑓(𝑥) if x in the domain X, XII. Eight Basic Functions then 𝑓(𝑥) is the result of applying the XIII. Transformation of Function function to X. XIV. Basic Types Of Transformations XV. Classification and Combinations of Functions Example: ○ 𝑖𝑓 𝑋 = {1, 2, 3}, 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑑𝑜𝑚𝑎𝑖𝑛 𝑜𝑓 𝑓 𝑖𝑠 𝑋. ○ This means you can input Function and Function Notation 1, 2, 𝑜𝑟 3 𝑖𝑛𝑡𝑜 𝑓. I. A function relation between two sets X and Y consists of order pairs (x, y) Range where: A. X is an element of X I. The range of a function f is a subset of II. In this context Y, which consists of all possible output A. X is the independent variable values 𝑓(𝑥) that the function can B. Y is the dependent variable produce from inputs in X. Example:  II. Relation to 𝑓(𝑥): The range is the set of At 𝑥 = 0, 𝑓(𝑥) = 4 − 0 = 2. all possible values of 𝑓(𝑥) as x varies Since the square root function produces over the domain X. non-negative values, 𝑓(𝑥) ranges from 0 to 2. Range [0, 2] Example: If 𝑓(𝑥) produces outputs 4, 5 𝑎𝑛𝑑 6 𝑓𝑜𝑟 𝑖𝑛𝑝𝑢𝑡𝑠 1, 2, 𝑎𝑛𝑑 3, Function Notation respectively, then the range is {4, 5, 6} I. The word Function was first used by Domain (X) = all possible inputs for the Gottfried Wilhelm Leibniz in 1694 as a function f term to denote any quantity connected with a curve, such as the coordinates of Range (subset of Y) = all possible outputs from a point on a curve or the slope of a the function f based on the inputs from the curve. domain X. II. Forty years later, Leonhard Euler used the word “function” to describe any Example: try to get the domain and expression made up of a variable and range: some constants. He introduced the 2 ○ 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛: 𝑓(𝑥) = 4 −𝑥 notation 𝑦 = 𝑓(𝑥) Solution: III. Function notation has the advantage of ○ Determine the Domain: clearly identifying the dependent 2 variables as 𝑓(𝑥) while at the same time ○ For 𝑓(𝑥) = 4 − 𝑥 , the telling you that x is the independent expression inside the square variable and that the function itself is root must be non-negative. “f”. The symbol 𝑓(𝑥) is read “f of x” ○ Set the expression inside the A. Form: 𝑓(𝑥) square root greater than or equal 2 1. F: the name of the to 0: 4 − 𝑥 ≥ 0 function. It can be any 2 ○ Solve for x: 𝑥 ≤4 letter or symbol, but f is −2≤𝑥≤2 commonly used. Domain [− 2, 2] 2. X. the input value, also called the independent Determine the Range: variable. At 𝑥 =− 2 𝑎𝑛𝑑 𝑥 = 2, 𝑓(𝑥) = 4 − 4 = 0.  A. Function name is f, independent variable Explicit Form is x. B. Function name is f, independent variable I. An explicit form of an equation is t. expresses one variable directly in terms C. Function name is g, independent of another variable. variable is s. A. 𝑦 = 𝑓(𝑥) Evaluating a function: 2 𝑥 +2𝑥+1 ○ 𝑓(𝑥) = 𝑥−1 2 (3𝑥−2) +2(3𝑥−2)+1 ○ 𝑓(3𝑥 − 2) = (3𝑥−2)−1 2 9𝑥 −12𝑥+4+6𝑥−4+1 Example: directly shows y as a function ○ 3𝑥−3 of x. 2 9𝑥 −6𝑥+1 ○ 2 Implicit Form: 𝑥 + 2𝑦 = 4; ○ 3𝑥−3 2 ○ Explicit Form: 2𝑦 = 4 − 𝑥 1 2 𝑦= 2 (4 − 𝑥 ) ○ Function Notation: 1 2 𝑓(𝑥) = 2 (4 − 𝑥 ) Note I. Although f is often used as a convenient function name and x as the independent variablem you can use other symbols. For instance, the following equations all define the same function. Difference Quotient 2 A. 𝑓(𝑥) = 𝑥 − 4𝑥 + 7 2 I. The difference quotient is a fundamental B. 𝑓(𝑡) = 𝑡 − 4𝑡 + 7 concept in calculus used to approximate 2 C. 𝑔(𝑔) = 𝑠 − 4𝑠 + 7 the derivative of a function. It measures the average rate of change o the function  over a specific interval. Here’s a Finding the Domain and Range of Function breakdown of what the difference quotient is and how it works. I. The domain of the function A. The difference quotient for a 𝑓(𝑥) = 𝑥 − 1 function 𝑓(𝑥) is given by II. Is the set of all x-values for which 𝑓(𝑥+ℎ)−𝑓(𝑥) 1. ℎ 𝑥 − 1 ≥ 0, which is the interval [1. ∞) II. Where: III. To find the range, observe that A. 𝑓(𝑥) is the function you are 𝑓(𝑥) = 𝑥 − 1 is never negative. So, analyzing. the range is the interval [0, ∞) B. X is a point in the domain of f. C. H is a small increment added to ➔ The domain of a function is the set of all x. possible values of x for which the function is defined Domain and Range of Function ➔ The range of a function is the set of all possible values the function can output I. The domain of a function can be described explicitly, or it may be Piecewise Function described implicitly by an equation used to define the function. I. A piecewise function is a function that is II. The implied domain is the set of all real defined by different expressions or numbers for which the equation is formulas for different intervals of its defined, whereas an explicitly defined domain. Instead of having a single domain is one that is given along with formula that applies to all input values, a the function. piecewise function multiple “pieces,” 1 A. 𝑓(𝑥) = 2 ,4 ≤ 𝑥 ≤ 5 each valid over a specific range of input 𝑥 −4 Has an explicitly defined domain given by values. {𝑥: 4 ≤ 𝑥 ≤ 5}. On the other hand, the function given by General Form: 1 A piecewise function is written as: B. 𝑔(𝑥) = 2 𝑥 −4 Has an implied domain that is the set {𝑥: 𝑥 ≠± 2}  Vertical Line Test I. The vertical line test is a simple method used to determine whether a graph represents a function. It helps to visually check if a relation between x - values is a function. ➔ If a vertical line drawn anywhere on the graph intersects the graph at no more than one point, then the graph represents a function. ➔ If any vertical line intersects the graph at more than one point, the graph does not represent a function Graph of a Function I. The graph of the function 𝑓 = 𝑓(𝑥) consists of all points (𝑥, 𝑓(𝑥)), where x is in the domain of f. In figure P.25, note that A. X = the directed distance from Eight Basic Functions the y - axis B. f(x) = the directed distance from the x - axis  Basic Types Of Transformations (𝑐 > 0) Transformation of Functions I. Function Transformations involve altering the graph of a function in Classification and Combinations of Functions specific ways to obtain a new graph. These changes affect the function’s position, shape, or orientation on the coordinate plane. Types of Transformations: Vertical Shifts: Moving the graph up or down. Horizontal Shifts: Moving the graph left or right. Reflections: Flipping the graph over a specific axis. Stretches and Compressions: Changing the size of the graph either horizontally or vertically.  EMath 1101 BSSE - 1 Week 3b LA Holleza NOTES BY: Sasa Inverse Function OUTLINE I. An inverse function is a function that I. Horizontal Line Test II. Inverse Function ‘reverses” the effect of the original III. Composite Function function. In other words, if you apply a IV. Function Addition function and then its inverse, you get V. Function Subtraction back to where you started. VI. Function Multiplication VII. Function Division VIII. Test for Even and Odd Functions IX. Even Function X. Odd Function XI. Zeros Function Horizontal Line Test I. If a horizontal line scanning from top to bottom only passes through the graph of a function 𝑦 = 𝑓(𝑥) in at most one place, then we say that the graph satisfies the horizontal line test and that the function f is one-one. This means that each potential output on the y-axis can correspond to at most one input on the x-axis. Composite Function I. Combining two functions where the output of one function becomes the  input for the other. For functions f and g, the composition is denoted as 𝑓 ◦ 𝑔, defined by (𝑓 ◦ 𝑔)(𝑥) = 𝑓(𝑔(𝑥)). Function Subtraction I. Substracting one function from another. For function f and g, the difference is II. The function given by denoted as: (𝑓 ◦ 𝑔)(𝑥) = 𝑓(𝑔(𝑥)) is called the A. (𝑓 − 𝑔)(𝑥) = 𝑓(𝑥) − 𝑔(𝑥) composite of f with g. The domain of 𝑓 ◦ 𝑔 is the set of all x in the domain of g such that 𝑔(𝑥) is in the domain of f. Function Multiplication I. Multiplying two functions. For functions f and g, the product is denoted as: A. (𝑓 − 𝑔)(𝑥) = 𝑓(𝑥) · 𝑔(𝑥) Function Addition I. Adding two functions together. For functions f and g, the sum is denoted as: A. (𝑓 + 𝑔)(𝑥) = 𝑓(𝑥) + 𝑔(𝑥)  Function Division I. Dividing one function by another. For functions f and g, the quotient is denoted as: A. (𝑓/𝑔)(𝑥) = 𝑓(𝑥)/𝑔(𝑥), 𝑤ℎ𝑒𝑟𝑒 𝑔(𝑥) ≠ 0 Zeros Function Test for Even and Odd Functions Even Function Odd Function  EMath 1101 BSSE - 1 Limits and Continuity LA Holleza NOTES BY: Sasa Example: OUTLINE A. What is the slope of a curve? It is the I. Limits limit of slopes of secant lines. A. The Limit Process B. What is the length of a curve? It limits II. The Idea of a Limit III. Theorem 1.1: Some Basic Limits the lengths of polygonal paths inscribed IV. Theorem 1.2: Properties of Limits in the curve. V. Theorem 1.3: Limits of Polynomial and Rational Functions VI. The Limit of a Polynomial VII. The Limit of a Rational Function VIII. Theorem 1.4: The Limit of a Function involving a Radical IX. Theorem 1.5: The Limit of a Composite Function X. Theorem 1.6: Limits of Trigonometric Functions XI. Theorem 1.7: Functions That Agree At All But One Point Limits The Idea of a Limit The Limit Process I. We start with a number c and a function I. The Limit Process (An Intuitive f defined at all numbers x near c but not Introduction) necessarily at c itself. In any case, A. We could begin by saying that whether or not f is defined at c and, if limits are important in calculus, so, how is totally irrelevant. but that would be a major II. Now let L be some real number. WE say understatement. Without limits, that the limit of f(x) as x tends to c is L calculus would not exist. Every and write single notion of calculus is a A. lim 𝑓(𝑥) = 𝐿 limit in one sense or another. 𝑥→𝑐 Provided that (roughly speaking) B. As x approaches c, f(x) approaches L Or (somewhat more precisely) provided that A. f(x) is close to L for all 𝑥 ≠ 𝑐 which are close to c.  Definition of Limit I. If f(x) becomes arbitrarily close to a single number L as x approaches c from either side, then the limit of f(x) as x approaches c is L, written as A. lim 𝑓(𝑥) = 𝐿 𝑥→𝑐 II. At first glance, this definition looks fairly technical. Even so, it is informal because exact meanings have not yet been given to the two phrases A. “𝑓(𝑥) 𝑏𝑒𝑐𝑜𝑚𝑒𝑠 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦 𝑐𝑙𝑜𝑠𝑒 𝑡 𝐿" And B. "𝑥 𝑎𝑝𝑝𝑟𝑜𝑎𝑐ℎ𝑒𝑠 𝑐" Theorem 1.1: Some Basic Limits Theorem 1.2: Properties of Limits  The Limit of a Rational Function Theorem 1.3: Limits of Polynomial and Rational Functions The Limit of a Polynomial Theorem 1.4: The Limit of a Function involving a Radical Theorem 1.5: The Limit of a Composite Function  Theorem 1.6: Limits of Trigonometric Functions Theorem 1.7: Functions That Agree At All But One Point  EMath 1101 BSSE - 1 Week 5 LA Holleza NOTES BY: Sasa OUTLINE I. Limits at Infinity A. Definition of Limits at Infinity II. Horizontal Asymptote Horizontal Asymptote A. Definition of a Horizontal Definition of a Horizontal Asymptote Asymptote III. Theorem 3.10: Limits at Infinity IV. One-sided Limites and Continuity I. Horizontal asymptotes describe the Limits at Infinity behavior of a function at the extreme ends of the x-axis, specifically as I. Describes the behavior of a function as 𝑥 −> ∞ 𝑜𝑟 𝑥 −>− ∞ the input approaches infinity or negative infinity. A. lim 𝑓(𝑥) 𝑜𝑟 lim 𝑓(𝑥). Theorem 3.10: Limits at Infinity 𝑥→∞ 𝑥 → −∞ Definition of Limits at Infinity One-sided Limits and Continuity