Applied Mathematics Chapter-wise Formulas and Rules PDF

FPMA0002 APPLIED MATHEMATICS Chapter-wise formulas and rules CHAPTER 1 – SYSTEM OF LINEAR EQUATIONS AND INEQUALITIES Solving System of Linear equations with two variables System of linear equations: When two or more linear equations are grouped together, they form...

FPMA0002 APPLIED MATHEMATICS Chapter-wise formulas and rules CHAPTER 1 – SYSTEM OF LINEAR EQUATIONS AND INEQUALITIES Solving System of Linear equations with two variables System of linear equations: When two or more linear equations are grouped together, they form a system of linear equations. An example of a system of two linear equations is shown below. We use a brace to show the two equations are grouped together to form a system of equations. 2𝑥 + 𝑦 = 7 { 𝑥 − 2𝑦 = 6 Solutions of a system of equations: The solutions of a system of equations are the values of the variables that make all the equations true. A solution of a system of two linear equations is represented by an ordered pair  x, y  Solve a System of Linear Equations by Graphing Page 1 of 17 STEPS TO SOLVE A SYSTEM OF LINEAR EQUATIONS BY GRAPHING. Step 1. Graph the first equation. Step 2. Graph the second equation on the same rectangular coordinate system. Step 3. Determine whether the lines intersect, are parallel, or are the same line. Step 4. Identify the solution to the system. If the lines intersect, identify the point of intersection. This is the solution to the system. If the lines are parallel, the system has no solution. If the lines are the same, the system has an infinite number of solutions. Step 5. Check the solution in both equations. Solve a System of Equations by Substitution STEPS TO SOLVE A SYSTEM OF EQUATIONS BY SUBSTITUTION Step 1. Solve one of the equations for either variable. Step 2. Substitute the expression from Step 1 into the other equation. Step 3. Solve the resulting equation. Step 4. Substitute the solution in Step 3 into either of the original equations to find the other variable. Step 5. Write the solution as an ordered pair. Step 6. Check that the ordered pair is a solution to both original equations. Solve a System of Equations by Elimination STEPS TO SOLVE A SYSTEM OF EQUATIONS BY ELIMINATION Step 1. Write both equations in standard form. If any coefficients are fractions, clear them. Step 2. Make the coefficients of one variable opposites. Decide which variable you will eliminate. Multiply one or both equations so that the coefficients of that variable are opposites. Step 3. Add the equations resulting from Step 2 to eliminate one variable. Step 4. Solve for the remaining variable. Page 2 of 17 Step 5. Substitute the solution from Step 4 into one of the original equations. Then solve for the other variable. Step 6. Write the solution as an ordered pair. Step 7. Check that the ordered pair is a solution to both original equations. Solving Systems of Linear Inequalities System of Linear Inequalities System of linear inequalities Two or more linear inequalities grouped together form a system of linear inequalities. A system of linear inequalities looks like a system of linear equations, but it has inequalities instead of equations. A system of two linear inequalities is shown here. 𝑥 + 4𝑦 ≥ 10 { 3𝑥 − 2𝑦 < 12 Solutions of a system of linear inequalities Solutions of a system of linear inequalities are the values of the variables that make all the inequalities true. Solve a System of Linear Inequalities by Graphing STEPS TO SOLVE A SYSTEM OF LINEAR INEQUALITIES BY GRAPHING. Step 1. Graph the first inequality. Graph the boundary line. Shade in the side of the boundary line where the inequality is true. Step 2. On the same grid, graph the second inequality. Graph the boundary line. Shade in the side of that boundary line where the inequality is true. Step 3. The solution is the region where the shading overlaps. Step 4. Check by choosing a test point. Page 3 of 17  CHAPTER 2: FUNCTIONS AND GRAPHS 2.1 Functions Definition: Rule form A function f : A B is a rule that associates each member of the first set A with a member of B. Domain/Codomain: The input variable to the function is called the independent variable, or argument. The set of all possible values for the independent variable is called the domain. Here, A is the domain of the function f and B is the co-domain of the function f. Range: The set of all possible values for the dependent variable is called the range (R) and it is a subset of the co-domain (B). Image and pre-image: The value f (x) is called the image of the element x  A. The elements in A are called pre-images whereas the elements in the range set are called images. Definition: Function as a set of ordered pairs A function can also be defined as a set of ordered pairs with the property that no two ordered pairs have the same first component and different second components. Domain : The set of all first components. Range : The set of all second components Definition: Functions defined by equations An equation represents a function if the equation assigns each input value (x) a unique output value (y). So, in other words, each x-value (independent variable) uniquely corresponds to one y- value (dependent variable). To check whether an equation represents a function, solve the equation for y, if possible. Functions defined by arrow diagrams: Given sets X and Y, we can define a relation from X to Y by listing the elements of the two sets and drawing arrows from each element in X to its corresponding element in Y. Page 4 of 17 To be a function it must have: (a) every element of X has an arrow coming from it and (b) no element of X has more than one arrow coming from it (to more than one element of Y). Types of functions One to one Function (Injective function) For every element b in the codomain B there is maximum one element a in the domain A such that f (a)  b. For an injective function, f (a)  f (b)  a  b. Onto function (Surjective function) For every element b in the codomain B there is at least one element a in the domain A such that f (a)  b. For onto function, range and codomain are the same. Bijection Bijection is the function which is both one to one and onto (Injective and Surjective). Vertical line test for a function Vertical line test is used to determine whether a relation is a function or not. A graph is said to be a function if the vertical line drawn does not intersect the graph more than one point. Horizontal line test for a 1-1 function A horizontal line test is used to determine whether a function is 1-1 or not. A graph is said to be a 1-1 function if the Horizontal line drawn does not intersect the graph more than one point. Finding the domain of a function Find the domain of the function if function written in an equation form. Step 1. Identify the input values. Step 2. Identify any restrictions on the input and exclude those values from the domain. Step 3. Write the domain in interval form, if possible. Find the domain of the function if function written in an equation form that includes a fraction. Step 1. Identify the input values. Step 2. Identify any restrictions on the input. If there is a denominator in the function’s formula, set the denominator equal to zero and solve for x. If the function’s formula contains an Page 5 of 17  even root, set the radicand greater than or equal to 0, and then solve. Step 3. Write the domain in interval form, making sure to exclude any restricted values from the domain. Find the domain of the function if function written in an equation form, including even root. Step 1. Identify the input values. Step 2. Since there is an even root, exclude any real numbers that result in a negative number in the radicand. Set the radicand greater than or equal to zero and solve for x. Step 3. The solution(s) are the domain of the function. If possible, write the answer in interval form. 2.2 Composite functions The process of combining functions so that the output of one function becomes the input of another is known as a composition of functions. The resulting function is known as a composite function. We represent this combination by the following notation: (f o g) (x) = f (g (x)) 2.3 Inverse Function Step 1. Make sure f is bijective function. If f is not one-to-one, then f −1 does not exist. Step 2. Solve for x and write x = f −1 (y). Step 3. Interchange x and y. Step 4. Find the domain of f −1. The domain of f −1 must be the same as the range of f. 2.4 Quadratic Equations Quadratic Formula The roots of a quadratic equation of the form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 where 𝑎 ≠ 0 −𝑏 ± √𝑏2 − 4𝑎𝑐 are given by the formula: 𝑥 =. 2𝑎 Step 1. Write the quadratic equation in standard form, 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0. Identify the values of a, b, and c. Step 2. Write the Quadratic Formula. Then substitute the values of a, b, and c. Step 3. Simplify. Step 4. Check the roots. Page 6 of 17 2.5 Graphing functions Graphing a linear function A linear function is a function whose graph is a line. Linear functions can be written in the slope intercept form of a line 𝑓(𝑥) = 𝑚𝑥 + 𝑏 where b is the initial or starting value of the function (when input, x = 0), and m is the non-zero constant rate of change, or slope of the function. The y-intercept is at (0, b). 𝑏 The x-intercept (− 𝑚 , 0) is calculated by solving f ( x)  0. To plot the graph, mark these intercepts on the axis and join it by straight line. Graphing a Quadratic function A quadratic function is a function of degree two. The graph of a quadratic function is a parabola. The general form of a quadratic function is 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 where a, b and c are real numbers and 𝑎 ≠ 0. The standard form of a quadratic function is 𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘. The −𝑏 vertex (h, k) is located as ℎ = and k = f(h). 2𝑎 Properties: The graph of f ( x)  a( x  h) 2  k is a parabola has:  (h, k ) as the vertex.  x  h as the axis of symmetry.  the domain  ,    , the set of all real numbers. If a  0 then the graph If a  0 then the graph open upwards open downwards has k as the minimum value has k as the maximum value has Range = [k , ) has Range = (, k ] decreases in the interval (, h] increases in the interval (, h] increases in the interval [h, ) decreases in the interval [h, ) Evaluate f (0) to find the y−intercept Solve the quadratic equation f (x) = 0 to find the x−intercepts, if exists. Page 7 of 17  CHAPTER 3 - EXPONENTIAL FUNCTION AND LOGARITHMIC FUNCTION Exponential Function: For real number 𝑏 > 0, 𝑏 ≠ 1 𝑓(𝑥) = 𝑏 𝑥 Property of Exponential Function: Let a and b be positives, a  1, b  1 , and x and y are real numbers: a x.a y  a x  y ax y  a x y a (ab) x  a x.b x x a ax    x b b Property of Exponential Function: For real numbers 𝑎, 𝑏, 𝑥, 𝑦 > 0, a ≠ 1, 𝑏 ≠ 1 𝑎 𝑥 = 𝑎 𝑦 if and only if 𝑥 = 𝑦 ; 𝑎 ≠ 1 Property of Exponential Function: For real numbers 𝑎, 𝑏, 𝑥, 𝑦 > 0, a ≠ 1, 𝑏 ≠ 1 and 𝑎 𝑥 = 𝑏 𝑥 if and only if 𝑎 = 𝑏 ; 𝑥 ≠ 0 How To Graph the exponential function of the form f ( x)  b x Logarithmic Function: Step 1. Create a table of points. For real2.numbers Step 𝑥, 𝑏3 > Plot at least 0, 𝑏from point ≠ 1the table, including the y−intercept (0,1). Step 3. Draw a smooth curve through the𝑓(𝑥) points. = log 𝑏 𝑥 Step 4. State the domain, (−∞,∞), the range, (0,∞), and the horizontal asymptote, y = 0. Page 8 of 17  Characteristics of the graph of the parent function f ( x)  b x An exponential function with the form f ( x)  b x , b > 0, b  1 , has these characteristics: one-to-one function horizontal asymptote: y = 0 domain: (−∞,∞) range: (0,∞) x -intercept: none y -intercept: (0, 1) increasing if b > 1 decreasing if b < 1 The figure compares the graphs of exponential growth and decay functions. Equivalent expression of Exponential and Logarithmic forms: For real numbers 𝑥, 𝑦, 𝑏 > 0, 𝑏 ≠ 1 𝑦 = 𝑏 𝑥 is equivalent to 𝑦 = log 𝑏 𝑥 Property of Logarithmic Function: For real number 𝑏 > 0, 𝑏 ≠ 1 log 𝑏 1 = 0 Property of Logarithmic Function: For real number 𝑏 > 0, 𝑏 ≠ 1 log 𝑏 𝑏 = 1 Property of Logarithmic Function: For real number 𝑏 > 0, 𝑏 ≠ 1 and 𝑥 be a real number log 𝑏 𝑏 𝑥 = 𝑥 Page 9 of 17 Logarithmic Function: For real numbers 𝑥, 𝑏 > 0, 𝑏 ≠ 1 𝒇(𝒙) = 𝐥𝐨𝐠 𝒃 𝒙 Equivalent expression of Exponential and Logarithmic forms: For real numbers 𝑥, 𝑦, 𝑏 > 0, 𝑏 ≠ 1 𝒚 = 𝒃𝒙 is equivalent to 𝑙𝑜𝑔𝑏 𝒚 = 𝑥 Property of Logarithmic Function: For real number 𝑏 > 0, 𝑏 ≠ 1 𝐥𝐨𝐠 𝒃 𝟏 = 𝟎 Property of Logarithmic Function: For real number 𝑏 > 0, 𝑏 ≠ 1 𝐥𝐨𝐠 𝒃 𝒃 = 𝟏 Property of Logarithmic Function: For real number 𝑏 > 0, 𝑏 ≠ 1 and 𝑥 be a real number 𝐥𝐨𝐠 𝒃 𝒃𝒙 = 𝒙 Property of Logarithmic Function: For real number 𝑏, 𝑥 > 0, 𝑏 ≠ 1 𝒃𝐥𝐨𝐠𝒃 𝒙 = 𝒙 Property of Logarithmic Function: For real number 𝑏, 𝑀, 𝑁 > 0, 𝑏 ≠ 1 𝐥𝐨𝐠 𝒃 𝑴 = 𝐥𝐨𝐠 𝒃 𝑵 if and only if 𝑴 = 𝑵 Property of Logarithmic Function: For real number 𝑏, 𝑀, 𝑁 > 0, 𝑏 ≠ 1 𝐥𝐨𝐠 𝒃 (𝑴𝑵) = 𝐥𝐨𝐠 𝒃 𝑴 + 𝐥𝐨𝐠 𝒃 𝑵 Property of Logarithmic Function: For real number 𝑏, 𝑀, 𝑁 > 0, 𝑏 ≠ 1 𝑴 𝐥𝐨𝐠 𝒃 ( ) = 𝐥𝐨𝐠 𝒃 𝑴 − 𝐥𝐨𝐠 𝒃 𝑵 𝑵 Page 10 of 17 Property of Logarithmic Function: For real number 𝑏, 𝑀 > 0, 𝑏 ≠ 1 and 𝑝 be a real number 𝐥𝐨𝐠 𝒃 𝑴𝒑 = 𝒑 𝐥𝐨𝐠 𝒃 𝑴 How To Graph the Logarithmic function of the form f ( x)  log b x Step 1. Draw and label the vertical asymptote, x = 0. Step 2. Plot the x -intercept, (1, 0). Step 3. Plot the key point (b, 1). Step 4. Draw a smooth curve through the points. Step 5. State the domain, (0,∞), the range, (−∞,∞), and the vertical asymptote, x = 0. Characteristics of the graph of the parent function f(x) = log b x For any positive real number x and a constant b > 0, b  1 , we can see the following characteristics in the graph of f(x) = log b x : one-to-one function vertical asymptote: x = 0 domain: (0,∞) range: (−∞,∞) x -intercept: (1, 0) and key point (b, 1) y -intercept: none increasing if b > 1 decreasing if 0 < b < 1 Population Doubling Time Model: t P  P0 2 d where P = population at time t , P0 = population at time t  0 and d = doubling time. Page 11 of 17 Exponential growth Model: A  A0 ekt where A = population at time t , A0 = population at time t  0 and k = relative growth rate. Half-Life Decay Model: t A  A0 2 h where A = amount at time t , A0 = amount at time t  0 and h = Half-life. Exponential Decay Model: A  A0 e kt where A = amount of radioactive material at time t , A0 = amount at time t  0 and k = a positive constant specific to the type of material. Page 12 of 17  CHAPTER 4 - STATISTICS Mean or Arithmetic mean or Average: 𝑆𝑢𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠 𝑥̅ =. 𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠 For grouped data ∑ 𝑓𝑚 𝑥̅ = ∑𝑓 , where 𝑓 = 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑚 = 𝑀𝑖𝑑 𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 Median: If 𝑛 is odd number, the median is the middle value of the ordered data (ordered smallest to largest). If 𝑛 is an even number, the median is equal to the two middle values added together and divided by two after the data has been ordered. Mode: The one that appears the most is the mode.  Unimodal Mode : One Mode  Bimodal Mode : Two Modes  Trimodal Mode : Three Modes  Multimodal Mode : More than three Modes Relative Frequency (R.F): 𝑓 𝑓 Relative Frequency = × 100 or Relative Frequency = 𝑛 𝑛 Where 𝑓 is a frequency, 𝑛 is the total number of frequency Cumulative Relative Frequency: Cumulative relative frequency is the accumulation of the previous relative frequencies. 𝑓 𝑓 Sector in angle = × 360𝑜 Sector in percentage = × 100 𝑛 𝑛 Range = Highest Value - Lowest Value Range Length of the Class Interval = Number of Classes Page 13 of 17 Bar Charts: Bar charts (Bar Graphs/ Bar Diagrams) are used to represent categorical data. Bar graphs consist of bars that are separated from each other. The bars can be rectangles or they can be rectangular boxes (used in three-dimensional plots), and they can be vertical or horizontal. Pie Charts: Pie charts are also used to represent the categorical data. This representation gives emphasis to the relative weightage of each category. In a pie chart, a circle is drawn and it is divided into sectors. Number of sectors will be the number of categories. The area of each sector is proportional to the frequency of the categorical variable it represents. Page 14 of 17 Histogram: A histogram consists of contiguous (adjoining) boxes. It has both a horizontal axis and a vertical axis. The horizontal axis is labeled with what the data represents (for instance, distance from your home to school). The vertical axis is labeled either frequency or relative frequency (or percent frequency or probability). The graph will have the same shape with either label. The histogram (like the stemplot) can give you the shape of the data, the center, and the spread of the data. Page 15 of 17  CHAPTER 5 - PROBABILITY 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑜𝑓 𝑒𝑣𝑒𝑛𝑡 𝐸 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑎𝑛 𝑒𝑣𝑒𝑛𝑡 𝐸 = 𝑃(𝐸) = 𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑃(𝐸′) = 1 − 𝑃(𝐸) Addition rule for Probability: 𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 𝑎𝑛𝑑 𝐵) Mutually exclusive events: If 𝐴 and 𝐵 be mutually exclusive events, then 𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 0 Conditional rule for Probability: 𝑃(𝐴 𝑎𝑛𝑑 𝐵) 𝑃(𝐴 / 𝐵) = 𝑃(𝐵) Tree diagram consists of “branches” that are labeled with probabilities Permutations: The number of ways to order(arrange) 𝑟 objects from 𝑛 different objects is 𝑛! 𝑃(𝑛, 𝑟) = 𝑛𝑃𝑟 = (𝑛 − 𝑟)! Permutations: The number of ways to order(arrange) 𝑛 non-distinct objects, 𝑟1 are alike, 𝑟2 are alike, …,𝑟𝑘 𝑛! are alike is = 𝑟1 !∙𝑟2 !∙…∙𝑟𝑘 ! Combinations: The number of ways to select 𝑟 objects from 𝑛 different objects is 𝑛! 𝐶(𝑛, 𝑟) = 𝑛𝐶𝑟 = 𝑟! ∙ (𝑛 − 𝑟)! Factorial: For a positive integer 𝑛 ≥ 2 𝑛! = 𝑛 ∙ (𝑛 − 1) ∙ (𝑛 − 2) ∙ … ∙ 2 ∙ 1 Factorial: 1! = 1, 0! = 1, 3! = 6 Page 16 of 17  CHAPTER 6 – MATHEMATICS OF FINANCE 6.1 - Simple Interest and Discount: Simple Interest: Simple interest 𝐼 = 𝑃𝑟𝑡, Here 𝑃 - Principal 𝑟 - Interest rate 𝑡 – Number of years Total or Accumulated amount 𝐴 = 𝑃 + 𝐼 = 𝑃(1 + 𝑟𝑡) Discount and Proceeds: Discount 𝐷 = 𝑀𝑟𝑡, Here 𝑀 - Amount borrowed 𝑟 - Discount interest rate 𝑡 - Number of years Proceeds or Actual amount 𝑃 = 𝑀 − 𝐷 = 𝑀(1 − 𝑟𝑡) 6.2 – Compound Interest: Compound interest: (Compounded for specific time period) 𝑟 𝑛𝑡 Total Amount 𝐴 = 𝑃 (1 + 𝑛) Here 𝑃 - Principal 𝑟 - Interest rate 𝑛 – Number of times compounded 𝑡 – Number of years Compound interest 𝐼 = 𝐴 − 𝑃 𝑟 𝑛 Effective interest rate (If compounded for specific period) 𝐸 = [(1 + 𝑛) − 1] Compound interest: (Compounded continuously) Total Amount 𝐴 = 𝑃𝑒 𝑟𝑡 Here 𝑃 - Principal 𝑟 - Interest rate 𝑡 – Number of years Effective interest rate (If compounded for specific period) 𝐸 = 𝑒 𝑟 − 1 Page 17 of 17