Free essays on Mathematics are written pieces of content on different aspects of mathematics that can be accessed without any cost. These essays cover various topics in mathematics such as algebra, geometry, calculus, probability, statistics, and more. They provide an in-depth understanding of the subject and help students learn new concepts and ideas. Free essays on Mathematics serve as useful study materials for students at any academic level and assist in preparing them for exams or other assignments. These essays are written by experts in the field and are available online on various platforms.
Tangent Lines and Instantaneous Rate of Change
Definition of Derivative A tangent line is the straight line that touches a curve at just one point. Tangent lines are central to calculus and have been since the 17th century. Today, we'll examine the problem of determining tangent lines for functions. The two fundamental questions addressed in calculus are: (1) What is the area under a curve, and (2) what is the slope of a line? In Algebra I, we learned that the slope of a function equals the…...
Calculus
Derivatives of Inverse Functions
Derivatives of Inverse Functions Let us assume that we have a function fox, and let us call it 3x - 1. We want to establish some sort of relationship between the slope of f and the slope of its inverse. To do this, we take these two functions and switch their x-variables and y-variables. This is all basic algebra. Next, we solve for y by adding x to one side of the equation and dividing by 3. After that, we…...
Calculus
Particle Motion. Example Problem 2
Particle Motion. Example Problem 2 An object moves along the y-axis such that its position is given by y(†) = t^3 - 12t^2 + 21t - 3 at what time does the object change direction? v(t)=0 y'(t)=3t^2-24t+21=0 t^2-8t+7=0 (t-1)(t-7)=0 t=1, 7 Given an object moving on the y-axis, its position is described by this equation. At what time does the object change direction? I believe you will find that the concepts of position, velocity, and acceleration-as they relate to particles--are…...
Calculus
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Linear Approximation and Perpendicular Vectors
Linear approximation You already know about about functions of several variables and linear approximation Also, you are familiar with vectors. Now we'll unite those concepts. Let's review linear approximation. Here is some complicated function, fof (x,y). F(x,y) The function is complicated, but if we take a little box in the xy- plane, then inside of this box the function is well approximated by something way less difficult. It is almost equal to an expression of the form ax + by…...
Calculus
Review Critical Points
Review critical points A critical point is a point where the partial derivatives are both O. In this lecture, we will continue to explore critical points and learn how to actually decide whether a particular point is a minimum, maximum, or sadale point. There are various kinds of critical points. Local minima, local maxima, which are like that, and saddle points which are neither minima nor maxima. And, of course, if you have a real function, then it will be…...
Calculus
Transitioning Between Rectangular and Polar Coordinates
Example Polar Coordinates To make this more concrete, consider the case in which you want to switch between rectangular and polar coordinates. In rectangular coordinates, you use x and y to locate a point in the plane: in polar coordinates. vow use r. the distance from the origin, and theta, the angle from the x-axis. [Graph] Thus, the change of variables for this equation is x equals r cosine theta and y equals r sine theta. x = r cos…...
Calculus
L’Hopital’s Rule: Using Derivatives to Evaluate Limits
L'Hopital's Rule L'Hopital's Rule states that if you have the limits of two functions, and the limit of the product of those functions is equal to the ratio of the limits of each function. then the derivative of that product will equal the derivative of that ratio. lim_(x->0) 3x/x=3 lim_(x->0)(x^2 -x)=0 lim_(x->0) [Graph] L'Hopital's Rule lim_(x->a) f(x)/g(x)=lim_(x->a) f'(x)/g'(x) if the limits an indeterminate form (0/0, inf/inf) We will conclude our discussion of limits by revisiting the concept and using derivatives…...
Calculus
Writing Equations as Dot Products
Writing equations as dot products Here is the question: find a vector v that is perpendicular to 1, comma, one-half, (1, 1/2). Find v perp to (1,12) The answer is we can check if it's perpendicular by taking the dot product. We want the dot product of v1 and v2 to be 0. If we write out this dot product, we get v1 plus one-half of v2. 0=*1,12=v1+12v2 So, there are many possible solutions to this problem. If you pick…...
Calculus
Solving Chain Rule Problems: Methods and Solutions
Chain Rule Problem Solution To begin, we must first recall that the total differential of z equals the partial derivative of z in the x direction multiplied by dx plus the partial derivative of z in the y direction multiplied by dy. a) dz = z_x dx + z_y dy Now, looking at our formula for z, we see that the partial derivative in the x direction is 2x, and the partial derivative in the y direction is 2y. dz…...
Calculus
Analytical Approaches to Finding Limits: Factoring, Rationalizing, and Algebraic Techniques
Finding Limits Analytically The limit of a function x approaches 3 as x approaches 3. We will call that x squared. If we graph this, it looks something like this. And there are tick marks, but we will not worry about them. The value 3 is over here and we have a point on the on curve approaching 9 Let f(x) = x2. Then f(3) = 9, so the limit of f(x) as x approaches 3 is also 9. This…...
Calculus
Level Curves and Partial Derivatives. Contour Plots and Level Curves
Level curves and partial derivatives. Contour plots and level curves The contour plot is another way to look at functions of two variables. It gives a slightly different view of the surface than the standard plot, and it's useful for visualizing regions where the function has extreme values. A contour plot is a way of representing the function of two variables by using a map. The same way that when you walk around, you have actual geographical maps that fit…...
Calculus
Simplifying Equations for Analysis
Rearranging the formula The equations we have to solve are as follows: <=>{sum_(i=1)^n (X_i^2 a+X_i b-X_iY_i)=0 {sum_(i=1)^n (X_i a+b-y_1)=0 So the first equation can be simplified by looking at the coefficients of a and b. You'll see that there are actually linear equations in a and b, so there's a lot of clutter with all these x's and y's all over the place. Let's simplify the formula by dividing out the factors of two. We can eliminate a and b…...
Calculus
Introduction to Mathematical Thinking. Tutorial for Assignment 2
Introduction to Mathematical Thinking. Tutorial for Assignment 2 1. Simplify the following symbolic statements as you can, leaving your answer in the standard symbolic form. (In case you are not familiar with the notation, I'll answer the first one for you.) (a) (π > 0) ^ (π<10) [Answer: 0 < π <10] (b) (p ≥ 7) ^ (p < 12) 7≤p<12 (c) (x>5) ^ (x < 7) 5<x<7 (d) (x < 4) ^ (x <6) x = 4 (e) (y…...
Mathematics
L’Hopital’s Rule. Example Problem
L'Hopital's Rule. Example Problem lim_(x->0) (cos(3x)-e^4x)/(x^2 -3x) lim_(x->0) (-3sin(3x)-4e^4x)/(2x-3)=-4/-3 =4/3 Let us consider the following example of L'Hopital's rule. If you evaluate the cosine of 0, you get 1. The exponential of 0 is 1, divided by 0 minus 0 - we get 0/0. Factoring probably won't work, since there are no algebraic methods or anything that will help us. We'll use L'Hopital's rule instead. Thus, we take the derivative of the numerator (3sin3x), times 3 because of the chain…...
Calculus
Simplifying Complex Logical Systems through Box Operations
Box Logic Box logic, though seemingly simple on the surface, is a sophisticated language used to create complex logical systems. In this segment we will investigate some of the latest innovations in logic technology by learning about box logic. Today our goal was to look at box logic. There are three constants: a large box, a medium box, and a small box. They're all oriented one way or another. It's easy to form compound sentences from our consonants—if we want…...
Mathematics
Calculus: Parallel Lines
Parallel lines Parallel and perpendicular lines are defined as follows: Parallel lines- two or more lines that do not intersect with each other, no matter what distance you move them. Perpendicular lines- two or more lines that intersect at a 90 degree angle. What if we don't have a line that's exactly equal to 0? Rephrasing, instead of x + 1/2 y = 0, what if we had x + 1/2 y = 1? Consider having the line x plus…...
Calculus
Statistics for IGCSE/ O-level students
Statistics for IGCSE/O-level Students Introduction Statistics is a very important section of mathematics. However, it can get quite deep, as it is, after all, statistics. However, this statistics course will not be too deep, and this lesson is literally a breeze for those with some exposure to statistics. Terms Mean In this case, we are referring to the arithmetic mean, which is basically the sum of all values in a data sample divided by the number of values. Example: 1,…...
Statistics
IGCSE Mathematics: How to find the quadratic functions with information given
IGCSE Mathematics: How to find quadratic functions with given information. IGCSE may provide you with: Vertex and another point x-intercepts and a point Vertex or intercepts with a = 1 Remember: y = a(x - h)² + k - this is known as the vertex form of a quadratic function. a is the constant. (h, k) is the coordinate of the vertex. Vertex forms have their uses, as you should be aware of: If a < 0, we have a…...
Mathematics
Partial Variation: Understanding Cost Relationships in Real-World Examples
Partial Variation Example 1: A banquet hall charges a flat fee of $4000 for an event plus a charge of $100 per person attending. Write a mathematical model to calculate the cost of an event. Does the cost of an event depend on the number of people that attend the event?Let C represent the cost of renting the hall and let p represent the number of people attending the banquet.C=100p+4000Costs depend partially on the number of people attending the party.…...
Mathematics
Functions and Properties of Protein
Proteins Proteins are made of small units called amino acids that are linked together. When the protein is heated, the links change, causing the structure of the food to alter. On example would be the solidifying of egg white as it is exposed to heat. Functions and properties of protein The use of eggs: Aeration: The whole egg is able to trap large amounts of air when whipped – this is called aeration. When whisked, the protein stretches and uses…...
Statistics
Unit 1 Test Review
Unit 1 Test Review Part A: 8 Multiple Choice Questions [8 marks] Refer to Pg.176 #1-9 (assigned last Friday) Part B: True or False Questions [3 marks] Ex: x3x5=x15 True/False, Explanation: x3x5=x8 Part C: 1. Exponent Laws [18 marks] 2. Simplifying Algebraic Expressions [19 marks] Distributive Property Adding and Subtracting Polynomials More Distributive Property 3. Application Problem [6 marks] Area and Perimeter of a rectangle Evaluate the area or perimeter...
Mathematics
Mastering Integer Multiplication: Rules and Examples
Mulplying Integers Example: ‐ Temperatures below freezing are indicated by negave integers; temperatures above freezing by posive integers. In general, numbers such as 1,2,3, ...are called positive integers; and numbers such as ‐1, ‐2,‐3,... are called negative integers. The negave integers, zero, and posive integers make up the set of integers, denoted by I.So, I=…,-3,-2,-1,0,1,2,3,…Integers such as ‐2 and 2 are called apposite integers. Integers can be represented on a number line. Note that any integer on the number line…...
Mathematics
Linear and NonLinear Relations
Linear and NonLinear Relations Recall: A line of best fit is a straight line that comes closest to the points on a scatter plot. Question: What if the relationship we are trying to model is not linear? Example 1: Plot each set of ordered pairs on the grids provided below. A line of best fit must be used on this graphb) A curve of best fit must be used on this graph. Example 2: Why is a line of best…...
Mathematics
Modelling with Algebra
Hwk.: Pg 215: #10 E=12mv2 2E=mv2 2Em=mv2m 2Em=v2 2Em=v2 2Em=v #11. Rocco: m=5 kg E=4.2 J Biff: m=5.5 kg E=5.2 J a)VRocco=2Em =24.25 =8.45 =1.296ms =1.3ms VBiff= 2Em =2 ×5.55.2 =15.2 =1.454ms =1.5ms Biff will reach the eucalyptus first since he has a greater speed. b) E=12m VBiff2 =1251.4542 =5.285 So, Rocco should have a kinetic energy of 5.285 J in order for the bears to reach encalyptus at the same time. Modelling with Algebra When problems are solved with…...
Mathematics
Surface Area and Volume of a Sphere
Surface Area and Volume of a Sphere Sphere: a round ball-shaped object; a set of points in space that are a given distance (radius) from a fixed point (centre). Surface Area =4r2 If the radius r is given d=2r d2=r Surface Are=4d24 =d2 if the diameter d is given Example 1: Calculate the Surface Area. Solution: r=3.5m ∴ The surface area is approx. 153.94m2 Example 2: A baseball has a diameter of 7.3 cm a) How much leather is needed…...
Mathematics
Statistical Data in Students Who Takes Business Administration Degree
Statistics is a very broad discipline that is defined as “the collection of informative data. interpreting these data and drawing conclusions about a phenomenon under study". Statistics has been perceived to be a technical subject since it is thought to involve the analysis of heavy computations. Whichever way one looks at it. statistics is a veiy crucial discipline that cuts across all manner of disciplines. Therefore. being a part and parcel of most areas of academia it is imperative for…...
Business AdministrationDataMarketingStatistics
Positivist Research for Change Management?
Discuss the statement that a positivist approach to research would be an inappropriate approach to researching change management. The generation of information is essential for the progress of our society, Knowledge and understanding of our natural, social, and economic environments have become the basis of modern societies and organizations and a firm grasp of research methods and processes have become an essential skill for modern managers. There are a few different approaches that may be taken to research and several…...
Change ManagementPhilosophyScienceStatistics
The Different Reasons for Choosing a Mechanical Engineering Course
I chose Mechanical Engineering because I enjoy solving problems, working with my hands, and I am good at mathematics. First, I chose Mechanical Engineering because I enjoy solving problems. This relates to my degree because an engineer's job is to solve a problem, from a customer, for known and unknown factors. In addition, I am required to think outside of the normal range of ideas and come up with unique solutions. Customers look for initiative solutions that will best answer…...
MathematicsMechanical EngineeringScience
Verbal Irony or Verbal Sarcasm
In their research, Abercrombie and Hovy touch upon a very interesting topic involving Twitter data sentiment analysis and sarcasm. As we all know, verbal irony or verbal sarcasm is not very difficult to detect as it is usually coupled with intensified/different tonal stress, hand and other gestures (eye roll), as well as the use of positive words to describe negative occurrences and exaggeration. On the other hand, written sarcasm cannot be detected so easily. In their study, Abercrombie and Hovy…...
IronyLearningStatisticsTechnology
I Lost My Desire To Be An Engineer
Some people may argue that if people are unsuccessful at something they’re trying to reach, it might not be a suitable thing for those people to complete, and failure can make someone lose trust in their progress. Having an optimal amount of involvement in something can drive the person to a profitable future. But if an individual doesn’t have any interest in what they are doing, the ideal thing to do is to search for another field or activity the…...
EducationMathematicsPassionPsychology
The Objective and Subjective Aspects of Psychology
I think that some aspects of psychology are objective while others are subjective Because psychology covers such a wide range of topics, it’s difficult to call the whole science objective or subjective. For example, many psychologists research the brain and what happens when you poke and prod at the different parts of it. That’s pretty straight forward and objective. A brain is a brain is a brain and most people have very similar ones. Many psychologists also use mathematics and…...
ExperimentPsychologyScienceStatistics
The Contrast between Increasing IQ Scores and Declining Creativity
According to the Flynn affect, the nation's IQ scores have been rising over the last century. This is probably happening because people want to get smarter. They also might be rising because the participants are motivated to get good grades because good grades can lead to scholarships and because the IQs are the broadest use of measuring mental ability. They are probably rising because you learn new things each year in school. If I was asked how to use the…...
CreativityLearningMathematics
The Development of My Model on Fantasy Football
A minute had already whizzed by. The countdown suddenly morphed from white to red as l frantically typed into the keyboard 30 seconds left. Sitting in my room on a mundane summer night could not have been more stressful. lnputting data left and right, I felt on the verge of reaching a conclusion. Either that or Microsoft Excel was ready to crash, I wasn't really sure. Only 10 seconds left, The apparent “conclusion" was not really a conclusion at all,…...
FantasyStatisticsTechnology
Different Ways of Measuring Information in a Data Set
There are many concepts 1 have learned so far in the course that has helped me un- derstand statistics better. Using concepts such as sample, population, mean, median, mode, standard deviation, coefficient of variation, 2 score, and probability. To start off with Sample size, Data sampling is defined as a statistical analysis technique primarily used to select, manipulate and also analyze a smaller portion of data points in order to identify similar or different patterns and trends in a larger…...
Data AnalysisMeanStatistics
A Study of Data Analysis
This is a process through which data is inspected. transformed and modelled in order to identify the most useful information. Data analysis leads to suggestions that point to conclusions and establishment of effective decisions that supports the data. Emphasizes that diverse techniques should be employed in the process of data analysis. Thus. various approaches are used depending on the nature of data being analysed. The field of study also determines the nature of data analysis. As a technique, data mining…...
DataData AnalysisStatistics
A Data Analysis and Assessment of the MCAS and Performance of Jake
I have worked with a student, from my small tutoring group which consists of 2 students, I am going to refer to the student’s name as “Jake" for this paper. Jake is a 10 year old boy, he is in the 4"‘ grade, His MCAS score shows that he is proficient in math, he performs according to the grade level. The data and assessment collected in the last five weeks, helps me indicate Jakes understanding about his geometric learning skills…...
Data AnalysisGeometryLearning
What Analysis Makes a Perfect Parent
Throughout your life, you may have closely watched your family members, relatives, or friends welcome a new addition to their families. It's an exciting and beautiful experience to witness them tackle this role as new Mothers and Fathers, You may have found yourself forming judgements and taking down mental notes while analyzing their parenting strategies. You’d think to yourself, "I want to raise my future children like them” or "Never in a million years would I treat my child like…...
ChildParentingRegression Analysis
CSR and Int’l Joint Ventures
This article aims at investigating organizational characteristics. It then develops a framework that relates to CSR practices of ”VS, It uses a regression analysis to produce a framework that facilitates a better understanding and identification of aspects that power CSR in local markets. The research findings indicate that consumers, competitors, and NGOs are the main people who determine responsible behaviors. The findings also indicate that there are considerate differences in line with CSR that is related to formative characteristics of…...
HypothesisRegression AnalysisResearch
Head and Neck Surgery: Advanced Clinical Otolaryngology
Abstract This study aimed at investigating the effectiveness of the undergoing an otolaryngology update training program in Saudi Hospitals in 2018. For that the researchers adopted the analytical description methodology by conducting a questionnaire as the study tool. The study sample consisted of (20) otolaryngology practitioners at the Saudi Hospital in 2018. The study results show that there is a statistically significant differences at the level of significance (α=0.05) of the effectiveness of the undergoing otolaryngology update training program in…...
HypothesisPlastic SurgeryStatistics
What Does Escapism Mean in Movies
Before I describe what escapism in movies means, I will explain what escapism means in general. Simply said, escapism is a way of avoiding boring things. You do this for example by thinking or reading a book. Another form of escapism is going on holiday, you are going somewhere and you are trying to have fun without having to think about boring things like studying or working. Escapism in movies means that movies offer you a way to escape reality.…...
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