Free essays on Algebra are educational resources that provide students with detailed information about various algebraic concepts and applications. These essays cover a diverse range of topics such as linear equations, quadratic equations, polynomial functions, inequalities, graphing, and more. They are designed to help students develop a deeper understanding of algebraic principles and assist them in honing their problem-solving and critical thinking skills. The essays are written by professional academics or experienced students who have excelled in the subject, and they are typically available online for free. They are excellent study aids for anyone seeking to improve their algebraic proficiency.
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
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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
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
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
Finding the Rotated Vector: Solving for w in Various Cases
Rotating a Vector Here's the problem: Let v denote a vector that has initial position v1 and final position v2. We know these two numbers. Now suppose we rotate v counterclockwise by an angle theta. The result is a new vector, denoted by w, that also has initial position v1 and final position v2. Let's say that we rotate vector v by an angle theta. If we do this, our new vector becomes w. The problem we are going to…...
Calculus
Derivatives of Exponential and Natural Log Functions
Derivatives of Exponential and Natural Log Functions We will introduce the concept of early transcendentals now, and then vou will learn more about them as we go along. We'll just incorporate them into our existing knowledge of products, quotients and derivatives. Now we will examine the derivatives of e to the x and natural log of x. Let's begin by doing e to the x. There are many ways to prove this. One definition of e is 1 plus x…...
Calculus
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
Calculus Basics: Limits, Functions, Derivatives, Integrals, and Series
Calculus is the study of change which focuses on limits, functions, derivaties, integrals, and infinite series. There are two main branches of calculus: differential calculus and integral calculus, which are connected by the fundamental theorem of calculus. It was discovered by two different men in the seventeenth century. Gottfried Wilhelm Leibniz – a self taught German mathematician – and Isaac Newton – an English scientist – both developed calculus in the 1680s. Calculus is used in a wide variety of…...
CalculusGeometryIsaac NewtonMathematicsPhysics
Allan Yashinski is currently a Temporary Assistant Professor
He received his B.S. in Mathematics with highest honors from the Bucknell University, Lewisburg, PA in 2007. Coursework is fulfilling the Computer Science major in the Bachelor of Arts degree program also has been completed. Later he received his Ph.D. in Mathematics from Pennsylvania State University in 2013, where his adviser was Nigel Higson. Mr. Yashinski is a co-organizer of the weekly Noncommutative Geometry Seminar at the University of Hawaii at Manoa. He also was a co-organizer of the…...
AlgebraGeometryMathematics
250 Words Essay About Fibonacci Sequence
Abstract: This paper gives a brief introduction to the famous Fibonacci sequence and demonstrates the close link between matrices and Fibonacci numbers. The much-studied Fibonacci sequence is defined recursively by the equation yk+2 = yk+1 + yk, where y1 = 1 and y2=1. By using algebraic properties of matrices, we derive an explicit formula for the kth Fibonacci number as a function of k and an approximation for the “golden ratio” yk+1 / yk. We also demonstrate how useful eigenvectors…...
AlgebraMathematics
Add Math Project 2019
Experience classroom environments which are Hellenizing, interesting and meaningful land hence improve their thinking skills. Experience classroom environments where knowledge and skills are applied in meaningful ways in solving real-life problems. Experience classroom environments where expressing ones mathematical thinking, reasoning and communication are highly encouraged and expected. Experience classroom environments that stimulates and enhances effective learning. Acquire effective mathematical communication through oral and writing, and to use the language of mathematics to express mathematical ideas correctly and precisely.Essay Example on…...
CalculusLearningMathematicsProject
Applying Math to Real Life
I think that many math topics have meaning and relevancy and are dependent on the path one takes in terms of finding real world application. For example, sports is largely dependent on sports. Decisions are made based regarding playing time as well as strategy based on percentages. In baseball, there is a strong use of math. Managers have to make decisions on which pitchers to start and, especially so in games of importance, those decisions are predicated upon statistical reality.…...
CalculusIsaac NewtonMathematicsTime
Matrices in Matlab
Matrices in Matlab You can think of a matrix as being made up of 1 or more row vectors of equal length. Equivalently, you can think of a matrix of being made up of 1 or more column vectors of equal length. Consider, for example, the matrix ? ? 1 2 3 0 A = ? 5 ? 1 0 0 ? . 3 ? 2 5 0 One could say that the matrix A is made up of 3…...
AlgebraMathematics
Application of Geometric Sequences and Series
A sequence is a set of real numbers. It is a function, which is defined for the positive integers. The value of the function at a given integer is a term of the sequence. The range of a sequence is the collection of terms that make up the sequence. If the sequence only be defined for the positive integers up to a given integer n is called infinite sequences. If a sequence defined only for positive integers up to a…...
CalculusMathematics
Cowling’s Rule Formula Analysis
The story of the history of the cubic equation illustrates a change in mathematics that came with the Renaissance: the understanding that mathematicians must publish their work to succeed; that instead of keeping mathematical knowledge as secrets within families, to be used as weapons in debates, or in mathematics contests, they must be available to the public. The solution of the cubic equation came at a time when both these conflicting views on mathematics were both in common practice; furthermore,…...
AlgebraMathematics
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