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.
Linear Approximations and Tangent Planes. Visualize in 3-dimensions
Linear approximations and tangent planes. Visualize in 3-dimensions We use the level curves to help visualize the graph of a function. The z-coordinate leaving the board is equal to the value of the function. As a visual aid, imagine a map of elevation levels, with zero representing the lowest point, and one through five gradually ascending to the highest point. All points in this line have an elevation of 0; all points on that line have an elevation of 1;…...
Calculus
Exploring Differentiability: Understanding Smoothness and Derivatives
Differentiability Differentiability is a measure of how smooth a function is. A function is differentiable at a point if its derivative exists at that point and is non-zero. The graph of a differentiable function is infinitely thin in the sense that it can be covered by an arbitrarily small rectangle. A simple way to think about differentiability is as follows: If you draw two tangent lines to a curve at two different points, and then draw all the possible curves…...
Calculus
Recitation video: Lagrange multipliers with 3 variables
Recitation video: Lagrange multipliers with 3 variables Lagrange multipliers can be used to optimize a function of several variables subject to a constraint. Find the maximum and minimum values of the function f(x,y,z)=x^2+x+2y^2+3z^2 as (x,y,z) varies on the unit sphere x^2+y^2+z^2=1 We have a function of the variables xv. equals × squared plus x plus 2y squared plus 3z squared. What we'd like you to do is find the maximum and minimum values that this function takes as the point…...
Calculus
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The Equation of Tangent Lines and Local Linearity
Tangent Lines We have been using the word "tangent" repeatedly. Now we are going to give an equation for the tangent line to a curve at a point and discuss how to determine whether a function is locally linear by examining its graph. We're going to create a function, fof x. We'll call it the slope of the tangent line. The derivative at any given point on this function will be the slope of the tangent line drawn to that…...
Calculus
Power Rule for Derivatives
Power Rule for Derivatives The power rule is one of a number of derivative rules in calculus. The power rule says that if a function is raised to a power, then the derivative is equal to that function raised to the same power times the original function's derivative. For example, suppose we are given a function f(x) and its derivative f'(x), then the power rule savs: If f(x) = xn, then f'(x) = nxn-1 The power rule is a shortcut…...
Calculus
Exploring Vector Decomposition: Unraveling Data Components and Force Analysis
Vector decomposition Vector decomposition is a technique used in statistical analysis to break down a dataset into its various components. The purpose of vector decomposition is to determine the variance of each component, which allows for a more accurate analysis of the data. Vector decomposition can be used to analyze stock prices, economic data and other types of data sets that contain multiple variables. Vector decomposition uses a series of mathematical equations to break down each variable into its most…...
Calculus
Comparison to the Quadratic Formula
Comparison to the quadratic formula It needs to be pointed out: Here we have a special quadratic expression. The quadratic formula is similar, but uses b2 - 4ac rather than b2 - 4bc. Let's see how the quadratic formula applies here. Let's present the same data differently to reach the same conclusion. Here, we have written the expression as y2 times a x over y2 plus b x over y + c. Since x and y are both non-negative, the…...
Calculus
Matrix-Matrix Multiplication: Combining Matrices and Unraveling Dot Products
Matrix-Matrix multiplication A Matrix-Matrix multiplication is a mathematical operation that multiplies the two matrices. The first matrix (the "a" matrix) is multiplied by the second matrix (the "b" matrix). The resulting third matrix iS called a "c' matrix. Matrix-Matrix multiplication is an operation that combines two matrices together to create a third matrix. The process involves multiplying every element of one matrix by the corresponding element of another, and then adding up all of these products. The result is a…...
Calculus
Power Rule for Derivatives. Example Problem 1
Power Rule for Derivatives. Example Problem 1 f(x)=2x^4 -7x^3 +5x-6sqrtx Find f'(x) 6x^1/2 f'(x)=8x^3 -21x^2 +5-3x^-(1/2) f'(x)=8-21+5-3=-11 Let's create a function f of x. We'll say, 2x to the 4th minus 7x cubed plus 5x minus 6 square root of x. Let's find the derivative off prime, given that f is equal to 1. Now that we know the power rule, we can find f'(x), which is just one of many derivatives. And by the way, it's really important that…...
Calculus
Warm up: Linear approximations
Warm up: Linear approximations Here is a little warm up. We are gonna to review of linear approximation. The task is to find the linear approximation of this function. f(1+Δx),1+Δy)~~-1+...Δx+...Δy If(x,y)near(1,1) F(x,y)~~...x+...y+... The main point of the warm-up is to practice writing both linear and quadratic approximations using two different methods. We will then check to see why there are two ways of writing it....
Calculus
Order Matters:AB is not BA in Matrix Operations
Caution: order matters. AB is not BA Note that AB and BA are not the same thing at all. From this interpretation, it is clear that the conversion of oranges to bananas is not the same as the conversion of carrots to oranges. The other hand--well, actually it may be worse than we thought. The thing may not even be well-defined. Cause if the width of A is equal to the height of B. then we can do this product.…...
Calculus
Infinity and Asymptotes: Exploring Limits and Vertical Asymptotes
Limits Involving Infinity In this section, we'll look at limits to infinity, positive infinity, and horizontal and vertical asymptotes. When we talk about horizontal asymptotes and limits to infinity, some functions might not have a horizontal asymptote. lim_(x-> infinity) f(x) [Graph] lim_(x-> -infinity) f(x) [Graph] [Graph] __________________________________________________ [Graph][Graph] lim_(x->a) f(x) [Graph] [Graph] A function could be represented by this equation. And as x approaches infinity, the value of y approaches 0. But instead of the asymptote being at 0, we…...
Calculus
Linear Approximations and Tangent Planes. Linear Approximation Review
Linear approximations and tangent planes. Linear approximation: review Given a function of one variable, g of x, linear approximation tells us how the function a changes if we change x by a small amount. If we look at the graph of g at x0 plus Delta x, we see that it is approximately the graph of g of x0 plus g prime of x0 times Delta x. This approximation works well if Delta x is small. Let's do simple example.…...
Calculus
Robot Arm Matrix Setup
Robot Arm Matrix Setup It would be good to work with you on this problem because it was challenging problem before, and many students were uncertain how to approach it. In addition, it is a part of our theme of matrices. The fact that mathematical expressions that are matrices keep coming up in various problems that don't sound related to each other should make us think about how such expressions might be used. Here we can think of matrice. Because…...
Calculus
Power Rule for Derivatives. Example Problem 2
Power Rule for Derivatives. Example Problem 2 f(x)=x^3 -9x^2 -48x+5 f'(x)=0 f'(x)=3x^2 -18x-48=0 3(x^2 -6x-16)=0 3(x-8)(x+2)=0 x=8, -2 So. Now you need to find the values of x for which f(x) = x3 - 9x2 - 48x + 5 has a horizontal tangent. If we have a horizontal tangent, that means there's going to be a slope of 0-that is, f(x) = 0. That means the derivative of f(x) is equal to 0. We will find f prime of x,…...
Calculus
The Geometry of Linear Approximation
The geometry of linear approximation A linear approximation of a function is a polynomial that passes through the sample points and has the same slope at each of them. The linear approximation of a function is often used in statistics; many statistical methods assume that the data are generated by some underlying model, and the linear approximation provides an alternative expression for the model's predictions. Let's go through it. So we take the x derivative of that. If we plug…...
Calculus
Finding Limits Analytically. Example Problem 3
Finding Limits Analytically. Example Problem 3 Find: lim_(x->2) f(x) [Graph] Let f(x)={x^2+4; 3x+1; x>2, x<=2 lim_(x->2) f(x) lim_(x->2) -3x+1=7 -> lim_(x->2)-f(x)/=lim_(x->2)+f(x) lim_(x->2)+x^2 +4=8 -> :. lim_(x->2) f(x) DNE We can see a piecewise function at work in the following example. So, x to the second power plus and we'll do this for values greater than 2. And perhaps we'll do 3x plus 1 for values less than or equal to 2. We're interested in finding the limit as x approaches…...
Calculus
Matrix Multiplication: Understanding its Origins and Order
Where does matrix multiplication come from? What AB Represents The product AB represents the transformation B followed by transformation A. That is a counterintuitive concept, which we are accustomed to writing from left to right. Unfortunately, matrices are defined by rules that require multiplication from right to left. If you think about it, when you write a function f times a different function g, what you're really saying is apply g, then f. Multiplying matrices is done in reverse order…...
Calculus
Decomposition of Vectors: Analyzing the General Case
General Case So it is this vector. Why? Well, if we consider the right triangle formed by this vector, we can read off its components from this right triangle. We begin by noting that the length of w is equal to the length of v. So the length of w is v2. This side is v2, the length of w. What about this side? The cosine of theta is v2 on this side, and the sine of theta is v2…...
Calculus
Derivatives of Inverse Trig Functions
Derivatives of inverse trig functions In the real world, trigonometric functions are used to model many trpes of motion. The derivative of these functions can be used to determine how quickly an object is accelerating, or how fast it is moving at any given time. Knowing the derivative and inverse relations of trig functions has practical applications in physics and engineering. We will develop the basic formulas for inverse trigonometric functions. To begin, let us consider the arc sine function.…...
Calculus
Cases for a Quadratic Function
Cases for a quadratic function There are three possibilities to consider, let's first eliminate the most complicated one. Suppose 4ac - b2 is negative. If this is the case, then it means that what is between the brackets is actually a positive quantity while the second term will be negative times y2. So it will be a negative quantity. One term will be positive, and one term will be negative. This tells us that we actually have a saddle point…...
Calculus
Finding Limits Analytically. Example Problem 1
Finding Limits Analytically. Example Problem 1 Find: lim_(x->-4) (x^2 -6x-40)/(x^2 +6x+8) lim_(x->-4) ((x+4)(x-10))/((x+4)(x-2)) lim_(x->-4) (x-10)(x-2)=-14/-2=7 We can find the limit of an algebraic expression in several ways. One way is to use simple substitution, but that doesn't work for indeterminate forms; evaluating numerator and denominator with negative 4 will yield 0/0. To clear up the 0/0, we need to factor. The top factors of 4x - 10 are 4x + 2 and 4x - 2 The denominator factors of 4x…...
Calculus
Definition of Derivative. Example Problem 1
Definition of Derivative. Example Problem 1 Order the slopes of the tangent lines at the given points for the graph below. [Graph] Let's consider a function f. We'll plot the function on the standard coordinate plane and consider the slopes of its tangent lines at various points. Basically, we are going to order the slopes of the lines tangent to the graph of fat given points from least to greatest. Of course, one can be negative and one can be…...
Calculus
Level Curves and Partial Derivatives
Level curves and partial derivatives. Partial derivatives: definitions Partial derivatives are a notation used to express the derivative of a function with respect to one of its variables, holding other variables constant. A function of several variables may have partial derivatives with respect to each variable, but it does not have a derivative in the usual sense. δf/δx(x_0,y_0)=lim_(Δx->0) (f(x_0+Δx,y_0)-f(x_0,y_0))/Δx partial δf/δx(x_0,y_0)=lim_(Δy->0) (f(x_0,y_0+Δy)-f(x_0,y_0))/Δy This symbol is a capital d with a curly line on the bottom. It is not a straight…...
Calculus
The Algebraic Intricacies of Dot Products
Warm up ' A first warm-up question involves the algebra of dot products. The dot product of the vector 2v with the vector 2w, when compared with y times w, is equal to..? There are two choices: one is to multiply V 2 times v dot w, or is it 4 times V dot w? (2v->)*(2w->)=2(v->*w->) or 4(v->*w->) The question remains. Which one is true? The dot product of 2v with 2w is 2w1 comma 2w2 times w2. What happens…...
Calculus
Partial Derivatives and the Gradient. Gradient and Theorem
Partial derivatives and the gradient. Gradient and theorem Here's the first cool property of a gradient: The gradient vector is perpendicular to the level surface corresponding to setting the function w equal to a constant. ∇w I level surface w ={constant} If we draw a contour plot of my function, then we will have a two-variable function with level curves representing the gradient at each point. If we then draw the gradient vector at each point on that contour plot,…...
Calculus
Matrices and Rotation. Matrix Multiplication
Matrices and Rotation. Matrix Multiplication Let's begin by discussing matrices. You should remember how to multiply a matrix times a vector. If we have a matrix, say a, b, c, d (or these could be numbers) and multiply it by a vector v, v1, v2 (or these could be numbers), what do we get then? A new vector. For the top component, we multiply times v1 by a plus b times v2. Somehow, if you put my fingers like this,…...
Calculus
Exploring the Definition of Derivative: Example Problem and Tangent Line Slope
Definition of Derivative. Example Problem 2 Use the definition of derivative to find f' (x) of the function: f(x)=3x^2 -4 Let's use the definition of a derivative to find the derivative of the function f where f(x) = 3x2 - 4. The derivative of a function f(x) is defined as the limit of the difference quotient as h approaches zero. C| If you recall, the derivative of a function f(x) is defined as lim h-+0 f(x + h) - f(x)…...
Calculus
Warm up 2
Warm up 2 Now, here is a question for you regarding the geometry of dot products, regarding kind of visualizing what dot products mean. Suppose I have a vector v over here and then I have vector u over here. And if I extend u as a straight line, I can extend it until it makes a right angle with this vector. This would be like making a right triangle here. Then I have another vector that goes in the…...
Calculus
Linear Relations and Matrix Multiplication
What Are Matrices? A matrix is a rectangular array of numbers or other data. In mathematics, it is a rectangular array of numbers whose elements are indexed by non-negative integers. Although the term "matrix" can refer to diverse mathematical objects, its most familiar meaning is that of an n-by-n table filled with elements whose ith entry is denoted by aij. So maybe you have had a little exposure to matrices in high school, but if you haven't, here is just…...
Calculus
Finding Limits Analytically. Example Problem 2
Finding Limits Analytically. Example Problem 2 Find: lim_(x->3) (sqrt(2x+10-4))/x-3 * (sqrt(2x+10+4))/(sqrt(2x+10+4)) lim_(x->3) (2x+10-16)/((x-3)(sqrt(2x+10+4))) lim_(x->3) 2x-6/(x-3)(sqrt(2x+10+4)) lim_(x->3) 2/(sqrt(2x+10+4))=2/8=1/4 Another algebraic technique we can use to clear up limits of indeterminate forms, 0/0, is to rationalize the denominator. Clearly, factoring the denominator is not going to work here. The square root of a negative number is also a negative number and therefore cannot be factored. Since we cannot factor the denominator, we must rationalize it. To rationalize a fraction, we multiply both…...
Calculus
Level Curves and Partial Derivatives. Computing Partial Derivatives
Level curves and partial derivatives. Computing partial derivatives There is a notation used in physics and applied math that allows one to compute these things. So how to compute? How to compute? to find δf/δx=f_x? treat y as constant x as variable Ex: f(x, y)=x^3 y+y^2; δf/δx=3x^2 y+0 δf/δy=x^3 +2y In Mathematics, the same symbol may be used for both a variable and one of its derivatives. For example, f(x) is the derivative of f(x), where y is treated as…...
Calculus
Equations of Lines and Normal Vectors
Equations of lines and normal vectors We use numbers now, but you can substitute any letter for each variable. If you have ax plus by plus c, the same story applies. By the same reasoning, if the line ax + by + c = 0, then the vector (a, b) is perpendicular to the line. ax + by + c = 0 That's it. Can zero be replaced with any number? Or, can it be like ax plus by plus…...
Calculus
Functions of 3 Variables. Linear Approximation
Functions of 3 variables. Linear approximation Suppose we want to find the tangent plane to the surface whose equation is x2+y2-z2=4 at the point (2, 1, 1). find the tangent plane to surface x^2+y^2-z^2=4 at 7. 1, 1 )? A way to do this is to solve for z in the equation z = f(x, y); once that is done, we can write z as a function of x and y. then find a tangent plane approximation of the graph…...
Calculus
Critical Point Type for a Quadratic Function
Critical point type for a quadratic function We will use the second derivative test to find a maximum or minimum value for an equation. By definition, if y'=0, then there must be a local maximum or minimum at that point. The main problem is that we have more possible situations and have several derivatives, so we need to think harder about how we will decide. However, it will involve the second derivatives again. Let us examine an example in which…...
Calculus
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
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