Writing a Persuasive Essay Topic on Dragonwings

Is the Derivative of a Function the Tangent Line

In calculus, we learn that the tangent line for a function can be found by computing the derivative. So theres a close relationship between derivatives and tangent lines. However, they are not the same thing. For starters, the derivative f(x) is a function, while the tangent line is, well, a line. Instead, the correct statement is this: The derivative measures the slope of the tangent lines. Think about this: a clock is not the same thing as time. But if you want to know the time of day, you can go look at a clock to find out. A clock measures the time at any particular point throughout the day. Lets take a closer look at tangent lines. What is a tangent Line? A tangent line for a function f(x) at a given point x = a is a line (linear function) that meets the graph of the function at x = a and has the same slope as the curve does at that point. Sometimes we might say that a tangent line just touches the curve, or intersects the curve only once,f but those ideas can sometimes lead us astray. The graph below shows the tangent lines (in red, purple, and magenta) at three different points on a curve y = f(x) (in black). Slope and Derivatives So how do we know what the slope of the tangent line should be? After learning about derivatives, you get to use the simple formula, m = f(a). In this formula, the function f and x-value a are given. Your job is to find m, which represents the slope of the tangent line. Once you have the slope, writing the equation of the tangent line is fairly straightforward. Finding the Tangent Line Suppose you are asked to find the tangent line for a function f(x) at a given point x = a. Here is a step-by-step approach: Find the derivative, f(x). Plug in x = a to get the slope. That is, compute m = f(a). If not already given in the problem, find the y-coordinate of the point. As always, you plug the x-value into the function in order to get the y-value. Let b = f(a). Use the point-slope form and solve for y to find the equation of the tangent line. In other words, plug in your values of m, a, and b into the equation, y = m(x a) + b. The Derivative Measures Slope Lets take another look at that first step, Find the derivative. Remember, the derivative is a function (of the input variable x). By plugging in different input values, x = a, the output values of f(x) give you the slopes of the tangent lines at each point x = a. This is what we mean when we say that the derivative measures the slope of the tangent lines. If I want to know the slope of f at x = 1, then I compute f(1). And if I want to know the slope at x = -352/13, then I compute f(-352/13). Simple as that! Example: A Polynomial Now lets look at an example function, f(x) = x3 + 3x2 + 1. Well find the tangent lines at a few different points. First of all, find the derivative: f(x) = 3x2 + 6x. (for practice finding derivatives, check out this Magoosh article about derivatives). The function value and derivative value at a few points are shown in the table below: x-3-2-101 f(x) = x3 + 3x2 + 115315 f'(x) = 3x2 + 6x90-309 For the points listed, we can easily find the equation of the tangent line. Note, on the AP Calculus Exam, the multiple choice answers may be simplified. So Ive shown you both the point-slope form and the simplified, or slope-intercept, form of the tangent line. xTangent line, point-slope formTangent line, simplified -3y = 9(x + 3) + 1y = 9x + 28 -2y = 0(x + 2) + 5y = 5 -1y = -3(x + 1) + 3y = -3x 0y = 0(x - 0) + 1y = 1 1y = 9(x - 1) + 5y = 9x - 4 Now lets see the graph of y = f(x) together with the tangent lines that we just found. Summary So now you know. The derivative is not the same thing as a tangent line. Instead, the derivative is a tool for measuring the slope of the tangent line at any particular point, just like a clock measures times throughout the day. With this in mind, youll have no trouble tackling tangent line problems on the AP Calculus exam! For more about slope, tangent lines, and derivatives, check out these related Magoosh articles: Is the Derivative of a Function the Slope? and How to Find the Slope of a Line Tangent to a Curve.

Essay Topics For College And University Admissions - How To Write One

Essay Topics For College And University Admissions - How To Write OneEssay topics for college and university admissions are crucial to ensure that you are writing a well written essay. However, many students struggle with essay topics that they know how to write but cannot find the time to write, or write about a topic they do not understand. When you struggle with essay topics, you do not gain the benefit of academic and verbal skills that your essay will need to reach your intended audience.Your college essay is your chance to share with your reader your interests, accomplishments, and passions. This part of your application will determine whether or not you are accepted into the college you have applied to. It is a rare occurrence that a student is accepted into their first choice school. Your essay should be representative of who you are as a person, and what you want to give to the college you have applied to.College and university essays can be written about any number of topic , but essay topics for college and university admissions are focused around your experience in school. Your essay is your opportunity to share your thoughts, opinions, and information about your life. You may need to write about major aspects of your life. If you have lived away from home for long periods of time, the past year and a half should be about this.Writing a good essay does not require your writing every single day; you may only write once a week if you are working on your college application. But, your college essay topics for college and university admissions are focused around one theme per week. This theme is a reflection of your life. For example, you may choose to write an essay on your vacation, work, school, school events, sports, politics, or literature. It is important to understand how the topic you choose will reflect who you are as a person and what you want to say to the college you have applied to.The topic you choose for your college essays is up to you. M any students do not really like the idea of writing each and every day. I cannot say that a more enjoyable activity is reading a book or taking a stroll. But, the task of writing a college essay can be very overwhelming. Therefore, if you do not like doing it, or are not ready to do it all the time, make sure that your essay topics for college and university admissions are written in advance so that they are ready when the time comes.In the essay topics for college and university admissions, you should include details about your life and current activities. If you are going away for the summer, your essay could be about the events you plan to experience and the places you plan to go during the summer. Your essay can also contain other information that relates to what you are studying. All of these details can be included in your essay, or if you feel you should add something else, you can mention this in your essay as well.One of the reasons why essay topics for college and universi ty admissions are such a challenge for many students is because students have trouble determining what they want to say. They may have a good idea of what they want to say but are just unsure of what will actually make a good impression on the college admissions committee. Another reason why many students struggle with their essays is that they do not find the time to write them. If you are not taking action, you may not be able to get yourself a scholarship.With all of the attention that essays receive in college admissions, it is important that you know how to write an essay. If you want to stand out from the competition, you need to start writing your essay topics for college and university admissions early on. Your essays will determine your chances of getting into the college you want. Essays that are written in advance will be more likely to show the admissions committee that you are ready to go, and to provide them with the information they need to make a decision.