Calculus Limits Problems And Solutions Pdf

1 Solving the Euler equation Theorem. The notes cover both MATH 453/653 and MATH 454/654 so you can see what the next class will cover if you want to look ahead. 1 The Tangent and Velocity Problems 1. 10 Sample Multiple-Choice Problems on Techniques of Integration. Use of symbolic differentiation and integration utilities is also included. 5 points If v w 1, 1, 1 and v w 2, find the angle between v and w. Following each midterm exam. Explains concepts in detail of limits, convergence of series, finding the derivative from the definition and continuity. If you are viewing the pdf version of this document (as opposed to viewing it on the web) this document contains only the problems themselves and no solutions are included in this document. But the example is important for the concept that there is no actual value of the function when `x = 3`, but if we get really, really close to `3`, the function value. Thirty years of AP exam problems with solutions. The solution are short and clear. Which cover almost all topics of mathematics. Calculus uses limits to give a precise definition of continuity that works whether or not you graph the given function. ∴ We can write ( ) ( )( ) ( ) x12 x1x1 fxx1 x1x1. Power Rule: If r and s are integers, s 0, then lim x→c f x r s Lr s provided that Lr s is a real number. In this text, there are approximately four challenging exercises per section. Find the zeros Find roots. An Intuitive Introduction To Limits Home › Math › Calculus › An Intuitive Introduction To Limits Limits, the Foundations Of Calculus, seem so artificial and weasely: “Let x approach 0, but not get there, yet we’ll act like it’s there…. Part 1: Single Part 4: Vector Analysis. com is sure that your order as One – Two- Three essay writing services online. All possible errors are my faults. Use a table of values to estimate the following limit: lim x!¥ x p. Practical problems have always motivated calculus. You can google him to learn more, but I highly recommend the (historical fiction) book. Be aware that L'H^opital's rule may not apply to every limit, and it may not be helpful even when it does apply. Your solutions will be graded by humans for mathematical. divergent if the limit does not exist. Solved introductory problems of limits of functions. You will learn fastest and best if you devote some time to doing problems every day. The results provide new and robust tools for the problem of probability model uncertainty arising in financial risk,. About Khan Academy: Khan Academy is a nonprofit with a mission to provide a free. To find limits of functions in which trigonometric functions are involved, you must learn both trigonometric identities and limits of trigonometric functions formulas. pdf) or Calculus: Systematic Studies with Engineering Applications for Beginners. Don't show me this again. and circles. The week of March 23rd we will be reviewing Limits and Continuity. 3 Surface Normals 8. I'm practicing for my multivariable calculus exam and I'm having some trouble mostly because I have no way of knowing if my solutions are correct or not. Steven Strogatz, The Calculus of Friendship In this class, we’ll discuss limits, derivatives, and integrals. 3 The Completeness Axiom 1. Not surprisingly the end result is the same. Use this limit along with the other \basic limits" to nd the. Local Behavior. In this video I do 3 examples of optimization or max/min word problems using calculus. edu [email protected] Chapter 5: Introduction to Limits Lesson 5. ap calculus problems calculus problems calculus problems and answers calculus problems examples easy calculus problems Hard Calculus Math Problems. Math 19: Calculus Summer 2010 Practice Problems on Limits and Continuity 1 A tank contains 10 liters of pure water. Worksheet Do problems 10-19, 26-27 only. This is the equation of the circle of radius r centered at the origin. Find materials for this course in the pages linked along the left. Roots of polynomials of degree greater than 2. At 8, the limit is 3. The fundamental theorem of calculus. Evaluate the following limits or state that the limit does not exist. Compare it with "Study and Solutions Guide for Precalculus Functions and Graphs : A Graphing Approach and: Precalculus With Limits : A Graphing Approach" (0669417297), which is older: 2/e, (c)1997. 1 The Tangent and Velocity Problems 1. The remainder theorem. FREE with a 30 day free trial. Our online Integral Calculator gives you instant math solutions for finding integrals and antiderivatives with easy to understand step-by-step explanations. In fact, you can use calculus in a lot of ways and applications. Utterly trivial problems sit alongside ones requiring substantial thought. 1 Introduction 9. Notes of BSC Complete Solution of Calculus. 5 Optimization Problems 4. Real number system, functions and limits, topology on the real line, continuity, differential and integral calculus for functions of one variable. a space of functions that is closed under limits and tame enough to integrate. After a formal definition of the limit, properties are established that make "finding limits'' tractable. rx) = + 2x — d. Math 122B - First Semester Calculus and 125 - Calculus I Worksheets The following is a list of worksheets and other materials related to Math 122B and 125 at the UA. Complete BSc Notes of Mathematics Download in PDF or View Online. dvi version, pdf version, return to list of problem sets. Not surprisingly the end result is the same. • Use the navigational buttons at the bottom of each page to go to the next or previous page. Plot the solution curve (usually columns are not adjacent, use CNTRL key!). 1: A Preview of Calculus pp 47 Exercises 100 complete 1. For example, a typical problem goes like t. The Fundamental Theorem of Calculus. Let’s look at common limit at infinity problems and solutions so you can learn to solve them routinely. In this video I do 3 examples of optimization or max/min word problems using calculus. AP Calculus BC - Problem Drill 16: Indeterminate Forms, L'Hopital's Rule, & Improper Intergals Question No. Modules with asterisks (including chain rule, Taylor polynomials, Riemann sums, and arc length) allow you to change values and see the effect. Our solution manuals are written by Chegg experts so you can be assured of the highest quality! Precalculus Mathematics for Calculus Solutions Manual than. Bring whatever supplies (loose leaf paper, notebook, pen, pencil, etc) you personally like to use to take notes. L Z 9M apd neT hw ai Xtdhr zI vn Jfxiznfi qt VeX dCatl hc Su9l hu es7. Use this limit along with the other \basic limits" to nd the. We say that lim x→a f(x) = L, which is read “the limit as x approaches a of f(x) equals L,” if we can make f(x) arbitrarily close to L by taking x values sufficiently close to a but not equal to a. The Fundamental Theorem of Calculus (FTC) says that these two concepts are es-sentially inverse to one another. Limit Review Solutions; Word Problems. Also the limit as x approaches negative infinity, the value of c is a very large and negative number, is expressed below. Following each midterm exam. 2 LIMITSAND DERIVATIVES 2. A particular boat can propel itself at speed $20$ m/s relative to the water. Write an equation to relate the given and the to find. As a historical note the - formulation actually came long after Newton and Leibnitz pioneered the subject of calculus. Cancel Anytime. MATLAB provides various ways for solving problems of differential and integral calculus, solving differential equations of any degree and calculation of limits. Limit Comparison Test. Based on the answers from the problems above, find a pattern for the behavior. I've collected tests, mid-terms and finals from all over the United States and placed the most likely programs a college student will face inside. If you are having any trouble with these problems, it is recommended that you review the limits tutorial at the link below. This will not simply be a lecture class. When you see "limit", think "approaching" It is a mathematical way of saying "we are not talking about when x= ∞, but we know as x gets bigger, the answer gets closer and closer to 0". Wherever appeared, lnx represents the natural logarithm of x with base e. Local Behavior. INTEGRAL CALCULUS - EXERCISES 42 Using the fact that the graph of f passes through the point (1,3) you get 3= 1 4 +2+2+C or C = − 5 4. Find Limits of Functions in Calculus Fifteen limit problems with complete solutions with explanations. Location: Dolciani Math Center Tutorial Rooms. Varberg focuses on the most critical concepts freeing you to teach the way you want! This popular calculus text remains the shortest mainstream calculus book available – yet covers all the material needed by, and at an appropriate level for, students in engineering, science, and mathematics. Synthetic division by x − a. Our online Integral Calculator gives you instant math solutions for finding integrals and antiderivatives with easy to understand step-by-step explanations. PROBLEMS 67 Chapter 4: Differential Equations 72 32. 1 Real functions of a real. Talk to you soon!. Statement (1) looks short, but it is actually fairly complicated, and a few remarks about it may be helpful. Lessons are practical in nature informal in tone, and contain many worked examples and warnings about problem areas and probable "trick" questions. (i) L is the left-limit of f(x) at c iff for any , there exists such that. Calculus I Exam 1 Review Fall 2016 Problem 1: Decide whether the following statements are true or false: (a) If f, g are differentiable, then d dx (f g) = f 0g0. The rigorous definition for the limit is the so-called - definition. Historically, two problems are used to introduce the basic tenets of calculus. Evaluate limits at infinity and use them to determine any horizontal asymptotes of a function. evaluating limits joke worksheet students solve limit problems and answer will reveal the punchline to a joke limit solution to example 14 pdf calculus limits. 11 Solve everyday problems that can be modeled using and apply to the solutions of problems. Calculate the limit:. Show that f() x is even Show that (−)= ( ) symmetric to y-axis 3. This problem set explores the idea of the set of limit points of a set. problems (2003 - 2006). How to Do Word Problems: Read the problem. 2 The Directional Derivative 8. You may use L'H^opital's rule where appropriate. a(x)=1 h(x),b(x)=1. Study the method of evaluating definite integrals by the method of substitution. Calculators Mika Seppälä: Limits and Continuity Calculators Mika Seppälä: Limits and Continuity Limits by Rewriting Limits by Rewriting x − 3x + 2 2 x3 + x2 + x + 1 Problem 1 lim Problem 2 lim x →2 x −2 x →∞ x3 + 3x 2 + 5x + 2 x 2 − 3 x + 2 ( x − 1)( x − 2 ) Solution Rewrite = = x − 1. Average rate between 40 s and 100 s = slope of secant line. a(x)=1 h(x),b(x)=1. It is im-portant not to get the impression that this is the whole story. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. In each case sketch a graph with the given characteristics. Study the method of evaluating definite integrals by the method of substitution. Online Calculus course at San Francisco State University for transfer to your university, or medical school Take distance Calculus course online class with video lectures, live help, forum. I have included a worksheet that students can use to record their. If you want to nd the tangent to the. By Ron Larson Free Easy Access Study Guide and. Thus, we see that as x approaches 4, f(x) approaches 21. AP Calculus BC – Problem Drill 16: Indeterminate Forms, L’Hopital’s Rule, & Improper Intergals Question No. Trigonometric limits. Schaum Calculus pdf contains most of the Chapters of calculus like Functions, Limits, Continuity, The Derivatives, Curvature, Differential Equations of First and Second Order etc. Subtopic (1) Left and right hand limit, (2) Algebra of limit, (3) Calculation of limit using L’hospital’s rule, (4) Algebraic limits, (5) Limit of exponential and logarithmic function, (6) Limit of trigonometric function, (7) Continuity of a function, (8) Problems on differentiability. Use features like bookmarks, note taking and highlighting while reading 50 Challenging Calculus Problems (Fully Solved). We’ll practice mathematical reasoning and develop problem solving skills. Mean Value Theorem. Find the limit. Questions on derivatives of functions are presented and their detailed solutions discussed. Haese and Harris Textbook PDF. DIFFERENTIAL AND INTEGRAL CALCULUS, I i Preliminaries Preparatory reading. Write an equation to relate the given and the to find. I Worksheet by Kuta Software LLC. , Wiley{Interscience, New York. Chapter 5: Introduction to Limits Lesson 5. Here are a set of practice problems for the Limits chapter of the Calculus I notes. The book guides students through the core concepts of calculus and helps them understand how those concepts apply to their lives and the world around them. 1 - Limits: A Numerical and Graphical Approach. 1 Limits & Continuity Summary of Tests for Convergence and Series Flow Chart with practice problems. Sometimes the function is not defined around the point c but only to the left or right of c. Available for Pre-Algebra, Algebra 1, Geometry, Algebra 2, Precalculus, and Calculus. The limit of a product of functions is the product of the limits of the functions. Trigonometric Limits more examples of limits – Typeset by FoilTEX – 1. The red and blue regions have the same areas of. For example, a typical problem goes like t. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. 1 Jan 26 Limits and Continuity Notes: PDF Answers to Problems: PDF Stewart: § 11. pdf Solutions Assignment 3 WA3. 3 The Function Extension Axiom. Calculate lim x→1 f(x) and lim x→2. Synthetic division by x − a. Download Theory and Problems of Differential and Integral Calculus - Schaum's Outlines. Logistic Growth. One solution requires cross-multiplication and simplification. Philosophical Problems with Calculus, Note. Get for free solution manual to advanced calculus by r creighton buck ebook in pdf/epub/online. We asked the interviewee to solve physics problems which were similar to their homework or exam problems and required use of simple integration or differentiation. 1The trigonometric functions The Pythagorean trigonometric identity is sin2 x +cos2 x = 1, and the addition theorems are sin(x +y) = sin(x)cos(y)+cos(x)sin(y), cos(x +y) = cos(x)cos(y)−sin(x)sin(y). You can focus on particular type of problem, or choose to do problems drawn from various AP Calculus BC topics; the choice is yours. 4 Limits at Infinity and Limits of Sequences 11. Introduction In what follows I will post some challenging problems for students who have had some calculus, preferably at least one calculus course. return to list of problem sets Limits of Sets Coming into this course, students are familiar with limits of sequences and limits of functions as x->a. 1: Tangent Line Problem Tangent Line Problem of each of the following using the limit definition. We also present Brownian motion under sublinear expectations and the related stochastic calculus of Itô’s type. The following represent the initial cut-offs for each letter grade. At 8, the limit is 3. More concretely, a function solves the initial value problem if it solves the differential equation and, i. View Notes - Limits at Infinity - Practice Problems from CALCULUS/C 202 at Benedictine College. (a) Find the equation of the plane through the points P, Q and R. It is a collection of problems and solutions of the major mathematical competitions in China, which provides a glimpse on how the China national team is selected and formed. sin! Solution:c=3 5-39. There are many things one could say about the history of calculus, but one of the most interesting is that integral calculus was first developed by Archimedes of Syracuse OVER 2250 YEARS AGO! He was a very interesting guy. Hey I have. It is to be completed CORRECTLY, NEATLY, and on a SEPARATE sheet of paper. Online Calculus course at San Francisco State University for transfer to your university, or medical school Take distance Calculus course online class with video lectures, live help, forum. The problems are sorted by topic and most of them are accompanied with hints or solutions. Calculus E. An average student should be able to do most of the exercises. The factor theorem. 1 Miami Dade College -- Hialeah Campus Calculus I Formulas MAC 2311 1. These problems will be used to introduce the topic of limits. Don't show me this again. In this video I do 3 examples of optimization or max/min word problems using calculus. Calculus without limits. When packged with ALEKS Prep for calculus, the most effective remediation tool on the market, Smith and Minton offers a complete package to ensure students success in calculus. Problem 1 A salesman sold twice as much pears in the afternoon than in the morning. Bring whatever supplies (loose leaf paper, notebook, pen, pencil, etc) you personally like to use to take notes. I heard from one of my friends that there's a pdf version of the calculus book just floating around, is it true and can someone share? do you have the solutions. PROBLEMS 67 Chapter 4: Differential Equations 72 32. CALCULUS 2015. For lots of values of a;b;c, namely those where b2 ¡ 4ac < 0, the solutions are complex. If you'd like a pdf document containing the solutions the download tab above contains links to pdf's containing the solutions for the full book, chapter and section. Calculus For Dummies Patrickjmt Pdf Read/Download Continuity and Limits Made Easy - Part 1 of 2 - Duration: 6:25. Limit Involving Infinity. In fact, you can use calculus in a lot of ways and applications. Drill - Evaluating Limits. And write it like this: In other words: As x approaches infinity, then 1 x approaches 0. Trakimas Math WHS. Solving Limits at Infinity. Here is a trivial example. 3 The Tangent Line Problem 11. You'll get used to this notation with some more examples. LIMIT WORKSHEET #2. 01 Exercises 2. 3, The Limit of a Function, Algebraic Methods Finding limits using factorization and rationalization 14 App 3. It is a tool to describe a particular behavior of a function. x= 5: y= 305, price is $3. In addi-tion, other people are not acceptable. When this happens we write. 7 Even More Maxima and Minima. Calculus I and II is taught at Portland Community College using a lecture/lab format. and thus serve as a limit-freefoundation for the differential calculus. 2: Finding Limits Graphically and Numerically pp 54. People actually think calculus is a hyper abstract algebra when in fact it is putting math into real world solutions. The Sandwich Theorem. Over 2000 Solved Problems covering all major topics from Limits and Continuity of Functions to Systems of Differential Equations Clear Explanation of Theoretical Concepts makes the website accessible to high school, college and university math students. In the tangent line problem, you saw how the limit process could be applied to the slope of a line to find the slope of a general curve. This will not simply be a lecture class. The two notations for the limit of a sequence are: lim. Otherwise it is impossible to understand. These are the books for those you who looking for to read the Rman Recipes For Oracle Database 12c A Problem Solution Approach Experts Voice In Oracle, try to read or download Pdf/ePub books and some of authors may have disable the live reading. The solution are short and clear. 109 Therefore we can not just drop some of the limit signs in the solution above to make it. This is a typical problem in the study of introductory limits. Credit is not allowed for both MA 425 and MA 511. us/precal/pretest. Applications of Differentiation. However, a row of all “0”s does not automatically imply that the corresponding system has infinitely many solutions. This page was constructed with the help of Alexa Bosse. View Notes - Limits at Infinity - Practice Problems from CALCULUS/C 202 at Benedictine College. Calculus of Variations solvedproblems Pavel Pyrih June 4, 2012 ( public domain ) Acknowledgement. Ideal for self-instruction as well as for classroom use, this text helps students improve their understanding and problem-solving skills in analysis, analytic geometry, and higher algebra. Worked-Out Solutions. (Euler) Suppose f(x;y;y0) has continuous partial derivatives of the. Fast Facts: 1. Weir, ISBN: 0321878965, Thomas' Calculus: Early Free ebook download AAzea is the biggest community for free ebook download, audio books, tutorials download, with format pdf, epub, mobi,…and more. ap calculus problems calculus problems calculus problems and answers calculus problems examples easy calculus problems Hard Calculus Math Problems. Calculus analyses things that change, and physics is much concerned with changes. The derivative of a differentiable function needs to be determined frequently in calculus problems with minimum, maximum or the inflexion points of a function. , guess and check, model, solve a simpler problem, use a chart, use diagrams, role-play). differentiate , integrate , limits , function problems will become easy. Modules with asterisks (including chain rule, Taylor polynomials, Riemann sums, and arc length) allow you to change values and see the effect. Preface This manual contains hints or full solutions to many of the problems in Chapters 1, 2, and 3 of the text: J. Calculus helps scientists, engineers, and financial analysts understand the complex relationships behind real-world phenomena. Trigonometric Limits more examples of limits – Typeset by FoilTEX – 1. 5 - Differentiation Techniques. This approach puts the ideas of the founders of the calculus on a mathematically sound footing, and is easier for beginners than the more common approach via limits. Calculus 2 Problems And Answers Pdf 2. Sometimes, we can use algebra to simplify a rational function to get the limit. (Solution). More References and links on Calculus Calculus Tutorials and Problems. In all limits at infinity or at a singular finite point, where the function is undefined, we try to apply the following general technique. The factor theorem. This is the equation of the circle of radius r centered at the origin. Understanding Calculus: Problems, Solutions, and Tips immerses you in the unrivaled learning adventure of this mathematical field in 36 half-hour lectures that cover all the major topics of a full-year calculus course in high school at the College Board Advanced Placement AB level or a first-semester course in college. All three books are great, my personal favorite is the flrst one. What I have in mind when I contrast the old method of using limits and the new method of a more abstract emphasis on functions is what I see in the contrast between two basic but closely related textbooks. Let f(x) = ˆ x +1 if x 6= 1 3 if x = 1. Differentiation rules 3. this is the collection where you can download for free solution manual download or books for free. This section focuses on limits that involve absolute value, rational. Cancel Anytime. Calculus of Variations solvedproblems Pavel Pyrih June 4, 2012 ( public domain ) Acknowledgement. The neat thing about limits at infinity is that using a single technique you'll be able to solve almost any limit of this type. Assignment 2 WA2. Creighton Buck? Sep 16, 2007 Does anyone know where to find the solution manual to. For homework, quiz, or test help, contact me on Whatsapp at +639178611702. and circles. Start Calculus Warmups. In this video I do 3 examples of optimization or max/min word problems using calculus. Calculus is part of the acclaimed Art of Problem Solving curriculum designed to challenge high-performing middle and high school students. The instructor may want to recall the function sin(π x) from Stewart, as x goes to 0, in order to discuss the problem. About Best eBook Download Library Our mission is to help people find and download e-books they love. Evaluate the following limits or state that the limit does not exist. you will likely nd a similar problem addressed. Calculate the limit: Calculate the side limits to determine the sign of ∞. Solution concept. PDF Classroom Demonstrations Last Update: 5/2/2000. Improve your math knowledge with free questions in "Find limits using graphs" and thousands of other math skills. Thus, we see that as x approaches 4, f(x) approaches 21. Other handy things you can do with complex numbers 65 31. pdf Solutions Assignment 4 WA4. Find the limit. Limit Review Solutions; Word Problems. In this third edition, clarifications have been provided. All about the book Calculus: Problems and Solutions - bibliographic data, summary, search for links to download an e-book in PDF, EPUB or read online. FREE with a 30 day free trial. c) p x+1 d) xsinx:. 1: The Power and Exponential Rules with the Chain Rule. This page has been designed as a means to support my Calculus I (MA 113) students. \wxMaxima for Calculus II" June 2015 as free. Topics: Derive by limit process, Equation of tangent line, Where does a graph have a given slope, Sketch the derivative of a graph and equation Quiz on 2. Rather than repeat standard and routine exercises in this text, we recommend the use of WeBWorK with its access to the National Problem Library and around 20,000 calculus problems. These lab manuals serves as guides for the laboratory component of these courses. Calculus I and II is taught at Portland Community College using a lecture/lab format. MATLAB provides various ways for solving problems of differential and integral calculus, solving differential equations of any degree and calculation of limits. The limit of 1 x as x approaches Infinity is 0. (Euler) Suppose f(x;y;y0) has continuous partial derivatives of the. Problems in Calculus of One Variable Free E-book PDF Download Problems in Calculus of One Variable is an exhaustive reference on Calculus for students studying a preliminary course on Calculus. It can be used as a base for a Moore method or inquiry based class, or as a guide in a traditional classroom setting where lectures are organized around the presentation of problems and solutions. Right side limit. ) Homework #1 (Due to servon's comment, the solution of Problem 2 is wrong. This is a typical problem in the study of introductory limits. Find materials for this course in the pages linked along the left. 1 Miami Dade College -- Hialeah Campus Calculus I Formulas MAC 2311 1. This Calculus Handbook was developed primarily through work with a number of AP Calculus classes, so it contains what most students need to prepare for the AP Calculus Exam (AB or BC) or a first‐year college Calculus course. This Solution Manual is complete guide for students of Engineering. A limit is defined as a number approached by the function as an independent function's variable approaches a particular value. However, it requires that you be able to ``squeeze'' your problem in between two other ``simpler'' functions whose limits are easily computable and equal. A collection of auto-scoring calculus modules organized into books, chapters, and sections, with help and hints for the problems. Calculus covers all topics from a typical high school or first-year college calculus course, including: limits, continuity, differentiation, integration, power series, plane curves, and elementary differential equations. Options parameters. 62 Chapter 2 Limits and Continuity 6. Limit Test for Divergence. Many types of AP calculus problems require knolwdge of derivatives of functions. Solving limits with substitution, solving limits that need simplification and solving limits that do not exist, are the three types of examples shown. The fact-checkers, whose work is more and more important for those who prefer facts over lies, police the line between fact and falsehood on a day-to-day basis, and do a great job. Today, my small contribution is to pass along a very good overview that reflects on one of Trump’s favorite overarching falsehoods. Namely: Trump describes an America in which everything was going down the tubes under  Obama, which is why we needed Trump to make America great again. And he claims that this project has come to fruition, with America setting records for prosperity under his leadership and guidance. “Obama bad; Trump good” is pretty much his analysis in all areas and measurement of U.S. activity, especially economically. Even if this were true, it would reflect poorly on Trump’s character, but it has the added problem of being false, a big lie made up of many small ones. Personally, I don’t assume that all economic measurements directly reflect the leadership of whoever occupies the Oval Office, nor am I smart enough to figure out what causes what in the economy. But the idea that presidents get the credit or the blame for the economy during their tenure is a political fact of life. Trump, in his adorable, immodest mendacity, not only claims credit for everything good that happens in the economy, but tells people, literally and specifically, that they have to vote for him even if they hate him, because without his guidance, their 401(k) accounts “will go down the tubes.” That would be offensive even if it were true, but it is utterly false. The stock market has been on a 10-year run of steady gains that began in 2009, the year Barack Obama was inaugurated. But why would anyone care about that? It’s only an unarguable, stubborn fact. Still, speaking of facts, there are so many measurements and indicators of how the economy is doing, that those not committed to an honest investigation can find evidence for whatever they want to believe. Trump and his most committed followers want to believe that everything was terrible under Barack Obama and great under Trump. That’s baloney. Anyone who believes that believes something false. And a series of charts and graphs published Monday in the Washington Post and explained by Economics Correspondent Heather Long provides the data that tells the tale. The details are complicated. Click through to the link above and you’ll learn much. But the overview is pretty simply this: The U.S. economy had a major meltdown in the last year of the George W. Bush presidency. Again, I’m not smart enough to know how much of this was Bush’s “fault.” But he had been in office for six years when the trouble started. So, if it’s ever reasonable to hold a president accountable for the performance of the economy, the timeline is bad for Bush. GDP growth went negative. Job growth fell sharply and then went negative. Median household income shrank. The Dow Jones Industrial Average dropped by more than 5,000 points! U.S. manufacturing output plunged, as did average home values, as did average hourly wages, as did measures of consumer confidence and most other indicators of economic health. (Backup for that is contained in the Post piece I linked to above.) Barack Obama inherited that mess of falling numbers, which continued during his first year in office, 2009, as he put in place policies designed to turn it around. By 2010, Obama’s second year, pretty much all of the negative numbers had turned positive. By the time Obama was up for reelection in 2012, all of them were headed in the right direction, which is certainly among the reasons voters gave him a second term by a solid (not landslide) margin. Basically, all of those good numbers continued throughout the second Obama term. The U.S. GDP, probably the single best measure of how the economy is doing, grew by 2.9 percent in 2015, which was Obama’s seventh year in office and was the best GDP growth number since before the crash of the late Bush years. GDP growth slowed to 1.6 percent in 2016, which may have been among the indicators that supported Trump’s campaign-year argument that everything was going to hell and only he could fix it. During the first year of Trump, GDP growth grew to 2.4 percent, which is decent but not great and anyway, a reasonable person would acknowledge that — to the degree that economic performance is to the credit or blame of the president — the performance in the first year of a new president is a mixture of the old and new policies. In Trump’s second year, 2018, the GDP grew 2.9 percent, equaling Obama’s best year, and so far in 2019, the growth rate has fallen to 2.1 percent, a mediocre number and a decline for which Trump presumably accepts no responsibility and blames either Nancy Pelosi, Ilhan Omar or, if he can swing it, Barack Obama. I suppose it’s natural for a president to want to take credit for everything good that happens on his (or someday her) watch, but not the blame for anything bad. Trump is more blatant about this than most. If we judge by his bad but remarkably steady approval ratings (today, according to the average maintained by 538.com, it’s 41.9 approval/ 53.7 disapproval) the pretty-good economy is not winning him new supporters, nor is his constant exaggeration of his accomplishments costing him many old ones). I already offered it above, but the full Washington Post workup of these numbers, and commentary/explanation by economics correspondent Heather Long, are here. On a related matter, if you care about what used to be called fiscal conservatism, which is the belief that federal debt and deficit matter, here’s a New York Times analysis, based on Congressional Budget Office data, suggesting that the annual budget deficit (that’s the amount the government borrows every year reflecting that amount by which federal spending exceeds revenues) which fell steadily during the Obama years, from a peak of $1.4 trillion at the beginning of the Obama administration, to $585 billion in 2016 (Obama’s last year in office), will be back up to $960 billion this fiscal year, and back over $1 trillion in 2020. (Here’s the New York Times piece detailing those numbers.) Trump is currently floating various tax cuts for the rich and the poor that will presumably worsen those projections, if passed. As the Times piece reported: