...Project Gutenberg EBook of Calculus Made Easy, by Silvanus Thompson This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org Title: Calculus Made Easy Being a very-simplest introduction to those beautiful methods which are generally called by the terrifying names of the Differentia Author: Silvanus Thompson Release Date: October 9, 2012 [EBook #33283] Language: English Character set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK CALCULUS MADE EASY *** Produced by Andrew D. Hwang, Brenda Lewis and the Online Distributed Proofreading Team at http://www.pgdp.net (This file was produced from images generously made available by The Internet Archive/American Libraries.) transcriber’s note Minor presentational changes, and minor typographical and numerical corrections, have been made without comment. All A textual changes are detailed in the L TEX source file. This PDF file is optimized for screen viewing, but may easily be A recompiled for printing. Please see the preamble of the L TEX source file for instructions. CALCULUS MADE EASY MACMILLAN AND CO., Limited LONDON : BOMBAY : CALCUTTA MELBOURNE THE MACMILLAN COMPANY NEW YORK : BOSTON : CHICAGO DALLAS : SAN FRANCISCO THE MACMILLAN CO. OF CANADA, Ltd. TORONTO CALCULUS MADE EASY: BEING A VERY-SIMPLEST...
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...Derivative=limf(x+change in x) –f(x) R=x*p P=R-C Change in x Limits: Point Slope form: y-y,=m(x-x,) Hole (removable discontinuity) Jump: Limit does not exist Vertical Asymptote: Limit does not exist Walking on graph at x=#, what is the y-value? Find the equation of a tangent line on f(x)=1/x at (1,1) Ex1: lim x^2+4x+3 = (-1)^2 +4(-1) +3 =0 point: (1,1) f(x)=x^-1 m=-1 = -1 = -1 = m x-1 x+1 -1+1 0 m=f’(x) f’(x)=-1x^-2 (1)^2 1 ------------------------------------------------- y-1=-1(x-1) y-1=-x+1 = y=-x+2= EQUATION Product Rule: f’(x)=u’v+v’u Quotient Rule: f(x)=h(x)g’(x)-g(x)h’(x)=lowd’high-highd’low [h(x)]^2 bottom^2 Chain Rule: derivative of the outside(leave inside alone)*derivative of the inside Implicit Differentiation: (1) take derivative of each term normally, if term has y on it, we will multiply it by y’ Critical Points: (1) Find f’(x); (2) Set f’(x)=0, solve it; (3) Plot points on # line; (4) Test points around the points in step 3, by plugging them into derivative. If positive: up If negative: down; (5) Write our answer as an interval Max and Mins (relative extrema): (1) Do all the up and down stuff from 3.1; (2) If you went up then down you have a max; if you went down then up you have a min; (3) Label the points (x,y) for max and mins to get the y, go back to f(x) Ex2: f(x) =1/4x^4-2x^2 a) Find the open intervals on which the function is increasing or...
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...University of Connecticut Department of Mathematics Math 1131 Sample Exam 1 Spring 2014 Name: This sample exam is just a guide to prepare for the actual exam. Questions on the actual exam may or may not be of the same type, nature, or even points. Don’t prepare only by taking this sample exam. You also need to review your class notes, homework and quizzes on WebAssign, quizzes in discussion section, and worksheets. The exam will cover up through section 3.2 (product and quotient rule). Read This First! • Please read each question carefully. Other than the question of true/false items, show all work clearly in the space provided. In order to receive full credit on a problem, solution methods must be complete, logical and understandable. • Answers must be clearly labeled in the spaces provided after each question. Please cross out or fully erase any work that you do not want graded. The point value of each question is indicated after its statement. No books or other references are permitted. • Give any numerical answers in exact form, not as approximations. For example, one-third 1 is 3 , not .33 or .33333. And one-half of π is 1 π, not 1.57 or 1.57079. 2 • Turn smart phones, cell phones, and other electronic devices off (not just in sleep mode) and store them away. • Calculators are allowed but you must show all your work in order to receive credit on the problem. • If you finish early then you can hand in your exam early. Grading - For Administrative Use Only Question:...
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...Differentiation Rules (Differential Calculus) 1. Notation The derivative of a function f with respect to one independent variable (usually x or t) is a function that will be denoted by D f . Note that f (x) and (D f )(x) are the values of these functions at x. 2. Alternate Notations for (D f )(x) f (x) d For functions f in one variable, x, alternate notations are: Dx f (x), dx f (x), d dx , d f (x), f (x), f (1) (x). The dx “(x)” part might be dropped although technically this changes the meaning: f is the name of a function, dy whereas f (x) is the value of it at x. If y = f (x), then Dx y, dx , y , etc. can be used. If the variable t represents time then Dt f can be written f˙. The differential, “d f ”, and the change in f , “∆ f ”, are related to the derivative but have special meanings and are never used to indicate ordinary differentiation. dy Historical note: Newton used y, while Leibniz used dx . About a century later Lagrange introduced y and ˙ Arbogast introduced the operator notation D. 3. Domains The domain of D f is always a subset of the domain of f . The conventional domain of f , if f (x) is given by an algebraic expression, is all values of x for which the expression is defined and results in a real number. If f has the conventional domain, then D f usually, but not always, has conventional domain. Exceptions are noted below. 4. Operating Principle Many functions are formed by successively combining simple functions, using constructions such as sum...
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...Computer Assignment Use Wolfram Alpha or any other technology to answer the questions below. Copy all relevant answers into this Word document. Save the Word document and send it to me via email attachment. Do NOT forget to type your name into the document, and include in your responses the commands you used to get the answers. 1. Consider the function f(x)=(e^2x-1)/x Find the limit of f(x) as x approaches zero. 2. Define the function Find the derivative of that function. Find f’(0.67) (the first derivative at 0.67). What does that mean for the function f at the point? Find f’’(0.67) (the second derivative at 0.67). What does it mean for the function f at that point? Find all points where the derivative is zero. A) B) C) D) 3. Define the function Find the derivative of the function and use Wolfram Alpha to confirm your answer. Find all points where the derivative is zero and classify them as local extrema, if possible Determine if f is increasing (going up) or decreasing (going down) between the points found in (b) A) B) Local extreme’s are listed C) Increasing 4. Find the following integrals: a) b) 5. Find the area between the graph of f(x) = (x2 – 4) (x2 - 1) and the x axis. Note that one simple definite integral won’t do it, you will need to carefully determine where the function is positive and negative and integrate accordingly, perhaps using multiple steps. 6. Solve the system...
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...Gateway Case Analysis As part of our analysis, we focused on the following goals: 1) Minimize the chance of aircraft crash losses and insurance costs combined exceeding $37 million in the next year, and 2) Obtain the insurance coverage at lowest cost over the five-year period. We performed our analysis in 5 stages, eliminating one prospective insurance plan or confirming our observations in each stage. Crystal Ball software served as an analytical tool for our analysis. The simulations conducted for our analysis consisted of 10,000 trials each. See Appendix for illustrations related to this analysis. Analysis Detail Stage 1: A comparative analysis of the maximum values of the total losses and insurance costs in the first year showed that there is only one insurance plan that guarantees that the next year losses are not going to exceed $37 million: HIC ($33.8 million maximum). RNCN1 has a maximum of $43.8 million, which is reasonably close to the $41 million limit that we are aiming to avoid in the first year. We also noted that the Self-insurance plan has minimal expected costs of $4.6 million in first year, but bears the highest volatility, which may lead to losses of up to $170.5 million (See Exhibit 1). At this stage, we exclude the RNCN2 plan from further considerations because even though is offers low expected costs, the probability of going over the limit of $37 million and incurring costs of $60 million is significant for the purposes of this analysis. We plan to...
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...Analytic functions Introduction I was interested in doing my Internal assessment in functions particularly in analytic functions as I was much fascinated with topic (functions)during the study of mathematics during the course. The theory part that includes Taylor series as well as its coverage on complex functionality. In this exploration I surveyed on the theorems associated with analytic functions as well as its functions. This has helped me widen my knowledge and mathematical skills on complex numbers, calcus and functions as topics studied during the course. I have always to kept correct justification on the theories and have used required mathematical models and correct interpretation in the results that I got in the theory processing. Analytic functions has a very wider application on Analytic modulated system whereas there is a general theory of analytic modulation system and they are developed in the transmitted signal σ(t) = Re {eiwctf(z(t)}.Due to this noticeable physical application in life, it motivated me to write a this maths exploration. In mathematics, an analytic function could be simply defined as a function that is locally given by a convergent power series. There exist two parts namely real analytic functions and complex analytic functions, functioning differently. These functions are infinitely differentiable, But as said above functions of each type are infinitely differentiable, having the complex analytic functions exhibiting properties that...
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...The Application of the Profit Function in a Business’s Growth Differential Calculus May 24, 2014 The application of calculus in a business is extremely important since calculus is considered as the study of changes. Its complexity in the study of changes has become one of the humankind’s greatest tool for analyzing changes in the marketplace. The profit function was created with the main purpose for businesses to understand how the changes in revenues and in costs would generate a profit or not profit at all. A profit can be generated when the amount of revenues is higher than the amount of costs. When a business starts, is normal to not gain any profit at all, in fact, most of the time, a business tends to lose money in its first year. However with that information, a business can analyze the money loss of that year and determine any gaps or holes that prevents the maximization of the profit and have a more prosperous result for the following year. For instance, if a company suffer a loss in profit, they can analyze the profit function to determine the main reason of why there was not a positive profit. If they see that the problem of the profit function was a low revenue then they can regulate the sale of products or services and price control, or if the problem lies in the cost function, they can adjust or lower the costs and expenses made by the business. The graph of a profit function can show at what time of the year the company tends to...
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...50=3x+5 50-5=3x 45=3x 45/3=x X=15 minimum 80=3x+5 80-5=3x 75=3x 75/3=x X=25 maximum F(x)=-x^2+2x+2 The horizontal distance, x, is greatest at f(x)=0 where f(x) is the height -x^2+2x+2=0 Using the quadratic formula X= (-b±√(b^2-4ac))/2a A=-1 B=2 C=2 X=(-2±√(2^2-4(-1)*2))/(2*(-1)) X=(-2±√(4+8))/(-2) X=(-2-2*√3)/(-2) X=1+√3 X=1+1.732 X=2.732 df/dx = -32x+200=0 -32x+200=0 -32x=-200 X=25/4 f(25¦4)=-16*(25¦4)2 + 200*(25¦4)+4 f(25¦4)=-625+1250+4 f(25¦4)= 629 feet F(x) = 2x +1. Where x is the 6th hour. Therefore the 6th hour, x=6. f(6) = 2*(6) +1 = 12 +1 =13 customers P(x)=x^2-4000x + 7800000 3800000=x^2-4000x + 7800000 x^2-4000x+4000000=0 (x-2000)^2=0 x=2000 P(2000)=3800000 number of items sold=2000 f(x) = 20000(1/2)x Value in 3 years = f(3) = 20000(1/2)3 = $2500 The max and min occur when the derivative is zero. Let's take the derivative: -2 * X + 3 = 0 -2 * X = -3 X = 3/2 = 1.5 the top is at 1.5 meters. Substitute x into the equation: f(x) = -x^2 + 3x + 6. f(1.5) = - 1.5^2 + 3 * 1.5 + 6 = - 2.25 + 4.5 + 6 = 8.25 meters x=y^2 x = 225, y = -15 x = 196, y =...
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...Academic Year 21250 Stevens Creek Blvd. Cupertino, CA 95014 408-864-5678 www.deanza.edu 2015 - 2016 Please visit the Counseling Center to apply for degrees and for academic planning assistance. A.A.T./A.S.T. Transfer Degree Requirements 1. Completion of all major requirements. Each major course must be completed with a minimum “C” grade. Major courses can also be used to satisfy GE requirements (except for Liberal Arts degrees). 2. Certified completion of either the California State University (CSU) General Education Breadth pattern (CSU GE) or the Intersegmental General Education Transfer Curriculum (IGETC for CSU). 3. Completion of a minimum of 90 CSU-transferrable quarter units (De Anza courses numbered 1-99) with a minimum 2.0 GPA (“C” average). 4. Completion of all De Anza courses combined with courses transferred from other academic institutions with a minimum 2.0 degree applicable GPA (“C” average). Note: A minimum of 18 quarter units must be earned at De Anza College. Major courses for certificates and degrees must be completed with a letter grade unless a particular course is only offered on a pass/no-pass basis. Associate in Science in Business Administration for Transfer A.S.-T. Degree The Business major consists of courses appropriate for an Associate in Science in Business Administration for Transfer degree, which provides a foundational understanding of the discipline, a breadth of coursework in the discipline, and preparation...
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...the prologue and chapter one of “The Calculus Diaries”, my perspective on calculus and its concepts have changed. “The Calculus Diaries” describes the history of calculus, such as who discovered it, when it was discovered, and how it can be used in everyday life. It starts out by describing the story of Archimedes, who invented devices to help fend off the Roman Empire from invading Syracuse. He was considered to be the first person to describe calculus concepts. The author describes that the two main concepts that make up calculus are the derivative and the integral. The author also describes his personal conflicts with calculus in the past and what it took for him to overcome his hatred for calculus and math in general. The...
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...Game Theory Math Review 1 1.1 Function Definition If we write down the relation of x and y as follows, y = f (x) this means y is related to x under the rule f . Also we say that the value of y depends on the value of x. This relation y = f (x) is called as a function if 1) the rule f assigns a single x value to single y value or 2) assigns multiple x values to single y value. 2 2.1 Shape of function When f (x) = ax + b Suppose that the function is given as follows. y = f (x) = ax + b ´ 1) Slope: f (x) = a and Y −intercept: b Y − axis is b.. b −a 2) Intersection with Y − axis: This is the case when x = 0. So from f (0) = b, the Intersection with 3) Intersection with X − axis: This is the case when y = 0. So the intersection with X − axis is from ax + b = 0. Example Suppose y = f (x) = ax + b. Draw this function in following each case. 1) When a > 0, b > 0 2) When a > 0, b < 0 3) When a < 0, b > 0 4) When a < 0, b < 0 Now suppose you want to find the linear function that passes through following two points,. x = (a, b) and y = (c, d). Then the linear function is defined as follows. ¶ µ b−d (x − a) + b f (x) = a−c µ ¶ b−d = (x − c) + d a−c ´ ³ ´ ³ b−d b−d So from f (x) = a−c (x − a) + b or f (x) = a−c (x − c) + d, ¶ (bc − ad) b−d x+ f (x) = a−c (c − a) | {z } | {z } µ 1 Here, the first term is the slope and the second term is Y-intercept. Example Find the linear function that passes through following two points. A = (2, 4) and B = (−4, −2) 2...
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...Quantitative Business Analysis, Mathematics part, Autumn 2012 Exercise set 1. The due date for this warm-up exercise set is Monday 17.9. at 9:45 (before the first exercise group). Please submit your exercise answers to the course box on the 2nd floor of the Chydenia building. Problem 1. (a) Consider the graph of f (x) = x2 and the line tangent to f (x) at (a, f (a)). The equation of the 1 2 tangent line is y = − 5 x − 25 . Find a, f (a), and the slope of the parabola f (x) at (a, f (a)). 6 (b) Consider the graph of y = x3 . Find the point(s) on the graph where the slope is equal to 11 . (c) The demand function for a commodity with price P is given by the formula D(P ) = a − bP . Find dD(P )/dP . (d) The cost of producing x units of a commodity is given by the formula C(x) = p + qx2 . Find C (x), the marginal cost. Problem 2. (a) Determine the limit limx→0 (3 + 2x2 ). (b) Determine the limit limx→2 (2x2 + 5)3 2 (c) Determine the limit limx→1 x +7x−8 (tip: modify the numerator using similar approach as in the x−1 lecture example) (d) Let f (x) = 4x2 . Show that f (5 + h) − f (5) = 40h + 4h2 . Hence, f (5 + h) − f (5) = 40 + 4h h Using this result, find f (5). Problem 3. Use differentiation rules (not the formal definition) to do the following: (a) Differentiate y = √ 3 − 6x2 + 49x − 54 2x 2 (b) Differentiate y = x − x 2 (c) Differentiate y = (x + 3x − 5)3 (d) Find the equation of the tangent line to the graph of f (x) = x/(x2 + 2) at x0 = 3. Tip: Given a point on a line and the slope,...
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...Creating a Job Analysis Abstract This essay will explain how to create a job analysis. It will explain what a job analysis is, and identifies the objectives of the job duties and roles for a position as a Human Resource Receptionist. Creating a Job Analysis The history of Job analysis was founded in 1900 by Frederick W. Taylor. His interest in improving the studying the principles of work and science developed into what is called a Job analysis. Human Resource Managers create a Job Analysis using task statements, Knowledge Skills Abilities, and create a job description. One method of developing a Job analysis is using the 7 stages of a Functional Job Analysis. The 7 stages are resources in helping to identify the task statements, KSA’s and job description. The following 7 stages are: things functions, work instructions, mathematical, data function, reasoning development, language develop and people function. (Pugh 2007) The thing functions are examples the ability to use office equipment anything tangible. Work instructions requires the ability to perform a given a task. Mathematical function is the ability to operate and handled business math. Data function is dealing with facts, statistics and information with business. Reasoning development show the ability to show comprehensive reasoning. Language develop require oral and written communication. People functions involve interaction with people. All 7 stages are used in developing the process of producing a Job analysis...
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...Economics Undergraduate Board (EUB) Math Review Autumn 2013 EXPONENT LAWS: xa+xb=xa+xb (xa)b= xab xa*xb=xa+b xaxb=xa-b DERIVATIVE NOTATION: Single variable: ddxfx=dfdx=f'(x)= “Taking the derivative of the function f with respect to x” Multiple variables: ∂∂xfx,y=∂f∂x=fx= “Taking the partial derivative of the function f with respect to x” ∂∂yfx,y=∂f∂y=fy = “Taking the partial derivative of the function f with respect to y” DERIVATIVE RULES: Product Rule: ddxfx*gx=f'x*gx+fx*g'(x) Quotient Rule: ddxfxgx=f'x*gx-fx*g'(x)gx2 Chain Rule: ddxfgx=f'x*g'(x) USEFL DERIVATIVES: ddxxn=nxn-1 ddxlnx=1x OTHER USEFUL THINGS: MUx=∂Ux,y∂x=∂U∂x= “Marginal utility with respect to x”: How much more utility you get from obtaining one more unit of x MUy=∂Ux,y∂y=∂U∂y= “Marginal utility with respect to y”: How much more utility you get from obtaining one more unit of y Endogenous Variables: Dependent variable explained by the model Exogenous Variables: Independent variables outside of the model and affect a model without being affected by it; given parameters Practice Problems → DERIVATIVES Single variable: e.g., given fx=2x0.2+x0.4+ x-0.3 , find f'x Exercises: for each of the following function, find f'x (i) fx=3x (ii) fx=6x19 (iii) fx=x2-2xx7+3 (iv) fx= 2ln(x2) Multiple variables (partial derivatives) : e.g., given fx,y=2x+3y+1 , find fx and fy...
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