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  • Title: Karl's Calculus Tutor: Starting Page for 1st Year Calculus Tutorial
    Descriptive info: .. Karl's Calculus Tutor.. Home Page.. 1996 through 2010 by Karl Hahn.. last update 11-Dec-2010.. Welcome to.. Greetings to Spring 2012 Semester Students.. Hello Twitter users -- Karlscalculus now tweets.. Wanted.. Your grandfather's math textbook.. If you have an out-of-copyright math text, please consider lending or donating it to Karl's Calculus Tutor so that I may transcribe into an online pdf file.. Use your own emailer.. Use form.. Enter the tutorial.. (below).. or.. search.. this website for a calculus topic.. You will find coverage of limits, continuity, derivatives, related rates, optimization, L'Hopital's rule, integration, and much more.. There are dozens of problems worked out for you step-by-step.. If you are having difficulty with a calculus topic, you are encouraged to go to the appropriate section, look at the text, and then follow along with the worked problems to learn how you can do similar problems on your own.. There is also remedial coverage of algebra topics, number systems, exponentials, logs, trig functions and trigonometry, if you are in need of  ...   of Integration.. 12) Applications of Integrals.. Interactive Features.. Special Interest Pages.. From B.. Cherkas at CUNY:.. A Word of Advice to the Math-Flustered.. Whenever a problem has you baffled, listen to your mind's quietest voices, for you will hear the solution in their whispers.. You can participate in a calculus discussion by posting to.. Karl's Calculus Forum.. Go to.. Acknowledgement:.. I am indebted to my high school physics teacher,.. Art Hovey.. , whose lectures decades ago had a profound influence on my own problem-solving thought processes, and whose methods of teaching I have endeavored to reproduce on many pages of this tutorial.. Click on his name to get to his own web pages, which include links to his high school physics class notes that he developed over 33 years of teaching.. You can email me by clicking this button:.. For information about.. to be emailed to you, use button.. or send a blank email to.. Note that your email address will not be added to.. any.. list if you do..

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  • Title: Karl's Calculus Tutor -- box 0.0: Copyright Notice
    Descriptive info: Karl's Calculus Tutor: Copyright Notice.. The entire contents of the web pages known as.. are copyrighted (1996 through 2010).. Web users are permitted to make personal copies, both electronic and/or paper, of any or all of the material contained therein.. Users (and especially math educators) are also permitted to distribute such copies provided that a) the material remains unaltered, b) no fee is asked for or paid in exchange for such copies, and c) that a copy of this notice is furnished with all copies exchanged.. All other rights are reserved.. Certain images including "the math instructor" (as shown to the left) and other clip-art images that are designated "Hemera" are licensed from Hemera, all rights reserved.. The author,  ...   me for permission.. Thank you.. Find out how you can get a copyright or.. trademark.. so you can have your ideas protected.. Web Links Policy.. Most websites, especially math education sites, are welcome to link to any page or pages of.. The expressed restrictions are that a website that links to.. must.. not.. contain sexually explicit material, hate rhetoric, depictions of violence, or encouragement of unlawful activities.. Web page designers are permitted but not required to use the.. logo image (upper right corner of this page) for creating links to.. Designers doing this will please make a.. copy.. of that image file that is resident at their site for use on their web pages.. Go To Table of Contents..

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  • Title: Karl's Calculus Tutor: Table of Contents
    Descriptive info: Table of Contents.. 1996-2009 by Karl Hahn.. last update 23-Oct-2010.. IMPORTANT VIEWING NOTE:.. To properly view these pages, open your view port out to at least->-->-->-->here.. If you can't see the word ".. here.. " in the line above, place your mouse-cursor on the right-hand edge of the frame, hold the left mouse button, and drag the right-hand edge of the screen to the right until you can see it.. See.. browser notes.. for more details.. Netscape 6.. x and 7.. x users who wish to view the Karl's Calculus Forum pages,.. click here.. Online Calculator.. Please note that Karl's Calculus Tutor is still a work in progress.. Expect a new unit to come on line every month or so.. Currently being drafted:.. Applications of Integration (Areas).. prependix a: Math Notation over the Web.. prependix b: Why Bother to Learn Calculus.. prependix c: Stuff You Should Already Know.. (this page also available in printer-friendly.. PDF.. ).. prependix d: Study Tips.. Karl's Calculus Tutor: Interactive Features.. Special Interest and Optional Pages.. where you can discuss calculus topics with other visitors to.. 0 Greetings and Welcome.. 1 Number Systems and Their Properties.. 1.. 0 Preliminaries.. 1 The Counting Numbers.. 2 The Integers.. 3 The Rationals.. 4 And At the Heart of Calculus is.. 5 The Real Numbers.. 2 Limits.. 2.. 0 Limits: A Day at the Races.. 1 Limits: How Close to the Edge Dare You Go?.. 2 Combining Limits: Two Tall Tales.. 3 The Sky Is the Limit.. 4 Limits on the Side.. 5 A Most Useful Rule.. 6 Putting the Squeeze on Limits (the Squeeze Theorem).. 3 Continuity.. 3.. 0 Places You Must Visit.. 1 Track to the Future.. 4 Derivatives.. 4.. 0 The Trials of the Animals (intro to derivatives).. 1 The Main Pillar of the Temple (the derivative defined).. 2 Rules to Live By (sum and product rules).. 3 More Rules to Live By (quotient and chain rules)..  ...   with Derivatives.. 8.. 1 The Cheshire Cat's Grin (L'Hopital's Rule).. includes.. Taking Limits of Differences.. 2 The Dance of the Derivatives (Related Rates).. Dimensional Checking (sometimes called Units Checking).. 3 Name That Function (inferring coefficients from clues about critical and inflection points).. 4 Little Red Riding Hood Goes to Town (approximation & intro to Taylor & Maclaurin Series).. 5 Power Tools for Taking Derivatives of Products (logarithmic differentiation and Leibniz' Rule).. 9 Summing Up Derivatives.. 9 Midterm Practice Exam.. 10 Integrals.. 10.. 1 Thoughts for a Wedding Reception (intro to integrals).. 2 Living Backward -- The Fundamental Theorem of Calculus.. 3 Grist for the Mill -- Definite and Indefinite Integrals.. 11 Methods of Integration.. 11.. 1 Intro to Methods of Integration.. 2 Integration Using your Rearview Mirror.. 3 Integration by Simple Substitution.. 4 Integration by Parts.. 5 Trigonometric Substitution.. 6 Hyperbolic Substitution.. 7 Partial Fractions.. (This.. page.. now available in PDF format).. 8 More Substitutions.. 12 Applications of Integrals.. 12.. 1 Motion Problems.. Still under construction.. 2 The Area of a Circle.. 3 The Area Between Curves.. Differential Equations.. Link to Differential Equations E-book.. Sections Coming to.. in the months ahead:.. Improper Integrals.. Areas and Volumes.. Convergent Series.. More on Taylor & Maclaurin Series.. Sample Exam Questions on Integral Calculus.. PC Users: To aid your studies you can.. download useful calculator programs.. that can do complex-number arithmetic and much more.. Quadratic Polynomials.. ||.. Cubic Polynomials.. Quartic Polynomials.. Hero's Formula.. Einstein's Formula.. Log Base 10 Tricks.. Measuring the Earth.. Long Ago in a Galaxy Far Away.. An Object Falling through Air.. Proof of The Intermediate Value Theorem.. Proof of The Extreme Value Theorem.. Rational Unit Circle Points/Pythagorean Triples.. Multi-variable Polynomial Long Division.. Intro to Lagrange Multipliers.. (vector calculus, PDF file).. Online Calculus Courses.. Accredited Online Colleges.. Return to top of Table of Contents.. You can email me by clicking this button:.. to be emailed to you, send a blank email to..

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  • Title: Karl's Calculus Tutor: Website Search Engine
    Descriptive info: Website Search Engine.. 2001 by Karl Hahn.. Search.. for a topic.. Reminder:.. Do check the spelling of your search words before you press the.. Find!.. button.. If you misspell a word, the search engine won't be able to find it.. FreeFind.. Search: Simply enter the word or words you want to search for and click the.. will find pages containing.. all.. the words or.. of the words in your query, depending upon what you select below.. Search Karl's Calculus Tutor.. powered by.. Find pages matching ALL words.. Find pages matching ANY word..

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  • Title: Karl's Calculus Tutor: Email Policies
    Descriptive info: Email Policies.. I try to answer all math emails, but that is not always possible.. Sometimes I am busy, out of town, or the volume becomes overwhelming.. But most of the time you will get a reply.. I cannot guarantee that replies will be within 24 hours for the same reasons cited above.. So if you are in panic mode the night before an exam, I'm sorry to say that it's too late for me to get you out of your jam.. It is important that, if the email you send me contains math notation, I be able to read and understand it.. This is not a trivial matter since it is not possible to type many of the symbols commonly used in mathematics notation.. You are urged to visit and read carefully at.. to learn more about how to send understandable math notation over email.. Please do not send attachments.. Because of the danger of computer viruses, any attachment you send is not likely to be opened.. Rather than scanning a handwritten problem and sending it to me, please try to cipher it into plain-text on your keyboard.. Make sure I can find your email address if you expect a reply.. Also make sure that the date is set properly on your computer.. My email reader sorts emails by date, and if your correspondence is dated 1969, I won't find it in the pile.. If you send me a homework problem that has you stumped, it would be unethical for me to do it for you.. I especially WILL NOT TAKE YOUR EXAM for you, whether it is a take-home or otherwise,  ...   problems are meant for you to do, not me.. Please keep your questions specific.. If you ask me to explain a general topic, I will likely point you to one of the pages of the online material.. The online material is indeed where I have put my best efforts at explaining the broad topics.. It is better for you to explore the Karl's Calculus Tutor web pages in search of the explanation you seek before you go asking me about it by email.. You'll save both of us a lot of time that way.. There are dozens of worked problems in the online material.. Many of them are or are similar to problems that I have seen many times in emails.. Again you save both of us time when you find a worked problem that answers your question.. I will also answer questions on high school level algebra, trigonometry, and statistics, as well as advanced topics like differential equations, topology, real analysis, complex variables, linear algebra, abstract algebra, etc.. But if you are taking graduate level math courses, there is a good chance that my understanding of the topic in question might not be any better than yours.. I also will answer first-year physics questions.. PRIVACY.. Your confidence is important to me.. I will not reveal the email address or the contents of any correspondence made to Karl's Calculus Tutor without the correspondent's permission unless I am ordered to do so by a court of law.. It is possible, however, that I might use a problem that you ask about in one of the web pages.. You can email me by clicking this button..

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  • Title: Karl's Calculus Tutor: Special Interactive Website Features
    Descriptive info: Special Interactive Features.. 2002 by Karl Hahn.. has a number of interactive features to help your math-learning along.. They are:.. This is a powerful calculator that can do multiple operations in one click of the mouse.. This calculator does operations on complex numbers.. Solve.. n.. equations in.. unknowns for.. equal 2 to 7.. Generate plot images of functions of your choice and have them emailed to you.. Generate polar plot images of functions of your choice and have them emailed to you.. Each interactive feature has an.. About.. Read Me.. Be sure to read it.. Return to Table of Contents..

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  • Title: Karl's Calculus Forum
    Descriptive info: 2006 by Karl Hahn.. Here is your chance to discuss calculus topics with your fellow students, teachers, and other visitors to.. Feel free to post problems that are challenging you as well as solutions to problems that others have posted.. You may also post comments or criticisms of the materials that you have found on.. I will be reading the postings here frequently.. I will also, from time to time, post problems here to stimulate discussion.. Please remember that these are family webpages, so keep your comments tasteful and in good spirits.. Archives of previous months' forum:.. See July-December 2006 entries.. See January-June 2006 entries.. See Summer/Fall 2005 entries.. See May/June 2005 entries.. See March/April 2005 entries.. See January/February 2005 entries.. See November/December 2004 entries.. See Sepember/October 2004 entries.. See Summer 2004 entries.. See Spring 2004 entries.. See February/March 2004 entries.. See December 2003/January 2004 entries.. See October/November 2003 entries.. See September 2003 entries.. See July/August 2003 entries.. See June 2003 entries.. See May 2003 entries.. See Apr 2003 entries.. See Mar 2003 entries.. See Jan-Feb 2003 entries.. See Dec 2002 entries.. See Nov 2002 entries.. See Oct 2002 entries.. See Sep 2002 entries.. See Summer 2002 entries.. See May 2002 entries.. See Apr 2002 entries.. See Mar 2002 entries.. See Feb 2002 entries.. See Jan 2002 entries.. See Dec 2001 entries.. See Nov 2001 entries.. See Oct 2001 entries.. The form-entry page contains instructions for entering math notation.. Please read them carefully, otherwise your posting might turn out illegible.. Post Your Message to the Forum.. Current Forum Discussion Begins.. plzzzz help me this question from application of derivative maximum and minimum word problem Q)what are the dimension of rectangle greatest possible area which can be inscribe in isosceles triangle with base 36cm and 12 cm.. hur.. mumbai, up india - Sunday, June 30, 2013 at 04:58:24 (EDT).. I'm 80 years old and retired and went to the university one-and-a-half years back in 1952/3 and could only pass the exams for Introductory Calculus.. Later on I got injured in an accident that left me paralysed and on a wheelchair.. I've remained interested in Calculus and Math in general since then.. My concrete question is: In which sense is integration the inverse function of differentiation.. Thank you very much for your kind answer.. Best regards, Walter.. Walter H.. Allende.. Buenos Aires, Argentina - Sunday, April 28, 2013 at 22:48:35 (EDT).. differentiate Sin2x.. davis munyao.. nairobi, kenya - Tuesday, April 02, 2013 at 02:40:51 (EDT).. Hallo, my name is Dimitris.. i have one straight line and one curve and i need to find the tungency point of them.. how am i going to do that? the staight line is r=0,0448+0,37σ and the curve is U=r-0,02σ.. 2 can you help me?.. Dimitrios Georgalas.. London, UK - Friday, November 23, 2012 at 08:03:00 (EST).. cos^-1(4x^3-3x).. please help me out and show me method.. let y = cos^-1 (x) i.. e x=cos y.. =cos^-1(4cos^3)y-3cos y).. =cos^-1(cos3y).. =cos^-1(-cos(Pi-3y)).. =cos^-1(-cos(3y-Pi)).. =Pi-cos^-1(cos(3y-Pi).. =Pi-(3y-Pi).. =2Pi-3y.. =2Pi- 3cos^-1(x).. =3/(1-x)^1/2.. can some explain me this trigonometrical substitution ?.. hussain.. rajkot, Gujarat INDIA - Tuesday, October 16, 2012 at 02:30:20 (EDT).. As a former teacher of 10 years, and best example I know for calculus is that of the speeding car.. Let's say a car crosses a stretch of 55 miles in one hour.. Given a speed limit of 55mph most people would say the car went the speed limit.. However if he had sped up and then decelerated, he could have broken the speed limit.. This is where Calculus comes in.. Fortunately this is not on the GRE test.. However, for good.. GRE prep.. please click on the link.. Best, Mike.. Mike.. Los Angeles, CA USA - Friday, September 28, 2012 at 20:58:56 (EDT).. solve please: the SQUARE sequence: a,b,c,d is sequence & x is a numbur that we dont know what it.. prove that (a*b*c*d)+(b-c)^4=x^2.. faezeh.. tehran, iran - Tuesday, July 31, 2012 at 15:00:45 (EDT).. What is the derivative of y=(1-x)^2 A Big thanx from a Bignewb.. Donald.. Toronto, Canada - Monday, April 30, 2012 at 17:52:20 (EDT).. Hi I'm new to this fourum can someone please help me?,.. I'm really stumped on this one problem could you help me?.. If dy/dt=ky and k is a nonzero constant, then y could be:.. A) 2e^kty.. B) 2e^kt.. C) (e^kt)+3.. D) kty +5.. E) (1/2ky^2)+1/2.. I would be very greateful if you could teach me how to solve this problem.. thanks,.. Peter.. Peter A.. Jacksonville, FL USA - Sunday, January 01, 2012 at 22:58:36 (EST).. Reply to Mike Frederick:.. You have.. f'(x) = 4qx^3 - 3rx^2 - 2sx + t.. Let the three points where max or mins are to occur be.. (x.. ,y.. ,.. , and.. So you have.. f(x.. ) = y.. and.. f'(x.. ) = 0 f'(x.. ) = 0.. Expand all of those equations out, put in the values for the.. x.. 's and.. y.. 's, and you'll find that 6 linear equations in the 5 unknowns,.. q.. r.. s.. t.. u.. Solve any 5 of them in the usual way.. Note that because this system is overspecified, it might not be possible to get a solution that satisfies all 6 equations.. If that turns out to be the case, then there is no solution to this.. Karl.. USA - Friday, July 08, 2011 at 13:28:20 (EDT).. Help! I don't know how to start this problem: f(x) = qx^4 - rx^3 - sx^2 + tx + u Find the values of q,r,s,t & u so that the function has local minimums at (-.. 398238, -.. 457621) and (1.. 20556, -3.. 46602) also so the function has a local maximum at (.. 130182,.. 069458).. Can you point me in a direction?.. Mike Frederick.. USA - Tuesday, April 12, 2011 at 23:12:26 (EDT).. Here is a fact that I always thought was interesting: It is easy to generate examples of differentiable functions with derivatives that are not continuous, e.. g.. f(x) = x*sin(1/x) for x not 0 and f(x) = 0 for x = 0.. But an interesting property that derivatives always have is the intermediate value property; that is, if f(a) = c and f(b) = d, with a.. Free Calculus Tutoring.. Cheers!.. Patrick Barrow.. Los Angeles, California USA - Thursday, December 23, 2010 at 05:09:07 (EST).. what is calculus?.. huttr.. californaia, dallas USA - Tuesday, December 07, 2010 at 03:21:56 (EST).. is this odd or even.. prove.. f(x)=(1+x)(1+x).. jonathan shiundu.. kenya - Thursday, October 14, 2010 at 09:18:46 (EDT).. 1/1-h - 1 lim = ---------- h ->0 h.. I need help trying to get rid of the h that's in the bottom of the faction, but not sure what I'm supposed to do to bring that 1-h denominator to a numerator.. Any help ASAP would be great! Another similar question is 1/x+h - (1/x) ------------ h.. Paul Wayne.. RI USA - Sunday, October 03, 2010 at 18:32:27 (EDT).. Newton discovered gravity, that was easy - F = GM1M2/RR Newton discovered the second law of motion, easy - F = MA Newton wrote the first differential equation, for a falling body R'' = - C / RR Now comes the hard part, he had to solve this DE.. Mathematica DSolve will solve the 1-dimensional DE as given above, giving an implicit solution.. shown below in Mathmatica notation.. DSolve[y''[x] == -a/(y[x]*y[x]), y[x], x] Solve[(-(( a Log[2 a + 2 C[1] y[x] + 2 Sqrt[C[1]] Sqrt[C[1] + (2 a)/y[x]] y[x]])/C[1]^(3/2)) + ( Sqrt[C[1] + (2 a)/y[x]] y[x])/C[1])^2 == (x + C[2])^2, y[x]] That is, Mathematica is producing an algebraic solution, and it says that there is no closed form solution.. I'd like to know how Mathmatica came up with that solution, it apparently cannot be persuaded to divulge this info.. Is this problem written up somewhere? Note: I have the derivation of the solution to the 2-d case, it's the standard 2-body problem.. All props to Newton ! Sorry for lack of paragraph runons, I can't get cr/lf s to appear.. Will.. USA - Tuesday, August 17, 2010 at 10:30:46 (EDT).. Hi, I'd like to start a study group that operates completely online for calculus and for other math subjects.. The current plan is to use free MIT videotaped lectures: http://www.. youtube.. com/watch?v=7K1sB05pE0A&feature=player_embedded and answer each other's questions about them in teamspeak 3 (http://www.. teamspeak.. com/?page=downloads), an online voice chat program, while being connected with a virtual whiteboard such as http://www.. scriblink.. com We would like both experienced and beginning students to join up.. A few of us have a number of ideas of how this could play out with more than just this subject but we would like to start as soon as possible with Calculus.. Contact me @ desk45z on skype or through the email button below.. Matthew Sullivan.. Bronx, NY USA - Monday, July 12, 2010 at 21:57:10 (EDT).. Hi, my name is Calvin and I am 60 years old.. 61 actually.. LOL.. What are the chances at this age I can learn calculus? I know basic math.. And some geometry.. What think ye? Calvin.. Calvin De Beverly.. Wichita, KS USA - Tuesday, June 01, 2010 at 00:52:14 (EDT).. I have two problems concerning the math: 1.. I have a problem with mixture problem, I can not solve the mixture problem.. I have a problem with motion problem.. Thanks a lot for understanding!.. Gohar.. Yerevan, Armenia - Wednesday, April 28, 2010 at 09:30:19 (EDT).. Cho hình chóp S.. ABCD có đáy ABCD là hình vuông cạnh a.. Có SA vuông góc với ABCD, góc tạo bởi SA và mặt phẳng đáy bằng 30 độ.. Tính thể tích hình chóp theo a?.. nguyen thi chung.. Quang tri, 12 viet nam - Monday, April 05, 2010 at 11:29:25 (EDT).. Hi, I am still stuck on a problem that I posted from Ireland - Monday, October 08, 2007 at 04:41:44 (EDT) I hope you can help me with differentiating a matrix or in my case the product of matrices.. I have written this out myself longhand, that is created instances of the matrices with values like x1, x2,.. ,b1, b2,.. , y1,, y2,.. etc then multiplied out and differentiated and put back into matrix from, I get the matrix result we were given in class without showing the steps.. I have not included my longhand workings because it is several pages handwritten which will be impossible (I feel) if I type out.. However if you feel that I should do so then I will do so, or is it possible to email a pdf of a scan? Notation: S is a scalar, Y is a matrix (n x 1) {n rows 1 column}, X is a matrix (n x p) {n rows p colums}, B is a matrix (p x 1) {p rows 1 column}.. the transpose is indicated by me to be "[]" that is X transpose is [X].. S = [Y]Y + [B][X]XB - 2[B][X]Y dS/dB = 0 + 2[X]XB - 2[X]Y Here are my thoughts: d([Y]Y)/dB = 0 as Y is independent of B d(2[B][X]Y)/dB = 2[d([B])/dB][X]Y as 2 constant can be pulled through, X & Y are independent of B so they are pulled through but to the postmultiply position to preserve associativity.. = 2[X]Y as d([B])/dB same way that d(z)/dz = 1, power rule multiply by the exponent and reduce the exponent by 1.. d([B][X]XB )/dB is where I am stuck coming up with the given result: 2[X]XB d([B][X]XB )/dB seems to require some form of a product rule.. Try d(UV) = Ud(V) + Vd(U) with U = [B][X], V = XB giving: [B][X]d(XB) + XBd([B][X]) [B][X]d(XB) = [B][X]X as d(XB) = X similar to earlier on note the dimensions of this are (1 x p)(p x n)(n x p) = (1 x p) XBd([B][X]) = XBd([XB]) using the rule the product of 2 transposes is the transpose of the reversed product = XB[d(XB)] guessing that the transpose can be pulled through = XB[X] note the dimensions of this is (n x p)(p x 1)(p x n) this is illegal and is only possible for the special case where p = 1 giving a (n x n) matrix, as otherwise the matrices are not conformable.. I think my problem is in the product rule or pulling the transpose through the derivative.. I do not just want the answer I want to be given the reasoning so that I can derive solutions to problems where the scalar has to be minimised or maximised by setting the first derivative to zero.. I have looked at the Wikipedia procedures and I was worse off as I was just unable to follow them, in fact I became more confused.. Please help, Peter.. Ireland - Monday, March 29, 2010 at 06:49:25 (EDT).. I have been away from conventional math classes since 1996 and am trying to teach myself Calculus.. I am stuck on the two following problems which may be considered easy by some, but I just don't know where to begin.. #1: Find two positive integers, the sum of whose squares is 16 and whose product is as large as possible.. (For this one, I know there has to be a way to actually determine this other than guessing.. All I have been able to do is come up with all of the numbers whose squares add to 16 and determine their product.. ) #2: Find the equation of the tangent line in slope interecept form of the curve given: (5x/y)+3y^2-x^2=-3 (-1,1).. Robert Reynolds.. Jacksonville, FL USA - Tuesday, January 19, 2010 at 22:18:19 (EST).. Jacksonville, FL USA - Tuesday, January 19, 2010 at 22:04:53 (EST).. Reply to Dain:.. Gardner has this to say about that paragraph: "Approximating a continuous curve by drawing smaller and smaller triangles under the curve is today called the 'trapezoidal rule' because the little triangles, joined to the narrow rectangles beneath them, form trapezoids as shown in figure 47.. ".. Once you understand that you are using trapezoids to approximate the area under the curve, the 1/2 factor arises from the formula for the area of the trapezoid, which uses the average of the two parallel sides times the width.. The 1/2 comes from taking the average.. Click to Send Email to Karl.. >.. USA - Thursday, December 31, 2009 at 10:09:27 (EST).. Hello.. To get ready to help my daughter learn Calculus, I have been reviewing Sylvanus P.. Thompson's classic "Calculus Made Easy" so see if I could find material that would impart a more intuitive understanding of Calculus to her.. I am stumped by one paragraph (Chapter XVII, "Integration", on p.. 189 of the pre-Martin Gardner revised edition) in which he briefly tries to explain the need for a "1/2" in the graphical depiction of the curve deduced from the derivative represented by (1/5)x.. In this paragraph he writes: "Now if x had been constant, S(1/5)x.. dx would have have been the same as (1/5)xS.. dx, or (1/5)x^2.. But x began by being 0, and increases to the particular value of x at the point P, so that its average value from 0 to that point is (1/2)x.. Hence S(1/5)xdx = (1/10)x^2; or y = (1/10)x^2.. " Here I have written "half", or 1/2, in parenthesis for clarity, and I used a capital "s" for the integration symbol, and "^" to symbolize the exponent.. It seems to me he has lost a golden opportunity to clearly explain the need for the "half" in the "integrated" function, which of course is (1/10)x^2, and which naturally emerges on getting the antiderivative of, or integrating,  ...   -sinx) * X))) / ((tanx^2)).. Then I'm really stuck on that one.. I mean, if I do plug in the "pi/3" to the "x" variables then it will just be a mess.. Unless that's the only way to get the answer? Or should have I used the product rule instead?.. 3).. If f(x) = sinxcosx, then f '(pi/6).. Product rule:.. y= f x g.. F'g + g'f.. = (cosx*cosx) + (-sinx*sinx).. = (cos^2x) + (-sin^2x).. = (cos^2(pi/6)) + (-sin^2(pi/6)).. = 0.. 633.. -Let * be a multiplication sign.. -Is that right?.. 4).. Differentiate: f(x) = x^2 + 2tanx.. So, I'll just get the derivative of the equation.. 2x + 2(???).. -Let ??? be "I don't know".. So, I'm stuck.. I have no idea what's the derivative of tanx have.. I already encountered this problem in question #3 and I assumed that it would be:.. tanx' = cox / -sinx.. Is that right? If yes, then I would get this equation:.. 2x + 2(cosx/-sinx).. Is that right? If yes, can I simplify it much more?.. 5).. Find the equation of the tangent line to the graph of f(x) = (x-1) / (x+1) when x = 1.. y=mx+b.. Chain Rule:.. = 1(x+1) - 1(x-1) / (x+1)^2.. = x+1 -x +1 / (x+1)^2.. = 2 / (x+1)^2.. = Plug in "x".. = 2 / (1+1)^2.. = 2 / 4.. = 1/2.. slope (m) = 1/2.. Now that I have the slope, I'll just get the x & y values from plugging in "1" to the equation.. y = (1-1) / (1+1).. y = 0/2.. y = 0.. So: x = 1 and y = 0.. Then reflect on the slope equation:.. y = mx+b.. Plug in the numbers from what I have gotten before:.. 0 = 1/2(1)+b.. 0 - 1/2 = b.. -1/2 = b.. So.. y = 1/2(x) + (-1/2).. I'll multiply the whole equation by "2" to make it more neater.. 2y = 2(1/2x) + 2(-1/2).. 2y = x -1.. Is that right?.. ------------------------------------------.. I know my solutions were kind of long.. But I hope that you could help me.. I really want to make this happen.. Or at least answer the questions correctly or in a much simplified way.. =).. PLEASE.. paola.. Canada - Wednesday, March 12, 2008 at 00:48:16 (EDT).. I've been working on this problem for a solid two, front and back, notebook pages.. Can someone please tell me if I'm on the right track with this? Differentiate: y=x^x.. I took the natural log of both sides.. ~ lny= xlnx.. I used the formula, [lnu]= u'/u, for the xlnx ~ lny = x(1/x).. I " e'd" both sides to get rid of the natural log of y.. ~ e^lny *******.. This is where I'm kind of stuck.. I dont know which of the following should be the outcome of my "e-ing".. a.. y= xe^(1/x) or b.. y= e^(x(1/x).. This could be an easy answer, but I'm a little confused.. Could someone please help me?.. Ariel.. Cambridge, MN USA - Monday, March 10, 2008 at 21:41:29 (EDT).. Can someone please tell me if I'm on the right track with this? Differentiate: y=x^x 1.. ~ lny= xlnx 2.. I used the formula, [lnu]= u'/u, for the xlnx ~ lny = x(1/x) 3.. ~ e^lny ******* 4.. y= e^(x(1/x) This could be an easy answer, but I'm a little confused.. Cambridge, MN USA - Monday, March 10, 2008 at 21:39:45 (EDT).. Hi, A question.. have you heard of the smoothstep function.. So if you have heard of this function then this question is for you.. Their exists a 3 degree polynomial function whose derivative is a solution to a smoothstep function.. i.. e.. f(x) = ax.. + bx.. + cx + d.. when drawn the derivative of this function gives a curve as shown in image Fig 1.. What is needed is a shown in Fig 2.. Is it possible.. How.. ? PS, Since it is not possible to load an image here.. Imagine Fig 1 as a less curvy 'S' and Fig 2 as a normal 'S' BRgds, kNish.. knish.. mumbai, MAH India - Friday, January 04, 2008 at 04:25:17 (EST).. Water is flowingat the rate of 6m^3/min from the tank shaped like a hemispherical bowl of radius 13m.. Answer the following questions given that the volume of water in a hemispherical bowl of radius R is: V = (pi/3)y^2(3R-y) when the water is y meters deep.. at what rate is the water level changing when the water is 8in deep?.. b.. what is the radius r of the water's surface when the water is y in.. deep?.. at what rate us the radius r changing when the water is 8m deep?.. Adam.. kingston, pa USA - Sunday, November 11, 2007 at 00:38:43 (EST).. General info.. I discovered a clear explanation of the Related Rate Problemin Calculus, J.. Stewart.. , 4th Ed.. Robert.. Dingmans ferry, PA18328General info: USA - Wednesday, November 07, 2007 at 15:19:40 (EST).. -1.. = y.. 1/2.. * ln(y.. ) Is there a way to rearrange this equation is terms of x? i.. x=?????? so I can solve it (without calculus) by just plugging in a y? Assume y >= 1, limit to natural numbers if necessary.. I only need positive, real solutions for x.. How do I solve for x when x is a root of a natural log? I think I could do it if there was only one x, or if there was no log.. But with two x's and one in a log, I don't know how to rearrange this so x is by itself on one side of the equation.. Surrey, BC Canada - Thursday, October 25, 2007 at 19:31:00 (EDT).. I need some help on the following to determine whether or not they are continuous and explain why or why not: Is f(x) = 5x-10 continuous at x=2? Help would be greatly appreciated.. Malga.. USA - Sunday, October 21, 2007 at 15:55:49 (EDT).. I'm having problems trying to graph these conditions.. I don't know where to begin.. Please HELP!! 1.. Sketch the graph of a function f(x) that satisfies the following conditions: a)f(-9)=f(1)=f(6)=0, b)f'(-4)=f'(3)=0, c)f"(-8)=f"(-1)=0, d)f'(x) positive only on (-4,3), e)f"(x) positive on (-8,-1) and (8,infinity).. On graph, label all the points of inflection, local maximum, and local minimums.. NETTE.. Denver, CO USA - Tuesday, October 09, 2007 at 03:43:30 (EDT).. given P = Y^tY - 2Y^tAC +C^tA^tAC Where Y is a column vector = a matrix with rows n and column =1.. A is a matrix rows = n cols = 2 the first col are x, that is x1, x2,.. ,xn; the second col has 1 in each entry.. C is a colum vector = a matrix with 2 rows and one column call the entries c1 & c2.. ^t symbolizes transpose The equation for P reads in English: (Y transpose by Y) minus (twice times Y transpose by A by C) plus (c transpose by A transpose by A by C) I want to find the partial derivative of P with respect to c using the rules of linear algebra.. I know the solution is -2A^tY +2A^tAC, in English minus (times A transpose by Y) plus (twice times A transpose by A by C) I am unable to locate a reference giving details on how to evaluate a derivative using the rules of linear Algebra with simple examples for me t work with.. I have expanded written out P using n=2 and then found the partial derivatives using the normal rules for functions of two variables.. It was most tedious and I would like to learn the linear algebra rules based on simple examples.. Many thanks, Peter.. Ireland - Monday, October 08, 2007 at 04:41:44 (EDT).. I am taking a distance learning Calculus course.. I have been out of school for 15 years and was never very good at the maths in the first place.. I am pulling my hair out trying to figure out a problem that should be simple.. How do I approach it? Here it is: A car rental agency has rates of $75/day (with partial days charged at the full day rate) and $330/week.. You are charged the daily and/or weekly rate, whichever is cheapest for you.. For example, if you rented a car that was $100 daily or %50 weekly, the charge for 4 days use would be $400; the charge for 6 days use would be $550; the charge for 8 days would be $650 ($550 + $100) Let A(t) represent the average cost to rent the car for t days, where 0.. Jordan Hyde.. Springville, UT USA - Tuesday, September 25, 2007 at 18:05:58 (EDT).. [Question] Hi, my teacher gave us this problem, and he couldn't figure out why method was incorrect and why I got the answer I did.. Given we know the gradient slope =.. at the point (2,0) on a surface f(x,y), in what direction, expressed as a unit vector, is f increasing most rapidly? [Difficulty] I solved the problem like this: Max slope = magnitude of gradient slope (gradient slope) dot (unit vector) = 56.. 03 -56.. 03Ux + 1.. 866Uy = 56.. 03 sqt(Ux^2 + Uy^2) = 1 Solving the system Ux= -.. 9977 Uy=.. 0672 The only problem is, by definition, I should be able to get the unit vector by taking the gradient slope vector and dividing by the magnitude of the gradient vector.. or,.. =.. The weird thing is, the correct Uy value is almost exactly half of mine.. what's going on??? [Thoughts] I tried a similiar technique for finding where the f = 0.. dot (unit vector) = 0 -56.. 886Uy = 0 sqt(Ux^2 + Uy^2) = 1 Solving the system this time I got the correct answer, why here and not there?.. John Smit.. USA - Wednesday, September 19, 2007 at 00:05:15 (EDT).. l would like to join your program, to send and receive calculus problems.. lam very good in math and l hope to enjoy your program with other students.. stephen nyamapfumba.. houston, texas USA - Saturday, September 15, 2007 at 19:52:41 (EDT).. Hi,.. I have not done calculus in a while.. Find the range for:.. g(z) = 1 / square of 4-z^2.. I can easily find the range for the other equations but I got stuck on this one since it's 1 / over a square, etc.. Martyn Reno.. Montreal, QX Canada - Thursday, September 13, 2007 at 22:17:51 (EDT).. hey, can anyone help with solving equations by extracting square roots.. varawdn.. USA - Saturday, September 08, 2007 at 19:32:08 (EDT).. Can ε = 1/δ ? I mean such function should not have a limit, because as delta gets smaller epsilon gets bigger so function is not converging.. Am I right?Thank you in advance very much.. felix.. USA - Friday, July 20, 2007 at 11:38:04 (EDT).. A piece of wire 5 inches long is to be cut into two pieces.. One piece is x inches long and is to be bent into the shape of a square.. The other piece is to be bent into the shape of a circle.. Find an expression for the total area made up by the square and the circle as a function of x.. I need this to be as detailed as possible.. I got the final formula.. I don't know if it's right, or if I have to put it into the y=^#$*&^$^ form.. So, if you could answer both of those questions, that would be EXTREMELY helpful.. Katie & Annamarie.. USA - Thursday, July 12, 2007 at 15:03:35 (EDT).. Reply to Chase.. If it is a simple graph here is what you need to think about: The graph of cos & sin repeat after 2pi or 360 degrees, the period The graph of cos & sin has a y distance from max to min of 2 units The graph of cos & sin is centred about y=0 look at your graph, think of it as a sin graph, what is its period? if it is 120 degrees then it is the graph of 3x if it is 720 degrees then it is the graph of 0.. 5x, call this number a.. Does it pass through 0 and it it increasing to the right of 0? if not it is the graph of a sin but shifted to the right or left so it needs to be written as sin(x + - some number), call this number b, if a from above is not 1 you need to write this as sin(ax +- ab).. What is the y distance from the max to the min? if it is not 2 then you need to multiply sin(ax +- ab) by some number C that is greater than or less than 1: Csin(ax +- ab).. Is the graph centered about y=0? if not you need to add or subtract some number D to get Csin(ax +- ab) +- D.. Hope this helps Peter.. Ireland - Monday, June 11, 2007 at 09:06:21 (EDT).. How do you write an sine and cosine equation from a graph?.. Chase.. fort smith, ar USA - Tuesday, May 29, 2007 at 23:17:14 (EDT).. I need help with finding simultaneous tangent lines for two equations f(x)= ln x and g(x)= e.. I have looked at the problem that was sampled but I am having troubles doing the substitution of f(x) and g(x) when trying to solve for one of the variables.. Josh.. md USA - Tuesday, May 15, 2007 at 21:38:24 (EDT).. Reply to Mark:.. Just because an expression seems simple, it doesn't mean that it's easy to integrate.. The antiderivative of your function cannot be expressed in any finite combination of elementary functions.. Indeed you need to use Jacobian elliptic integral functions in order to integrate this.. That is a topic that is usually not covered in first and second year courses.. USA - Friday, March 30, 2007 at 15:25:35 (EDT).. This is a great site, and has kept me entertained for countless hours rediscovering my "A" level maths (I guess that equates to High School for those of you [no doubt the majority] on the other side of the pond!).. I've set myself increasingly complex problems until I've finally stumped myself! I'm trying to integrate the following pretty simple expression:.. f(x)= (1+kcos(x)).. I've tried the route of substituting x=arccos(w), x=arccos(w/k) and various mungeing ideas.. I've tried expressing the resultant as logs, but no matter what twists and turns I try, I can't get it to fall out.. If anyone has any clever ideas I would be very grateful to hear from you.. Many thanks,.. Mark.. Croyde, Devon UK - Thursday, March 29, 2007 at 10:23:13 (EDT).. I need help with calculus (integrate), out of school too long.. Charles Carter.. Vine Grove, KY USA - Monday, March 26, 2007 at 14:02:26 (EDT).. Reply to Jesse:.. There is an almost identical problem to this in the online webpages.. Please.. to see the worked solution.. Then see if you can apply the same method to your problem.. USA - Friday, January 05, 2007 at 12:53:05 (EST).. A spotlight on the gound shines on a wall 12 m away.. If a man 2 m tall walks from the spotlight towards the building at a speed of 1.. 6 m/s,how fast is the shadow on the biulding decreasing when he is 4 m from the building?.. Jesse.. Los Angeles, California USA - Thursday, January 04, 2007 at 19:11:23 (EST).. Scripts and Message Board created by Matt Wright and can be found at.. Matt's Script Archive..

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  • Title: Karl's Calculus Tutor - Sending Math Notation over Email
    Descriptive info: Math Notation via Email.. © 1998, 2006 by Karl Hahn.. ">.. Contents of this page.. Basic Guidelines.. Raising Something to a Power.. Products.. Square Roots.. Using Parentheses.. Absolute Values.. Derivatives.. Limits, Summations, and Integrals.. Quotients and Division.. Subscripting.. Inequalities and Relational Symbols.. Infinity.. Greek Letters.. Vector Operators.. Special Functions.. Also see these examples from The Math Forum at Swarthmore.. If web page options leave something to be desired with regard to rendering math expressions, email can be a nightmare.. But on various math discussion usenet groups, some standard notations seem to have evolved.. Basic Guidelines:.. First, before you compose any math expressions for sending over the email,.. you should set the font of your email composition window to an equal-space font like Courier.. This will assure that columns will line up the same way for both sender and receiver.. Also it is.. very important.. to remember to press the enter key before you reach the 80th column of each line.. Do.. count on the auto-wrap to insert the end-of-lines --.. please.. explicitly press the enter key at the end of each line.. Many mail readers render email very poorly if it contains no hard end-of-lines, and your colums will not line up for me the same way as they did for you if you fail to follow this guidline.. Please do NOT use special symbols.. that you cut and paste from web pages or select from font-tables.. When they arrive in my email, I cannot read them.. And please,.. do NOT ever use tabs anywhere.. If you do you will very nearly assure that columns will not line up for the recipient as they did for you.. So always use spaces instead of tabs.. The notation.. x^n.. is almost universally taken to mean.. It is much easier to do exponentiation using the.. ^.. symbol than to try to line up the exponent in the correct column of the preceding line.. If the exponent is an expression, be sure to use parentheses --.. x^(n+1).. for example.. The function,.. y = e.. , you can render either as.. y = e^x.. or as.. y = exp(x).. Either notation is widely recognized.. Most of the time you can indicate the product of two things simply by listing them, one after the other, with no operator symbol in between.. For example,.. 2a.. is taken to mean two times.. There may be cases, though, where this can be hard to read or ambiguous.. asin(x).. could be taken to mean.. times sine of.. , or it could be taken to mean arcsine of.. It would be better to write the former as.. a sin(x).. a*sin(x).. a * sin(x).. The.. *.. symbol is always taken to mean multiplication unless specified otherwise.. If you put the.. in to indicate multiplication even where it isn't necessary, there is no harm done.. To render the square root of an expression, you can use.. sqrt(expression).. (expression)^(1/2).. For higher roots, like the cube root, for example, use.. (expression)^(1/3).. Please don't bother trying to construct those fish-hook radical symbols using characters.. Use parentheses wherever there is any opportunity for ambiguity.. ambiguous unambiguous unambiguous.. 1/2a 1/(2a) (1/2)a 2^n-1 2^(n-1) (2^n) - 1 e^x^2 e^(x^2) (e^x)^2.. Generally, if no parentheses are shown, exponentiation is assumed to be done first, followed by multiplication, division, subtraction, and addition, in that order.. Even if you follow that convention, when leaving out the parentheses, it often behooves you to put an additional blank to offset the later-applied operators:.. y = x^2 + a x + b.. This  ...   an easy way to write them in email, though, that is self-evident to the reader.. lim{x- 0} x ln(x).. is "the limit as.. goes to zero of.. times.. ln(x).. integral{0 to r} f(x) dx.. is the definite integral taken from.. 0.. to.. of.. f(x) dx.. If you leave off the curly-brackets and their contents, you have an indefinite integral.. If you like, you can abbreviate the word, "integral" with "int.. " For double and triple integrals you would have something like.. double integral{r=0 to 1; theta=0 to 2pi} f(r, theta) r dr dtheta.. For summations, use, for example.. sum{k=0 to infinity} x^k/k!.. You can see that the behavior of the counting variable is described here inside the curly-brackets.. No easy way to do these.. For small expressions you can use something like.. 1 / (x^2 + 1).. For longer expressions, simply use a row of dashes to make the line, and try to keep your columns aligned properly.. x^2 - 2x + 1 y = ------------------- x^3 - 3x^2 + 3x - 1.. There are two commonly used ways of doing this.. If you are going to be putting expressions into the subscript, use something like.. b[n] = n(b[n-1] + b[n-2]).. If you are simply going to be using the subscript as a label, for example taking a difference between.. v.. -final and.. -initial, you can use either the square-bracket notation or something like.. delta v = v_f - v_i.. =.. for "less than or equal" and.. for "greater than or equal".. You can use either.. >.. !=.. as a symbol for "not equal".. The squiggle,.. ~.. , is a good symbol for "approximately equal".. Also,.. means "much less than", and.. means "much greater than".. For the equivalence symbol,.. , simply use.. equiv.. Please do not send a special symbol like ".. " or "oo".. Instead simply use the word, ".. infinity.. ," as shown in the.. example.. shown above, or you can abbreviate it as ".. inf.. You can either spell them out, like.. delta.. epsilon.. omega.. , or you can use the Roman equivalent, which in these cases would be,.. e.. w.. respectively (use.. for.. theta.. if you don't want to spell that most popular of Greek letters out).. Usually the context will make it clear what you mean.. If you are sending stuff that involves vectors, you can represent the dot product of two vectors,.. v dot w.. Likewise, their cross product would be.. v cross w.. The norm or length of a vector,.. , is simply.. |v|.. If you are dealing with vector fields, use the following equivalents for the "del" operator:.. expression email version.. f grad f.. ∙f div f.. f curl f.. f del^2 f.. The expression,.. [x].. , is usually used to mean the.. floor.. function of.. Another phrase for this function is the.. greatest integer.. function.. It means, take the greatest integer that is less than or equal to.. So, for example:.. [1.. 5] = 1 [0.. 99] = 0 [-1.. 1] = -2.. You can use the.. notation to represent this, or you can use.. floor(x).. sgn(x).. , is.. for positive values of.. for negative values of.. , and zero for.. step(x).. , zero for negative values of.. , and is undefined for.. sinh(x).. cosh(x).. tanh(x).. coth(x).. sech(x).. csch(x).. are the hyperbolic sine, hyperbolic cosine, hyperbolic tangent, hyperbolic cotangent, hyperbolic secant, and hyperbolic cosecant respectively.. For the inverse of any of these, just precede its name with.. arc.. (e.. ,.. arctanh(x).. Return to Front Page.. Return to table of contents..

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  • Title: Karl's Calculus Tutor: Browser Notes
    Descriptive info: Browser Notes.. Browser Notes:.. If all or part of the word, ".. " is beyond your right-hand margin, then part of certain equations will also be beyond your right-hand margin.. Recommended Browsers:.. Microsoft Internet Explorer versions 5.. x or 6.. x and Mozilla FireFox, version 1.. These pages also work with Netscape 4.. x.. You can also use Microsoft Internet Explorer 4.. x but you will find some misrenderings of certain equations.. For example, many versions of Explorer do not render.. -x.. correctly (the above should appear with two distinct levels of superscription).. Versions 5.. x and 6.. x have corrected this problem.. Note that Explorer 3.. x and earlier has some real problems with these pages.. Microsoft Internet Explorer.. (including all known 5.. x versions) has a bug.. (big surprise there) regarding rendering of the horizontal lines used in Karl's Calculus Tutor to indicate quotients.. Sometimes (and there seems to be no real pattern) the horizontal line will fail to appear, even though it does appear reliably with other browsers such as Netscape and Firefox.. The problem has been in Microsoft IE pretty much since its inception, and MS appears to have no intention of ever fixing it (shame on you Bill Gates).. As of 2-Oct-2004 I am still working on tracking down all the places where this is a problem and recoding those pages so that Microsoft IE renders them correctly.. Sorry for the inconvenience.. x (Mozilla Gecko) do not support the symbol font.. As of 2-Oct-2004, the symbol font has been eliminated from all of the pages of Karl's Calculus Tutor except for the Karl's Calculus Forum pages.. So if you are a Netscape 6.. x or 7.. x user, unless you want to view the Forum pages, you need not take any of the steps below.. Everything should render properly for you.. If you use those versions of Netscape to view these webpages.. (this caveat now applies only to Karl's Calculus Forum pages).. , many of the math symbols will not render correctly in your display.. You can fix the Mozilla versions using instructions available.. Another Netscape alternative for viewing these pages is to use version 4.. 8 or below.. If need be, you can  ...   the size of the fixed width font is important for alignment of equations and to ensure that they will fit onto the width of your screen.. To make the renderings of equations fit best with the various graphics, choose the font size of the fixed width font that most closely makes the two x's just below the same size:.. x.. The size that does that is usually 12.. If you have a small screen, your choice of fixed width font size might have to be a compromise between making the x's the same size and making the ".. " fit on your screen.. If that's the case, experiment a little.. Font size adjustment is also available in Microsoft Internet Explorer, and the same applies about coming as close as possible to having the two x's the same size.. In Netscape, also set the radio buttons for either "Use document-specified fonts, but disable dynamic fonts," or "Use document-specified fonts, including dynamic fonts.. Attention AOL subscribers.. : If the browser that comes with the standard AOL package appears to be inadequate to view these pages properly, you can still run Netscape under your AOL account, and it will give you a much better view of the tutorial.. If you do not know how to set up Netscape under your AOL account, consult with your AOL technical representative via email or phone to find out how.. Or you can.. to go to a Netscape archive download site and fetch the Netscape Browser on your own.. Attention Unix Users:.. So if you are a Unix user, unless you want to view the Forum pages, you need not take any of the steps below.. If you are running Netscape on an X-Windows platform, you may need to enable the.. symbol font.. in order to view these pages properly.. For.. Red Hat.. Sun.. users,.. Click here.. and either follow the instructions or find somebody who knows how Unix works to do it for you.. This solution might also work on other Unix platforms, but I don't know for sure.. You will know that you are lacking the symbol font if you see something other than the Greek letter, omega (which looks like a horseshoe), here:..

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  • Title: Karl's Calculus Tutor: Interesting Links
    Descriptive info: Interesting Links.. � 1999 by Karl Hahn.. Mohamed A.. Khamsi, Helmut Knaust, Nancy Marcus, Michael O'Neill,.. Luis G.. Valdez Sanchez.. in association with.. The University of Texas at El Paso.. have developed a most excellent math site that covers topics ranging from basic algebra to complex variables, and very much includes calculus.. They call it.. S.. O.. S.. Mathematics.. Click the logo to get to their main page.. M.. I.. T.. has an.. Open Courseware.. series on line.. It includes some excellent class notes and video on a number of math and science topics.. Calculus-Help.. com.. by.. Michael Kelly.. offers multimedia tutorials with especially clear explanations of limits, continuity, and basic derivatives.. He also offers books and a weekly practice problem.. He has also given Karl's Calculus Tutor his.. Headbutt of Excellence Award.. A math teacher, Mr.. Koh, in Singapore has assembled a tutorial website,.. White Group H2 Maths.. , that has great learning material on a number of topics, including calculus topics.. On his website, click the question locker tab.. He also offers tutoring services.. Susan Cantey.. , a math teacher at a Cincinnati high school, has prepared these.. Power-Point slides.. to drill you on basic calculus concepts and procedures.. Free math books!.. Just follow the link to.. and download all you like.. Also available are books on other topics such as physics, chemistry, and engineering.. "World Class Online Tutoring, Just A Click Away.. " That's the service that.. TutorVista.. offers.. "TutorVista's online tutoring is done using an interactive whiteboard to work problems, simulations, assessments and other tasks.. all subjects at an extremely affordable price!".. The Math Forum.. has a great list of calculus links as well as a host of other resources.. Click on their star-logo to go there.. They also provide the.. Ask Dr.. Math.. tm.. pages, which I very much recommend.. has some useful graphing and algebra functions that can not only save you some grunt-work, but also illustrates how it does the problems you give it.. To see common math mistakes made by students and their corrections, see.. mathmistakes.. info.. The University of Akron.. also has an excellent calculus tutorial, complete with many examples and problems for you.. to get to.. e-Calculus.. You will need an.. Adobe Acrobat.. ® plug-in for your browser in order to view their material.. Gerardo Mendoza.. Dan Reich.. Rutgers University.. have created an interactive tutorial:.. COW:.. C.. alculus.. O.. n.. eb.. It quizzes you (and not just multiple guess questions -- you have compose your answers) then checks them -- even making suggestions on what you did wrong when you goof.. Highly recommended.. It also has units on linear algebra, number theory, and abstract algebra.. Stuck on a homework problem from your textbook?.. Hotmath.. org.. has step-by-step worked solutions to the odd-numbered problems in many popular algebra, trig, and calculus textbooks.. Eric Schechter.. Vanderbilt University.. has prepared a litany of common math mistakes made by undergraduate students.. to see what they are so that you won't also fall victim.. He also has plenty of other math pages that you can get to by.. clicking here.. How about downloading a.. free.. calculus reference booklet?.. Someone who chooses not to post his or her name has made something called.. The Calculus Bible.. available as a Microsoft Word document.. It's a handy reference guide with all the formulas and methods you have  ...   If you are in need of a tutor,.. Chaim Lazar.. offers a free service for matching tutors with students over the internet.. To find out more,.. Another fine tutor-matching service (and it's also free of charge) is.. FindaTutor.. , offered by.. Chris Baldwin.. Just click on the name.. Click on.. for an excellent calculus tutorial by.. Larry Husch.. at the Math Department of the.. University of Tennessee.. At.. InteractiveMathTutor.. (a paid subscription service) you can get personal attention by phone, email, or two-way blackboard.. Help is offered for a wide range of math topics, including calculus.. Harvey Mudd College.. has a well organized tutorial that covers calculus together with much of the prerequisite material that you should know before taking calculus.. to go there.. If you need a review of precalculus material that you learned last year but have now become rusty on,.. It will take you to.. FHS Precalculus Study Page.. where they have coverage of topics such as linear and quadratic equations, inequalities, trig functions, polynomials, functions, etc.. A generous soul who identifies herself only as.. Karen.. has a cool math site called.. Click on the logo to go there.. She offers lots of useful algebra and precalculus pages if you need review.. I also especially like her page on.. limits.. And there are plenty of math-entertainment pages at her site too.. Check it out.. has lots of pages of both calculus and precalculus.. Worth a visit.. Some folks at the.. University of Toronto.. have made class notes and past examinations available on the web.. You will need to download an.. ® viewer in order to read them (and they take some time to download).. There is a large volume of material here, and it is available to you by.. The Math Department at.. has a webpage of math tools called.. MathServ.. It can do things like factor polynomials, graph equations, find derivatives and antiderivatives, and lots of other neat stuff.. Dr.. Thomas I.. Vogel.. of Texas A&M University has a.. gallery.. of calculus oddities that are of interest once you have some calculus concepts under your belt.. Kenny Felder.. has some illuminating and useful essays, some of which might help you with your calculus and some with physics, if you are taking that as well.. Check it out by.. Doug Shaw.. has a page of calculus brain-teasers where you are asked to.. find the error.. And don't ask me for the answers -- I'm not at liberty to reveal them.. You have to hash these puzzles out for yourself.. Eric W.. Weisstein.. has compiled a vast online math reference in association with Wolfram.. Mathworld.. Wolfram.. to have a look.. Duane Kouba.. at.. UC Davis.. is building a solved calculus problems site that you can check out by.. also has its own.. calculus site.. that.. J.. Haas.. edit.. Mark Brittenham.. , an assistant professor of math at the.. University of Nebraska.. , has a list of.. study tips.. , written by.. Susan Hermiller.. Melanie Martin.. , that you would be wise to look at.. Websites with more links to math websites:.. www.. dmoz.. directorykids.. calculus.. math.. utk.. edu.. For some biographical material on.. great mathematicians.. to find out more about.. the history of mathematics.. Here is a link to.. my own.. personal page.. The math section has a few interesting brain ticklers on it if you are interested..

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  • Title: Karl's Calculus Tutor - Recommended Books
    Descriptive info: Recommended Books.. © 1999,2002 by Karl Hahn.. Karl's Calculus Tutor is completely nonprofit.. Commissions from book sales are the primary source of funding with which to offset the expenses of keeping Karl's Calculus Tutor on line.. The difference between a clear and lucid math book and one that is, to coin a politically correct phrase, organizationally challenged, can be the difference between success and failure for the math student.. I can't tell you that I have read through even a sizeable fraction of the calculus texts and study guides that are on the market, but I have looked at a bunch of them.. If you are taking a college calculus course, you probably have already been assigned a required text.. You will, of course, have to buy that text, whether it is any good or not, just so that you can do the homework assignments.. But it doesn't hurt anything besides your pocket book to have other references at your fingertips.. So here I shall make my recommendations from among the books I am familiar with that are still in print.. If you click on a book, it will take you to the page of.. where you can order that title.. The prices listed here are those posted by Amazon.. com at the time I prepared this page.. Study Guides.. Here is a book I highly recommend for anybody who tends to fall asleep while reading math books.. The style in.. How to Ace Calculus: The Streetwise Guide.. by Joel Hass, Abigail Thompson, Colin Conrad Adams is so lively that I had to put it down in order avoid succumbing to the urge to plagiarize it.. In the actual calculus material, you don't have to wade through a lot of stuff to get to the critical points.. This book specializes in critical points -- that is the stuff you must know to pull down a good grade.. And it describes them language so plain it will have you chuckling from time to time.. It also has some great practical advice on how to select a good teacher, how to ask questions, how to study, etc.. The examples worked in the text are not the really difficult ones that sometimes might show up on an exam, but you have to learn the not-so-hard ones before you can do the hard ones.. To those of you who thought you could never understand math, this book is worth many times its price.. by Hass, Thompson, and Adams, Paperback:.. $11.. 96.. Hass, Thompson, and Adams have written a sequel to the above guide,.. How to Ace the Rest of Calculus.. You will be wanting this if you are going on to take calculus 2 and multivariable calculus.. Once again it takes the hard concepts and puts them into plain and lively language.. Note:.. Titles shown in box ads are Amazon.. com's recommendations based upon topic.. How to Ace Calculus the Rest of Calculus.. $10.. 47.. An outfit called.. Schaum's Outline Series.. has been publishing study aids in a wide variety of subjects for decades now.. I once had a coworker who was fond of saying that he got his degree from Schaum's Institute of Technology.. There are several calculus volumes that they publish.. One of them I am quite familiar with, and that is the one by Frank Ayres and Elliot Mendelson.. There are brief, clear explanations of the concepts and methods.. But the emphasis is upon worked examples.. There are over 1000 of them.. For what it's worth, this book has sold over a million copies in the 30 years it's been in print.. Theory and Problems of Differential and Integral Calculus.. by Ayres and Mendelson, 3rd ed.. , Paperback:.. Here's another from the.. Schaum's.. series.. It has fewer worked problems, covers fewer topics, but has much more explanatory text.. And the explanations are organized in a consistent way.. Each section is divided into the concepts of Approximation, Refinement, and Limit.. The author, Eli Passow, shows how these three stages are effective at attacking a wide variety of problems.. Understanding Calculus Concepts.. by Eli Passow, Paperback:.. 16.. More Good Schaum's Titles.. Easy Outline: Differential and Integral Calculus.. by Frank Ayres Jr.. and Elliott Mendelson.. Outline of Trigonometry.. by Robert E.. Moyer and Frank Ayres Jr.. Outline of Precalculus.. by Fred  ...   who has fallen in love with calculus can know its poetry.. Just as any musician can tell you how music is far deeper than mere spots on a score sheet, John C.. Sparks leads you to understand how calculus is far deeper than mere cryptic symbols on a blackboard.. The sad truth is that too many calculus classes and texts teach students only the rules by which you manipulate the cryptic symbols.. But those cryptic symbols, like all symbols, stand for something greater.. Spots on a score sheet stand for melodic sounds, and mathematical symbols stand for beautiful ideas.. It is these ideas that.. Calculus Without Limits.. teaches.. Without grasping the ideas and concepts of calculus, a student is left only with the grind of applying pencil lead to paper in this or that prescribed manner and has thoroughly missed the point.. So it's hardly a surprise that so many calculus students are frustrated.. But for a student who reads John C.. Sparks' explanations of how the symbols assemble themselves into something meaningful, the symbology becomes just a tool -- as it should have been all along.. The real knowledge and beauty behind the symbols glimmer through.. And for a student who makes the special effort (and all mathematics learning requires special effort, even from those who love it) to follow that light where it leads -- that student too might very well fall in love and know the poetry.. Calculus Without Limits: Almost.. by John C.. Sparks, Paperback:.. $22.. 95.. Here's another book that has emphasis on readability and an exposition style that effectively glues the concepts to your memory.. Calculus (College Review Series).. by Elliot Gootman.. Textbooks.. Of all the beginning calculus texts that have been foist onto students, there is one that, in my opinion, stands head and shoulders above the rest.. Apparently I am not the only one to think so.. Enough others feel this way to have kept this book in print for nearly 40 years.. This is.. Calculus and Analytic Geometry.. by George B.. Thomas Jr.. at M.. It is now in its 9th edition.. Later editions have some contributions from Thomas' associate, Ross L.. Finney.. The development is logical, there are loads of worked examples, the explanations are clear, and the authors have had 9 editions and several decades to bring this work very near perfection.. by Thomas & Finney, 9th ed.. , Hardcover:.. $102.. 67.. Calculus and Analytic Geometry Study Guide.. by Thomas & Finney, Paperback Vol 1:.. $30.. 40.. by Thomas & Finney, Paperback Vol 2:.. For anyone planning a career in science or engineering, I recommend that you.. eventually.. get.. Mathematical Methods in the Physical Sciences.. by Mary Boas.. It is a broad survey of theory and methods that you will be using to solve problems that involve physics or engineering.. It is quite accessible to anybody who has had two semesters of calculus, and the explanations of the methods are clear and concise.. When you take your advanced undergrad courses, this book may prove to be invaluable.. by Mary L.. Boas, 2nd ed.. $106.. 50.. If your second or third semester calculus course covers differential equations, you will want this book.. It begins by explaining concepts in linear algebra, which is a skill you will need if you plan to continue your math education.. It then goes on to the basics of differential equations, with emphasis on linear differential equations.. There is plenty of explanatory text and worked examples.. Introduction to Linear Algebra and Differential Equations.. by John W.. Dettman, Paperback:.. 36.. Reference books.. Here are two very useful reference books for those of you who will be using math in your future studies or careers:.. CRC Standard Math Tables and Formulae.. 30th ed.. Hardcover.. $59.. Handbook of Mathematical Functions.. by Milton Abramowitz and Irene Stegun Paperback:.. $23.. 07.. Recreational Math Books.. 1 2 3 Infinity.. by George Gamow, Paperback:.. $8.. 76.. The Mathematical Tourist.. by Ivars Peterson, Paperback:.. Math Trek: Adventures in the Mathzone.. by Ivars Peterson (for younger readers), Paperback:.. Math Trek 2: A Mathematical Space Odyssey.. 80.. The Language of Mathematics: Making the Invisible Visible.. by Keith J.. Devlin, Paperback:.. $12.. 00.. More books will be appearing on this page soon, as I get a chance to review them.. Need a Calculator for Class?..

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