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This book is a revised and expanded version of the lecture notes for Basic The Fundamental Theorem of Calculus (several versions) tells that differentiation.
Table of contents
- Navigation menu
- List of calculus topics
- A Gentle Introduction To Learning Calculus
- A Gentle Introduction To Learning Calculus – BetterExplained
And there are so many others that could be told as well. I had a hard time accepting the reality of elliptic geometry until one author made the comparison with latitudes and longitudes. I am 49 years old, been a housewife most my later adult life, office manager in younger years, and am planning to go back to college next Fall. I wish to major in physics. I took algebra and geometry in high school and did fairly well, but that was many years ago.
I love math but due to Parkinsons, tend to have memory issues. Anyway, in preparation for this endeavor, I wish to re-educate myself to prepare for college calculus. It has been very interesting for me as a teacher to use a little calculus gadget to teach them a new way of seeing things. For example, with the formulas you write perimeter, area, volume of a sphere of radius, I always get the most surprised faces when I show them how the derivative of the volume function leads to the surface area function, and the derivative of the area function leads to the perimeter function.
It is quite exciting for them to see that and they start asking themselves questions, which is great. The counting of syllables, like numbers in math, is critical to the practice and appreciation of verse. Will be using this site more in the Fall when I start my first of several Calculus classes for my physics degree. Very glad I found this site. Can we find the function from the integral? If so, how can we find the integreal with so little data?
Hi Tim, great question. If we only have 2 data points the start and end , then we have to assume a linear progression from 0 to 60mph over the course of 6 seconds i. In this case, integrating to figure out how much distance was traveled may not be accurate. As we gather more data points, we can get a better idea of the actual shape of the acceleration curve. I seriously love you man.
I have hated math my whole life and failed calculus miserably. I felt like something was always missing, and that was insight. You nailed everything on the hammer and gave such a helpful guide. I actually know what is going on in class now, instead of starting at the board and zoning out at the giant mass of information. It is like studying a language before you can speak it, or study the physics of art before art itself. But this is honestly the best thing ever, and thank you. In college, I enrolled in calculus and dropped it within my first week. I was lost by the end of lesson 1 and drowning by day 3.
Everything is within our grasp when explained properly. Very, very happy the approach is clicking for you. Dear friend, Your text was so fascinating. I am a student at Engineering faculty in Afghanistan. Please help me up. Really appreciate your great work and intuitive to help people understand things better. Will you try to explain some concepts in Linera Algebra in future? I think that is a weakness in some colleague students, i am one of it honestly: Hi Keith, thanks for the note.
Yep, I have a quick intro to linear algebra http: I thought I hated math for a long time, but as with many other things, it turned out I just hated how humans were approaching it. Hi Kallid, I love the way you derived the formula for the area of a circle using the circumference of a circle. My question is on how to derive the surface area of a sphere in a similar manner.
I also have ADHD which makes the classroom setting a nightmare, especially in math. Explanations like this are greatly appreciated because it takes the horror away from math and helps me to understand it in a logical and practical way. Wow, thanks for the thoughtful comment! I always knew I was smart but never could get even the basics of math.
Well, until we got to the end of the unit, or the next math up. I needed to actually see the practical application to appreciate what I was doing. Even graphing calculators stunk because those curves served no practical purpose. I found his website in eighth grade and it was really helpful in teaching me to love math, even if calculus seemed like a kind of fascinatingly foreign idea at the time. Beautifully explained i like the way…. I am a ninth grade student, trying to learn stuff ahead of our syllabus, and calculus was my first pick.
The way you have given an intro to calc is just epic. I understood everything as well as possible. I will surely continue on your series. This is actually very interesting Im 11 and Dad is trying to teach me calculus and I can understand you, unlike my prealgebra book. Hope you enjoy the rest of the calculus series! Both grown adults, are exceptional in mathematics, to this day. Love the way you have explained the fundamentals of calculus. I love the insights. Not understanding the essence of mathematics makes the majority of people not appreciate it.
In order to understand what an abstract word really means, one must get a hold first of its manifestations in the concrete world, and then how the abstract thereafter relates to the concrete. I feel moved to share some facts, inferences and insights regarding its validity. Our scientific formulae are so predictive only because each scientific formula represents a scientific generalisation that has been based on factual observations.
We keep on observing sets of phenomena in this way. However, that does not explain how they can be consistent. Therefore one is left with two general categories to explain the consistency of each of them: What do we call these certainties in the universe? What intuition do you think drove us to call physical laws laws? Nothing comes from nothing. The law of conservation of energy signifies this.
This therefore makes us conclude that the universe has always existed from eternity past. However, the universe began. Our universe is characterised by cosmic expansion. The second law of thermodynamics indicates that the longer time has elapsed, the greater the overall entropy of the universe shall be. Given that the universe is currently not at a state of maximum entropy, the first and second laws of thermodynamics indicate that the universe must not have always existed from eternity past.
Matter, energy, space and time, which constitute the universe, have not always existed. Therefore, because the universe began to exist, either some Being or something must have caused it. This cause of the universe must be immaterial, because the cause of the universe cannot be the universe itself, which is the totality of all material things, as nothing can cause itself that has not arisen from nothing. In other words, something causing itself is like saying that it appeared out of nowhere.
Something arising out of nothing can only be true if that thing is not under the law of conservation of energy, or, if some Being xor some other thing caused it that, being able to create energy, is above the law of conservation of energy. Because of laws such as the laws of thermodynamics, only the Creator can and will create the universe from nothing.
The theory of evolution holds that millions and millions of years ago, fish began evolving by means of little cumulative changes over long periods of time. Over approximately years, fish managed to evolve to amphibians. Over approximately years, amphibians evolved to reptiles. Some of these reptiles evolved to nonmonkey mammals, still over a long period of time, which then evolved to monkeys—simply put, our ancestors.
Of course, fish came all the way from a common ancestor. This is what Darwin has proposed. After the discovery of DNA, however, the theory of evolution itself evolved to include nonliving chemicals that happened to live by time and incredible luck. There is no substantial evidence, however, to support this. The assertion that genus evolves to another genus over a very long period of time is contrary to science genome is the total of all the genetic possibilities for a given species, and should not be confused for genotype. I understand that, in order to appear as though it was falsifiable, and thus be convincing, this assertion depends on natural selection.
One purpose of natural selection is to eliminate the abnormal mutations cause abnormalities. Too much of this and extinction would occur. Living beings adapt to their surroundings because of the way they were designed — not because of natural selection; without design in the first place, natural selection would be meaningless.
No one has ever observed actual evolution happen naturally. One only sees supposed evolution in some man-made books with pictures and in man-made realistic 3D animation movies. All proponents of the theory of evolution can show are some fossil remains with similarities, which have already undergone decomposition. The lips, the eyes, the ears, and the nasal tip leave no clues on the underlying bony parts. You can with equal facility model on a Neanderthaloid skull the features of a chimpanzee or the lineaments of a philosopher. These alleged restorations of ancient types of man have very little if any scientific value and are likely only to mislead the public… So put not your trust in reconstructions.
The fact that one language was used to design, and to dictate all the functions of, all living beings on Earth is just undeniable. After all, all living beings on Earth have one thing in common— life. If one has ever used a programming language before, one would understand the necessity of reusing a set of specific codes to a number of different programs.
Computer programmers though have a way of converting lengthy codes to just a short one by saving codes in header files because it would be tiring for humans to retype lengthy codes over and over again. Information is contained in our DNA, and our bodies were designed, and functions, as well, according to the specifications of this information. What happens when a living being is exposed to harmful things such as radiation? Mutations are alterations that take place in the DNA—damaging the information in it.
Information never originates by itself in matter; it always comes from an intelligent source. The outdated microscopes of their time made the very complex structure of the cell look so simple. However, if we would subscribe to the current scientific discoveries, as well as the technologies, of our time, we would begin to apprehend that the indications never really pointed to the theory of evolution. As science progresses, intelligent design becomes more evident. What else is the meaning of evidence?
Everywhere we look, the more attentive we are to the details, the more evident intelligent design becomes.
Mathematical formulae are symbolic representations of mathematical ideas, and ideas can only be conceived by the mind. We experience this whenever we conceive mathematical ideas. The Fibonacci numbers is one such idea. Fibonacci numbers and golden section often occur in nature, even in our bodies, and this repetition goes against mere coincidences.
It seems to me then that just as we humans can make something only out of that which has already been created, we can not conceive mathematical ideas other than that which has already been thought by an immaterial intelligent Being prior to the universe, as we humans rely upon the universe to derive our conclusions and mathematical ideas from. All mathematical ideas that we know of are embedded throughout the whole universe. As a matter of fact, mathematics is so pervasive it even permeates science. This does not contradict intelligence prior to the universe, but rather, proves it.
I understand that not all religions can be trusted to teach one what is true, but lies exist not only in religion. One should learn upon the insights of the reasonable, rather than calling the untaught ignorant without even educating them. I love math because to me there is nothing more beautiful than the truth, and math to me is also the realisation of the quantitative objective aspect of the truth algebraic logic counts truth value — 0, 1. The Internet Archive has a copy, though. You could replace that link with this one: The base is just the x axis in that graph….
This diagram may help: When limits ranges and domains met my math brain I thouht I was done and I and I amost lost hope in maths! Finally a site that explains the understanding and ideas behind maths rather than just robotically regurgitating rigid formulae. I have a degree in mathematics a singular noun from an American university that I earned in the s. We did have superscripts for exponents and normal algebraic notation for products.
Maybe a sentence of explanation would be in order. Hi Steve, thanks for the great feedback! Glad you enjoyed this post! Thanks for a great explanation, Kalid. Coming from an evolutionary biologist background it was very easy to follow. Just wanted to let you know that fat actually has the most calories. Professors rarely teach the bigger concepts.
They teach you the nit picky problems that will be on the test. THIS is why I hate math or at least learning it in college. Mr azad you Can blame me for My lagard intuition but i want to correct myself the. Newton did understand the mathematical rigor behind the calculus but only intuitive part became popular and Cauchy got the credit for mathematical rigor. It is difficult to believe that a robust theory like calculus originated purely out of intuition.
For people struggling with math I created powerful derivative calculator in addition to steps derivative can be evaluted at point and integral calculator with steps shown. Also, I looked for good free online graphing calculator, but all they lack customization, so I created another one. You can check them out here: Graphing calculator — http: Still, I love those moments where I get mind blown at how he math all comes together. So, thank you for this incredibly helpful site. Thank you for your help!
I am about to take calculus and was really discouraged about it until I read this. Though I have not the slightest clue on what calculus is. That notion scared me as I hate to be not prepared for anything, and more importantly I hate not being able understand how something works, let alone how to do it. This little introduction cleared my worries on the fact. I wish you had been my math teacher all of my school days and maybe I would have loved it instead of fearing it. Its actually called Maths but as you are probably american and the article is very good I will just add the s on as i read through your stuff.
Amazing, your explanation is easy to understand and just like calculus itself you explain the little things which helps to work with the bigger things. If you are not a teacher already then you should consider teaching it. By far the best explanation I have ever seen. Kalid, thanks so much for this and your other articles.
At school I enjoyed maths but lost interest when it became a mass of meaningless formulas. I came across your articles while digging into neural networks and discovering horror that exponentials, logarithms and calculus lay at their heart. After 30 years of avoidance the subject has become tangible again: And one think i really appreciate that you replied to every single comment you possibly could.
Could tell you that how awesome this site is. I really wished I had discovered it before. I would never know such site which is solely based on intuitive exist. Consider i am a engineering student in India you know that how bad condition the education system here is. I really had to depend on internet for learning.
List of calculus topics
You can also email me the list Thanks. Thanks Rohit, really glad it helped! For resources, I like anything by Richard Feynman the famous physicist. There are a bunch of books and lectures of his online and a great place to start. Never even heard of such things as Algebra, calculus and all the other things until I was an adult.
So I thought; can a 86 year old learn these things and that led to your article and no, I got as far as your coloured in circle then I was lost. I hope that beside being an author you are an up front teacher, for if you are your students must be absorbed. PS, excuse the English spelling. Is it because you chose not to learn maths, unless it is naturally comming to you like drawing or painting, kept you inquisitive, even at your age? My daughter is a multi tasker. She suggested me to masticate some regular things for her so that she could digest them later. As a result, I have to visit pages like this one almost on a daily basis.
I cant blame her since she is too busy with important assignments like Instagram, twitter, DIY zombie makeup, nailpaintings etc. Can we Go Further????. Are They Satisfied NOw?????. JUst Trying To Help???. Because programming always starts with a known objective i.
I now teach higher mathematics at degree level. The reason is that I understand the algorithms, rather than just knowing how to mindlessly repeat them like a parrot. Your gentle introduction to Calculus is very good. Anyhow, thank you, and well done.
A Gentle Introduction To Learning Calculus
Is it fine for me to want to learn this? Why does Calculus insist on functions, which are limited to a single output, e. I enjoyed math from a very young age. When the math classes focused on memorizing steps of the process, I struggled and intimidation settled in. I realize now that I enjoyed it when it was intuitive. I look forward to learning more your work.
Thanks for this help for me and others like me. You Seem To Understand Then???. Your argument is faulty in several ways. For example, poetry is not about the form meter, stanzas, etc. Nobody — appreciate what the writer and do not try to nick pick on things. May be you can counter him by writing better stuff on Calculus? Kalid, thank you for this article. The way you used rings and made a right triangle of circumferences is very imaginative and helpful in understanding how the actual formula for area relates to circumference. I had one question for you. I am returning to school after graduating with a bachelor of science degree in criminal justice in I want to further my education and pursuing an engineering degree to go along with my criminal justice degree.
I also took trigonometry in high school but that was even farther back. I just wanted your opinion about what you think my best options are going forward. Thank you in advance. It was invented centuries earlier by an 11th century Indian mathematician Bhaskara, and later refined by the scholars at Kerala school in the 14th century notably Madhava.
Like a true snake western scholars presented works already introduced by others as their own. In actuality the albino race is relatively stupid and dull.
BetterExplained helps k monthly readers with friendly, insightful math lessons more. Calculus is similarly enlightening. Imagine studying this quote formula: But calculus is hard! Calculus does to algebra what algebra did to arithmetic. Arithmetic is about manipulating numbers addition, multiplication, etc. Using calculus, we can ask all sorts of questions: How does an equation grow and shrink?
How do we use variables that are constantly changing? Heat, motion, populations, …. And much, much more! Realize that a filled-in disc is like a set of Russian dolls. Here are two ways to draw a disc: We get a bunch of lines, making a jagged triangle. But if we take thinner rings, that triangle becomes less jagged more on this in future articles.
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For each possible radius 0 to r , we just place the unrolled ring at that location. Image from Wikipedia This was a quick example, but did you catch the key idea? A note on examples Many calculus examples are based on physics. A note on rigor for the math geeks I can feel the math pedants firing up their keyboards. Discovering Pi A Calculus Analogy: Integrals as Multiplication Calculus: The Calculus Camera Abstraction Practice: I look forward to whatever article you come up with next in the series.
BTW, where are you from? I wud love to meet a genius like you sometime! Wow, thanks for the comments guys! That would be awesome. Always appreciate an interesting discussion! Many Thanks for Sharing, such a valuable information. Many thanks, glad you enjoyed it! Another great article from a great writer. Hi Ferenc, thanks for the support!
I also appreciate your efforts in replying to each of the comments. Thanks, glad you liked it. To start, forgive my english, its my third language. Hi, I am an engineer by profession. With lots of love and respect, Ferose Khan J. Again, excellent article and I look forward to reading more from you. Great thx for pictures. Glad you liked it. You have no idea how much this has helped me.
Hi Brendan, thanks for the note — always happy to help! Still waiting for that next article. Thanks for the encouragement, the next one is in the works as we speak: Thanks, glad you enjoyed it. Kalid — You are a gifted teacher. Kalid, you are the man. The first illustration is perfect for a beginner. Thank you for the kind words! Running the site is a lot of fun. Thanks so much for that. As soon as you said unroll the rings I got it, fucking brilliant! Another great article — keep on changing the world one article at a time. Thanks for the encouragement Em!
Glad you enjoyed it. Keep up the fantastic explanations! Your article was quite insightful and what I needed, thank you! Math has some very good discussions on math that may help. Thank you, really glad it was useful! Glad you enjoyed it: Idk, i thought i would let you know how I feel about the accusations of subtle religious bias. Derivatives of Inverse Trig Functions — In this section we give the derivatives of all six inverse trig functions.
We show the derivation of the formulas for inverse sine, inverse cosine and inverse tangent. Derivatives of Hyperbolic Functions — In this section we define the hyperbolic functions, give the relationships between them and some of the basic facts involving hyperbolic functions. We also give the derivatives of each of the six hyperbolic functions and show the derivation of the formula for hyperbolic sine.
Chain Rule — In this section we discuss one of the more useful and important differentiation formulas, The Chain Rule. With the chain rule in hand we will be able to differentiate a much wider variety of functions. As you will see throughout the rest of your Calculus courses a great many of derivatives you take will involve the chain rule! Implicit Differentiation — In this section we will discuss implicit differentiation.
Not every function can be explicitly written in terms of the independent variable, e. Implicit differentiation will allow us to find the derivative in these cases. Knowing implicit differentiation will allow us to do one of the more important applications of derivatives, Related Rates the next section. Related Rates — In this section we will discuss the only application of derivatives in this section, Related Rates.
In related rates problems we are give the rate of change of one quantity in a problem and asked to determine the rate of one or more quantities in the problem. This is often one of the more difficult sections for students. We work quite a few problems in this section so hopefully by the end of this section you will get a decent understanding on how these problems work. Higher Order Derivatives — In this section we define the concept of higher order derivatives and give a quick application of the second order derivative and show how implicit differentiation works for higher order derivatives.
Logarithmic Differentiation — In this section we will discuss logarithmic differentiation. Logarithmic differentiation gives an alternative method for differentiating products and quotients sometimes easier than using product and quotient rule. More importantly, however, is the fact that logarithm differentiation allows us to differentiate functions that are in the form of one function raised to another function, i. Critical Points — In this section we give the definition of critical points.
Critical points will show up in most of the sections in this chapter, so it will be important to understand them and how to find them. We will work a number of examples illustrating how to find them for a wide variety of functions. Minimum and Maximum Values — In this section we define absolute or global minimum and maximum values of a function and relative or local minimum and maximum values of a function.
We also give the Extreme Value Theorem and Fermat's Theorem, both of which are very important in the many of the applications we'll see in this chapter. Finding Absolute Extrema — In this section we discuss how to find the absolute or global minimum and maximum values of a function.
A Gentle Introduction To Learning Calculus – BetterExplained
In other words, we will be finding the largest and smallest values that a function will have. The Shape of a Graph, Part I — In this section we will discuss what the first derivative of a function can tell us about the graph of a function. The first derivative will allow us to identify the relative or local minimum and maximum values of a function and where a function will be increasing and decreasing.
We will also give the First Derivative test which will allow us to classify critical points as relative minimums, relative maximums or neither a minimum or a maximum. The Shape of a Graph, Part II — In this section we will discuss what the second derivative of a function can tell us about the graph of a function. The second derivative will allow us to determine where the graph of a function is concave up and concave down.
The second derivative will also allow us to identify any inflection points i. We will also give the Second Derivative Test that will give an alternative method for identifying some critical points but not all as relative minimums or relative maximums. With the Mean Value Theorem we will prove a couple of very nice facts, one of which will be very useful in the next chapter. We will discuss several methods for determining the absolute minimum or maximum of the function.
Examples in this section tend to center around geometric objects such as squares, boxes, cylinders, etc. More Optimization Problems — In this section we will continue working optimization problems. The examples in this section tend to be a little more involved and will often involve situations that will be more easily described with a sketch as opposed to the 'simple' geometric objects we looked at in the previous section.
Linear Approximations — In this section we discuss using the derivative to compute a linear approximation to a function. We can use the linear approximation to a function to approximate values of the function at certain points. While it might not seem like a useful thing to do with when we have the function there really are reasons that one might want to do this. We give two ways this can be useful in the examples. Differentials — In this section we will compute the differential for a function. We will give an application of differentials in this section. However, one of the more important uses of differentials will come in the next chapter and unfortunately we will not be able to discuss it until then.
Newton's Method is an application of derivatives will allow us to approximate solutions to an equation. Derivatives of Trig Functions — In this section we will discuss differentiating trig functions. Derivatives of Exponential and Logarithm Functions — In this section we derive the formulas for the derivatives of the exponential and logarithm functions.
Derivatives of Inverse Trig Functions — In this section we give the derivatives of all six inverse trig functions. We show the derivation of the formulas for inverse sine, inverse cosine and inverse tangent. Derivatives of Hyperbolic Functions — In this section we define the hyperbolic functions, give the relationships between them and some of the basic facts involving hyperbolic functions. We also give the derivatives of each of the six hyperbolic functions and show the derivation of the formula for hyperbolic sine.
Chain Rule — In this section we discuss one of the more useful and important differentiation formulas, The Chain Rule. With the chain rule in hand we will be able to differentiate a much wider variety of functions. As you will see throughout the rest of your Calculus courses a great many of derivatives you take will involve the chain rule! Implicit Differentiation — In this section we will discuss implicit differentiation. Not every function can be explicitly written in terms of the independent variable, e.