Page 1 of 6 Target audience: Someone who understands the previous Overview chapters and can comfortaly: 2. Divide common fractions: 3 4 5 = 2 3 5 4 = 10 12 a = a 1 c c = a = ac c 1. Find common denominators: 2 + 3 5 = 1 2 5 + 3 5 5 2 = 5 2 10 + 6 10 = 11 10. State that x 3 means x x x,. Cancel common factors in a fraction : 2a2 3 4a = 2 a a 2 2 a. Evaluate : ( a + ) 2 = (a + )(a + ) = (a + ) a + (a + ) = a 2 + a + a + 2 = a 2 + 2a + 2 On the Language (Notation) of Mathematics. = a2 2 For most adults the notation of mathematics is a foreign language in which they know, perhaps, a few words. If children spent as much time practicing the language of mathematics as they do reading and writing their native language, we d have a surfeit of mathematical geniuses. there are algorithms I think we can safely discard in the modern world. We don t need to teach students how to extract square roots y hand,. Calculators are also useful tools that people worked hard to make we should use them, too, when the situation demands! I don t even care whether my students can divide 430 y 12 using long division though I do care that their numer sense is sufficiently developed to reckon mentally that the answer s a little more than 35. Ellenerg, Jordan (2014-05-29). How Not to Be Wrong: The Power of Mathematical Thinking (p. 57). Penguin Group US. Kindle Edition. Working an integral or performing a linear regression is something a computer can do quite effectively. Understanding whether the result makes sense or deciding whether the method is the right one to use in the first place requires a guiding human hand. When we teach mathematics we are supposed to e explaining how to e that guide. A math course that fails to do so is essentially training the student to e a very slow, uggy version of Microsoft Excel. And let s e frank: that really is what many of our math courses are doing. Ellenerg, Jordan (2014-05-29). How Not to Be Wrong: The Power of Mathematical Thinking (p. 56). Penguin Group US. Kindle Edition. Page 1 of 5
Page 2 of 6 Reverse derivatives (antiderivatives). Oserve the following pattern. ( x 2 +1 ) = ( x 2 ) + 1 = 2x + 0 = 2x ( x 2 + 2 ) = ( x 2 ) + 2 = 2x + 0 = 2x ( x 2 + 3 ) = ( x 2 ) + 3 = 2x + 0 = 2x ( x 2 + C ) = ( x 2 ) + C = 2x + 0 = 2x for any constant C There are many function rules (which on this page we generically call) F(x) such that F (x) = 2x If we pick one of these function rules and call it F 0 (x) then all the function rules in the set { F 0 (x) + C C is a real numer} are called reverse derivatives (antiderivatives) of f (x). Here are a few more examples. ( 3x 2 1) = 3x 2 + 0 ( ) = 3x 2 + 2 = 3x 2 + π ( ) = 3x 2 + e ( ) ( ) = ( 3x 2 + 5) = ( 3x 2 + 6) = 6x For any value of x these reverse derivatives (antiderivatives) all have the same slope computed with 6x. Here are their graphs. https://www.desmos.com/calculator/njunesw7m The meaning of C. Different values for C produce the different graphs in the example aove. Page 2 of 5
Page 3 of 6 Working with C.. Suppose that we start with 6x as aove and somehow determine that ( 3x 2 + C) = 6x. If we choose C = 4 then we have defined the function rule 3x 2 + 4 for the lack graph aove.. Suppose that we want to find a C that makes the graph of 3x 2 + C pass through the point 1, 2 ( ) that is, 3( 1) 2 + C = 2. Just solve the equation for C. C = 2 3 = 1. 3x 2 1 is the function rule for the ottom red graph aove. Another example of the relationship of antiderivatives with different C s https://www.desmos.com/calculator/vulls06ozv Move the points around and see how the slopes compare. ( x 3 + 4x 2 + x + C) = 3x 2 + 8x + 1 determines the slope for each of the graphs aove (at each value of x ) ecause it s the derivative rule for all of them. Page 3 of 5
Page 4 of 6 Notation, notation, notation. Indefinite Integrals An interesting choice of notation. One could use the following notation to represent the set of all antiderivatives of f (x) : A f (x) def = { F(x) F (x) = f (x)}. Better still is { A f (x) def } = { F(x) F (x) = f (x)}. { A } helps us to rememer that this symol string refers to a set of A ntiderivatives. Or we could write A f (x) dx def = { F(x) F (x) = f (x)}. Better still is { A f (x) dx def } = { F(x) F (x) = f (x)}. However, this notation is seldom used. For historical reasons the following notation 1 is the most commonly used to represent the set of all antiderivatives of f (x). { F(x) F (x) = f (x)} A set of function rules. The symol is called an indefinite integral. One could have reasonaly defined that symol like this: F 0 (x) a single function (exactly one antiderivative). But our mathematical ancestors chose to define it to represent a set of functions. def = { F(x) F (x) = f (x)} which means the same thing as = { F 0 (x) + C C is a real numer}. Many people think of F 0 (x) as the antiderivative where C = 0. We always do. The following areviation is the most commonly used notation. { } F(x) + C F 0 (x) + C C is a real numer The following is the notation that you will see in most textooks. F(x) + C In these notes we write it like this { F(x) + C} to remind you that an indefinite integral evaluates to a set of functions. Do not try to assign any meaning whatsoever to dx all y itself in this context! Think of and dx together as one indivisile symol (even though it is written as if it were not). A susequent chapter shows a mechanical procedure where you can manipulate the symol dx as if it were a separate oject. But y definition it is not. Someody had to prove a theorem that you could do those manipulations and get the right answer. We are tempted to call it ause of notation ut it s so common it won t go away. 1 Maye it s ecause you can write = ( ) a { ( F 0 () F 0 (a)) }. Most people are happy to just call it a numer. which is a set consisting of one element a Page 4 of 5
Page 5 of 6 Manually finding antiderivatives can e tricky except in the simplest cases ut these days symolic calculators allow us to find antiderivatives (evaluate indefinite integrals) swiftly and correctly. Indefinite integral calculators are availale from the main menu of this wesite. Advanced chapters under Scalars 4.1 show various manual techniques for evaluating indefinite integrals if for some reason you need to do this. You have made great progress! You have now learned enough Calculus to start solving some very interesting prolems. Initially you can use the calculators to perform the mechanics of differentiation and reverse differentiation (finding antiderivatives, indefinite integration). But pretty soon you will proaly want to start doing the differentiation yourself to speed things up. That means practicing the asic differentiation rules and learning the extremely important Chain Rule. Except for the simplest cases you will proaly want to continue using the calculators to find antiderivatives when solving real life prolems. We ll start with position, velocity and acceleration prolems the sorts of things that interested Sir Isaac Newton when he was inventing the early forms of the Calculus. Then we ll develop more ackground for the most useful functions and use them to model an even larger set of interesting prolems using differential equations (equations that contain a derivative rule). After that, we ll define the Definite Integral and use it to solve yet an even larger class of prolems. During all of this we can use calculators from the main menu to do the grunge work so that we can focus on the interesting aspects of modeling and solving real world prolems, the sorts of things that human eings still do etter than computers. Now things really get cool! Page 5 of 5