Preview of "The Core: Introductory Calculus…As It Should Be"

8/7/2019 Preview of "The Core: Introductory CalculusAs It Should Be"

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The Core: Introductory CalculusAs It Should Be

Written by Ugochukwu C. Akwarandu


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Copyright 2010 by Ugochukwu C. Akwarandu, Publishing by Widget Flux LLC,P.O. Box 922091, Norcross, GA 30010-2091. All rights reserved. Printed in the UnitedStates of America. This publication is protected by copyright, and permission should beobtained from the publisher prior to any prohibited reproduction, storage in a retrievalsystem, or transmission in any form or by any means, electronic, mechanical,photocopying, recording, or likewise. For information regarding permission(s), write to:Permissions and Rights Department.


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About the Author:

Ugochukwu C. Akwarandu received a B.A. in chemistry from Duke University in 2010.During college, he served in a volunteering group for 3 years in which he tutored localhigh school students in the math and sciences. He also worked as a teaching assistant ingeneral chemistry in which he helped freshmen digest difficult concepts. Mr. Akwaranduspends his free time trying to connect the dots, or make sense of problems that cannotseem to be solved.

Table of Contents

Note : the subtopics within each chapter are a general guideline. Some ideas are veryintricately entwined, so it was difficult to separate one subsection from another.Therefore, one will not find a specific title within the book for every subsection titlewritten in the table of contents.

- Preface

- Chapter 1: Twisting algebra to understand the full concept of a given equation

- Manipulations- Further Manipulations: Equations 1-5- Ratios- Examples- Problems

-Chapter 2: The Duality of the Differential (key to being able to jump between calculusand algebra)

- Gelled/Un-Gelled Differential

-

1=

- Making Links in a Chain- Problems

- Chapter 3: Basic calculusas it should be taught

- the derivative operator- the integral operator- Jumping the Fence: the transition b/t Calculus operations (ie SeparableDifferential Equations)- Relation of differentials to algebraic variables- Treating like a variable that can be crossed out- The duality of (or d ) as an operator and a pseudo-variable- Calculus Operator Hierarchy and Sub Operators


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- Multi-Differential Equations and beyond- Examples- Problems

- Chapter 4: multiple variable calculus can be treated like multiple variable algebra

- Breaking down the meaning of a multivariable calculus equation- Direct differentiation & implicit differentiation- Problems

- Chapter 5: Multivariable differential equations; the multivariable chain rule(introduction to high degree differentials/derivatives and mixed differentials/derivatives)

- double derivatives- mixed derivatives- mixed integrals- problems

- Chapter 6: Graphs and equations

- Some rules and definitions- Differentials and graphs- Orbs & arrows- A jumble of slightly confusing but fully logical thoughts- Problems

- Chapter 7: Manipulating calculus equations- A couple of sets- Examples- Problems

- Solutions to end of chapter problems


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--------------------

Some material has been excluded from the preview

--------------------

Chapter 2: The Duality of the Differential (key to being able to jump betweencalculus and algebra)

Un-gelled differential = b the operator and the variable act as two separate variables;its almost like two variables being multiplied together. In this case, can be called apseudo variable.

gelled differential = b the operator and the variable acts as one variable; that is, theycan be thought of as being attached (or bound) to one another during differentialmanipulations

Set 2.1:

r = wh

hwr = follows the rules of an equations because one delta is pseudomultiplied to each side of the equation; the deltas are then gelled to twodifferent variables ( r and h)


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Manipulations of hwr = :

r h

w

=1

divided both sides of original equation by r (gelled manipulation)

whr

=

divided both sides of original equation by h (gelled manipulation)

hwh

wr

=

=

1

divided both sides of original equation by ; h remainsgelled (Un-gelled manipulation)

whr

=

divided both sides of original equation by ; r remains gelled(Un-gelled manipulation)

means is the same as algebraically

=

hw

r wh

r =

=

hwr r = wh

Note : 1=

this relation makes the un-gelled manipulation possible

Note : in algebra 1 and 2, students could not multiply or divide the + or - signs acrossthe equation. That is, these operators could not be treated like a variablethey couldnt


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be treated like the is treated in calculus. Most people dont understand calculusbecause this fact is not mentioned until Calculus 3 (when a majority of people who had agrasp on algebra were weeded out)

Note :

1= ; Understand that is called the integral sign. Youll see this quite a bit in

calculus. Just like the , it can be moved around like a variable before its operation is

activated. From

1= , one can see that (using algebraic goggles) 1= ( this happens

when

1= is multiplied by ). This can also be written as 1=

. Understanding all

of these variations is the key to gaining a conceptual understanding of manipulations of the elements of calculus.

One must also realize that substituting with d does not change how one manipulatesvariables (or differentials). Understand that d represents a very small differencebetween two points while represents a difference that is not considered to be as smallcompared to that of d . The main time to deal with these different symbols within oneequation will be when we reach multivariate calculus. If youre confused between thetwo symbols, dont worry. You just have to remember that they are both manipulated thesame way when dealing with basic differential equations. So just keep this in mind. Thenext set compares manipulation of differentials (using the d symbol) with analogousmanipulations in algebra.

Set 2.2:

Differentials Algebra Analogy

dhdR

(dh ) = dR good

dydR

(dy) = dR good

hR

(h) = R good

yR

(y) = R good


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dk dR

( dk ) = dR good

dhdR

( db ) = dR bad

dhdR

( de ) = dR bad

k R

(k ) = R good

hR

(b) = R bad

hR

(e) = R bad

--------------------

Some material has been excluded from the preview

--------------------

Set 2.7

I would like to emphasize what I mean by the term activation in the next paragraph.And even though I have explained the difference between a gelled and ungelleddifferential, I wanted to introduce an additional visual heuristic that the student might useto differentiate between the two. Lastly, I wanted to introduce the phrase final form.Its my hope that the combination of these terms will help me to reveal all of thepreviously implied properties that exist when one manipulates a calculus equation. Usingthese terms within an example should aid new students in fully grasping the movementbetween a derivative and an integral. Now, I know that we have not gone overderivatives and integral equations at this point the in the book (this will be done inchapter 3), but I have designed the following section in a way that focuses only on


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differential manipulation. So in theory, one doesnt actually have to understand what aderivative operation (or integration operation) does, one just needs to follow themovement of the differential or its components (ie the delta and the variable).

Activating a term is the action of solving a problem that contains a given operator (ie + , -signs or some other operators). It can be as simple as having the term 3+2 and thencreating the equation 3+2=5 in order to show the final solution that the + operatorproduces. I will be using it to notify the student when a calculus operator should be usedwithin a given term in order to create an equation that will then be further manipulated.

In order to emphasize the difference between a differential in its gelled and ungelledform, it might help to use single quotes around the gelled form and not use those quotesaround the ungelled form. The single quotes will act like braces that hold the gelleddifferential together when it is being multiplied or divided across the equation sign.Therefore, the term T would represent a gelled differential (because it surroundedby single quotes) and T (without single quotes) would represent a ungelleddifferential. There are some examples below which differentiate the two forms adifferential.

1) T T

=

ungelled differential (can be divided by a delta)

2)

'' T gelled differential (cannot be divided by the delta because T is treated as

one variable)

3) 1''''

=

T T

two identical gelled differentials can simplify each other (same concept as

with the division of two variables in algebra)

4) 21

''2

''

''''

'' =

=

+

T

T

T T

T

5) 23 =

T T


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6) 3''

T cannot be simplified

7) =

T T

example of an ungelled differential

The single quotes used are just a heuristic for one to explicitly note when a particularform (ie gelled) of a differential is being used. That means, one can actually change theform of a differential in order to reach a required solution. Indeed, this type of manipulation is needed in order to transform a derivative into an isolated function. Andby doing this, the isolated function becomes an integral of another function. Ill just use

two variables below and perform differential manipulations with these variables in orderto show what I mean.

Given that: A = R 3 and B = 3R 2

1) BRA

=

''''

initial equation in its heuristic form because the quotes are applied to

atleast one differential in the equation (the term on the left is what I will refer as one of two arbitrary final forms in calculus; this will be explained more later)

2) '''' RBA = multiplied BRA

=

''''

by the gelled differential , '' R

3) '' RBA = I converted the gelled differential, '' A , to its ungelled form so thatI could divide both side of the equation by delta and therefore isolate the variable A.

4) ''1

RBA

= divided '' RBA = by delta on both sides as I alluded to above

5) = '' RBA used the 1 = substitution ; this is also another final form

6) = RR 23 3R after substituting A and B we reach our final solution atwhich point one can remove the braces because we are no longer involved in any calculusmanipulations


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Now, I will refer to two main final forms of an equation which some of you might not

have learned yet. One final form is the derivative operator (ieR

) applied to some

variable/function,RA

. In this case, the variable (representing a function) is A.

Students will activate the derivative operator in order to create an equation. There is ageneral formula that leads to logical solution from the activation of the derivativeoperator. This formula will be introduced in chapter 3. Another final form in calculus is

the integration operator (ie

Ror R ) applied to some variable/function, RB . In

this case, the variable (representing a function) is B. Students can also activate the

derivative operator in order to create an equation. There is a general formula that leads tological solution from the activation of the integration operator. This formula will beintroduced in chapter 3.

Another example:

Given that: A = R 3 and B = 3R 2 (same as the previous example)

1) RB need to activate this term and isolate B

2) ARB = activated form of RB

3) ARB = '' added the braces (ie single quotes) to prepare for the movement of thegelled differential

4) ARB =

''1

substituted the integral sign with

1

5) ARB = '' multiplied ARB =

''1

by delta

6) '' R

AB

= multiplied the previous equation by the gelled differential (ie '' R ) in

order to fully isolate the function B

7) RR

R

=

)(3

32 removed braces from R and substituted A and B for their real

values; the is the final form


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Note : to those who know about the integration operation, I have purposely left out theconstant that is created when a function is integrated. I just wanted things to be simpleand symmetrical for now.

Manipulating equations from one final form to the other (and then reversing the initialprocess) will help the student become used to manipulating calculus equations. Practingthis transition/manipulation on paper should help the idea of calculus stick. Even if one doesnt understand the operation resulting from each activation now, one shouldcome back to this section after reading chapter 3. This review after chapter 3 shouldhelp engrain the necessary steps of calculus manipulation in the students head

I feel like its important that individuals view repetition of these ideas so I am going topresent the same examples above with different values. This way, the student canbecome more comfortable with calculus based manipulations. The examples will bepresented in a free flowing format while still offering notes when necessary. For anyonewho really gets the concept of manipulating equations from one final form to another, Iwould advise you to just glance at the following example as much as necessary and thenmove on.

Another example:

Given that: A = E 6 and B = 6E 5,

1) E A

2) BE A =

the derivative final form

3) BE A =

''only have to add braces to E because A (as a whole) will not be

multiplied across the equation sign- only the from A will be divided later on

4) '' E BA =

5) == ''''1

E BE BA


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6) = E E E 56 6 the integration or integral final form

Another example:

Given that: A = E 6 and B = 6E 5,

1) E B

2) AE B=

the integration final form

3) AE B =

1

4) AE B =

''1

5) AE B = ''

6) '' E

AB

=

7) E E

E

=

)(6

65 the derivative final form

Yet Another example^__^

Given that: A = H 2 + H 4 and B = 2H + 4H 3,

1) H A


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2) BH A

=

3) BH A =

''

4) '' H BA =

5) '' H BA =

6) ')'42()( 342 H H H H H +=+ substituted in order to show the distribution inthe next step

7) )''4''2()( 342 H H H H H H +=+ distributed '' H on the right side of theequation

8) )''4''2(1

)( 342 H H H H H H +

=+

9) )''41

''21

( 342 H H H H H H

+

=+

10) )''4''2( 342 H H H H H H +=+

11) H H H H H H +=+ 342 42

Another example:

Given that: A = H 2 + H 4 and B = 2H + 4H 3,


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1) H B

2) AH B =

3) AH B =

1one can also add the braces before the integral sign is substituted; it

doesnt really matter as long as you know that the deltas should cancel out at this pointthen your fine

4) AH B =

''1

5) AH B = ''

6) )(')'42( 423 H H H H H +=+

7) )(''

)42( 423 H H H

H H +

=+

8) )''''

(42 423 H H

H H

H H

+

=+

9) '')(

'')(

4242

3

H H

H H

H H

+

=+

10) H H

H H

H H

+

=+

)()(42

423

Note : the student might need to revisit the above set of examples after reading throughchapter 3. This is the last section the heuristic braces will be used to explicitly showwhen a differential cannot be divided by one of its components. However, the studentcan continue to use the braces ( in your notebook ) to represent a gelled differential inorder to cement this property in his/her head.


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Problems

No problems necessary for this section. The next section will make up for it. I promiseyou that ^___^.

--------------------End of Chapter 2--------------------

Chapter 3: Basic Calculusas it should be taught

Basic calculus can start with either integrals or derivatives. Basic calculus should betaught with the knowledge of how to integrate and take the derivative of a function. Thatway one learns the complete cycle of calculus. Integrals and derivatives shouldnt betaught across two semesters of a school year. Why would anyone do that? Think back toalgebra. After learning multiplication, didnt you learn how to divide almost immediately(ie in the same lesson or within days)? The two operations of multiplication and divisionwere taught simultaneously and this should be the case with the two operations of integration and differentiation. And once each calculus operation is introduced one mustlearn how to move back and forth between them. The transition between integrationand differentiation is the crux of calculus (in my opinion). Without proper knowledgeof this transition, one will not have the ability advance in calculus-based fields with muchsuccess. But dont worry, this transition is quite easy. The problem is that its notexplicitly introduced to students until calculus 3. It goes by the name of separabledifferential equations. Again, not teaching separable differential equations in thebeginning of calculus is a big FAIL.--------------------

The general idea is that the derivative operatorW

acts on a function R = W n where R

and W are just variables and n represents a whole number. When the derivative of afunction is taken, the resulting answer is equal to n*W n-1 . That is, the original number inthe exponent position (ie n) is multiplied by W to the power of one less than theoriginal number (ie n-1 ) in the exponent. Examples of this can be seen in Set 3.1.


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Set 3.1: Learning the general derivative

If R = W 3, then 23

3)()(

W W

W W R

=

=

If S = 2N 5, then 445

1052)2()(

N N N N

N S

==

=

If Y = X2

, then X X X

X Y

2)()( 2

=

=

For the derivative of a set of variables raised to a power s uch as (3B + B) n where B is avariable and n is a number, the derivative of this general equation is n(3B + B) n-1

* )3( BBB

+

.

So if U = (4L 2 + L 4)3 , then )48()4(3 3242 LLLLLU ++=

348 LL + is the derivative of the contents within the parentheses of the original function.--------------------

The general idea for integration is that the integral operator

W or W (remember

that =1

) acts on a function D = W N and the resulting function is equal to

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11

C W N

N ++

+ , where C 1 is a constant. One should take note that

W is the inverted

form of the derivative operatorW

. That is, the integral operator has the inverted form

of the derivative operator. These two forms can be reached by moving each term in theoperator to the opposite side of the equation through methods of division andmultiplication. The methods in Set 3.2 explain the previous sentences.

Set 3.2: Jumping the Fence


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W W 23 substituted G

13

3

33 C W

W +=

Note : C 1 represents a constant not a variable.

Part 2:

H = W3

+C 1

W H

W C W + 13

W C W W +

13

distribution of the integral sign and the differential, W

21

4

4C W C

W ++

Note : C 1 and C 2 are both constants

--------------------

Relation of differentials to algebraic variables :

Examples of differentials: x, b, n, e, s etc

The evolution of algebra to calculus (via differentials):


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A random, completely arbitrary equation is made up(below this) to explain the idea of how algebra mutates into calculus.

n=abc algebra equation

n = ab c differential equation (ie algebra/calculus mix)

n = ab c calculus equation

Set 3.4:

Differentials Algebraic Analogy

V t x

=

t V x = multiplied by t

22 )()( t V x = squared both sides of

equation

1)()(

2

2

=

t V x

divided by (V t)2

22 )(1

)(1

xt V =

divided by ( x)2

V t x

=

Vt x = multiplied by t

22 )()( Vt x = squared both sides of

equation

1)()(

2

2

=Vt

x divided by (Vt) 2

22 )(1

)(1

xVt = divided by (x) 2

Note : I advised you to manipulate algebraic equations before. And now, I will also adviseyou to thoroughly manipulate equations with differentials. If you can manipulateequations with differentials, then you should have no problem manipulating calculusequations because the manipulation principles are exactly the same. This step is essentialfor one to be able to alter a derivative into the change of its original function. One cansee this in the first two steps of the differentials/algebra box above.


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Treating like a variable that can be crossed out

The Set 3.5 may be considered mathematically improper ( that is, if you consider thetruth to be improper), but it works so I would suggest you learn it. Sometimes I think of like I would think of a variable (ie a pseudo-variable) when I cross out things.Examples below:

Set 3.5

Algebraic Exponents Delta Analogy

AAA

=2

3

25

7

BBB

=

79

16

E E

E =

=

2

3

25

7

=

79

16

=

The examples above are how I remember that the division of one differential by anotheris equal to a constant (as opposed to another differential).

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Preview of "The Core: Introductory Calculus…As It Should Be"