Todorovich / Walker Calculus

Maths of the Gods
1. Auflage 2024
ISBN: 978-1-923078-25-3
Verlag: Vivid Publishing
Format: EPUB
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 200 Seiten

Reihe: Calculus

Discover the intriguing world of calculus with this book, which aims to demystify and enrich your understanding of this fundamental branch of mathematics. While many find calculus both fascinating and intimidating, this guide strives to make it more accessible and engaging. Written in an informal tone, every effort has been made to maintain mathematical integrity, ensuring clarity without sacrificing depth. Through a range of tools and techniques, readers will uncover the reasons behind phenomena such as the elliptical orbit of Earth around the sun, the story of Newton's apple, and Einstein's insights into the atom bomb. Encouraging readers to unleash their imagination, this book invites contemplation of the infinitely small and large, fostering a deeper understanding of the universe. Ultimately, it underscores the vital role of imagination in scientific discovery, with calculus serving as the language that quantifies our boldest imaginings.

Ed Walker is the author of 'Calculus - Maths of the Gods'.

Autoren/Hrsg.

Todorovich, Bill

Walker, Ed

Leseproben

Chapter 1 Trigonometric Identities and Logarithms

It is impossible to study calculus without knowing the basics of trigonometry and logarithms.

sinesin ?=OppositeHypotenusecosinecos ?=AdjacentHypotenuse

tangent tan ?=OppositeAdjacent=sin?cos?

From the graphs and diagram above, it is apparent that sin is positive in the first and second quadrants, and negative in the third and fourth.

Cos is positive in the first and fourth quadrants, and negative in the second and third.

Tan is positive in the first and third, while negative in the second and fourth quadrants.

secant sec ?=1cos?cosecant csc ?=1sin?cotangent cot ?=1tan?

sin?=cos90-?tan?=cot90-?

sin-?=-sin?cos-?=cos? tan-?=-tan?

sin2?+cos2?=11+tan2?=sec2?

The inverse functions arcsine, arccosine, arctan, arcsec, arccosec and arccot may be written:

asin, atan or denoted by sin-1 tan-1.

Example: Express cos ? in terms of x, where arctan x = ?.

In the diagram above, the angle is represented as ?.

As ?= atan x?tan?=x1 from which?????cos ?=11+x2

The Unit Circle Showing Six Trigonometric Identities

(1) AB = sin ?

(2) OB = cos ?

(3) AD = tan ?

(4) OE = cosec ?

(5) OD = sec ?

(6) AE = cot ?

Note:i???????OE2+OD2=ED2?cosec2?+sec2?=cot?+tan?2Also ii?????BD2+AB2=AD2?sec?-cos?2+sin2?=tan2??iii ??OE-AB2+OB2=AE2?cosec?-sin?2+cos2?=cot2?iv?? Area ?OAE+Area ?OAD=Area OED

12 cot ?+12 tan ?=12sec ? cosec ??cot ?+tan ?=sec ? cosec ?

Double Angle formulas

OQ=cos ßPB=cos a sinßAQ=sin acos ßsina+ß=BC+PB=AQ+PBsina+ß=sin a cos ß+sin ßcos asina-ß=sin a cos ß-sin ßcos a

cosa+ß=OC=OA-CA=OA-BQcosa+ß=cos a cos ß-sin a sin ßcosa-ß=cos a cos ß + sin a sin ß

tana+ß=sin a cos ß+sin ß cos acos a cos ß-sin a sin ß

Dividing both the numerator and the denominator by cos a cos ß we get

tana+ß=tan a+tan ß1-tan atan ßtana-ß=tan a-tan ß1+tan atan ß

sin 2?=2 sin? cos ?

cos2?=cos2?-sin2?tan2?=2tan?1-tan2?

Pythagoras’s Theorem

sina=ACAB=ADAC?AC2=ADABsinß=BCAB=BDBC?BC2=BDABAC2+BC2=ABAD+BDAC2+BC2=AB2

Sine Rule

d=b sin C=c sinB?bsinB=csinC Similarly asinA=bsinB?asin A=bsin B=csin C

Area of Triangle

Area=12hd+e=h2a=b sin C2a=12ab sin CSimilarly??????? Area=12bc sin A=12ac sin B

Cosine Rule

a2=c+AD2+CD2=c2+2cAD+AD2+CD2=b2+c2+2cAD=b2+c2+2cb cos180-A?a2=b2+c2-2bc cosA

Similarly b2=a2+c2-2ac cosB and c2=a2+b2-2ab cos C

The Radian

A radian is vital for calculus as many of the formulas for differentiation and integration only apply when angles are given in radians.

A radian is the angle which subtends an arc length equal to the radius.

The ratio CircumferenceArc Length subtended by one radian=2prr=2p

Therefore there are 2p radians in a circle.

12p=One radian expressed in degrees360?Radian=3602p ˜ 57.29577951 degrees

Exercise: a=2 b=1.2 angle B=30o Find angle A

asinA=bsinB?sinA=asinBb=2121.2=56?A=asin 56=56.443 or 123.557 degrees

This example illustrates that when using the sine rule, if the side opposite the given angle is the lesser of the two then there may be no solution or two solutions.

Exercise (2) Cover over the right side and have a try before checking the answer. Do not use your calculator. Turn the radians into degrees, and then work out which quadrant we are in.

sinp4=sin45o=22sin-p4=sin315o=-22sin3p2=sin270o=-1tan3p4=tan135o=-1cos-p4=cos315o=22cos8p=cos0o=1

sin7p=sin180o=0

sin5p2=sin90o=1cos2p3=cos120o=-12

Exercise (3) x = cos ? y = sin 2? Express y in terms of x.

y=sin2?=2sin? cos?=2x1-x2

Exercise (4) y = atan x Express sec y in terms of x

secy=1+x2

Exercise (5) Find the three sides of the triangle if a=2Area=1B=p6

Area=12ac sinB?1=122c12?c=2b2=a2+c2-2ac cosB=4+4-832?b=22-3˜1.035276

The three sides are a=2b=1.035276c=2

Exercise (6) In triangle ABC a = 4 b = 2 c = 3 Solve for angle A

a2=b2+c2-2bc cosA?16=4+9-12 cosA

?cosA=-14?A˜104.477512 Degrees

Note: The cosine rule only gives one possible answer.

Logarithms and Exponentials

y = loga x This means y is the power by which we need to raise “a” to equal the value of x. “a” can be any number; however, there are three particularly useful bases.

1???logx=log10x???Common Logarithm2???lnx=logex???Natural Logarithm3???lgx=log2x??Binary Logarithm

Common Logarithms are the most useful for calculations.

Example: As 103 = 1,000 ? log10 1000 = 3

Multiplying

100×1000=100,000102×103=105log100+log1000=log100,000The exponents are?????? 2+3=5

Dividing

10,00001,000=100105103=102log100,000-log1,000=log100That is?????? 5-3=2

The Power Function

1003=1023=102×3=1063log100=log1,000,000That is 23=6

Summary of the rules of logarithms

loga×b=loga+logblogab=loga-logblogab=b log a

The Natural or Napierian logarithm loge written ln is extensively used in calculus.

“e” is the transcendental number 2.71828…

If ln 7.38905 = 2 ? e2 ˜ 7.38905 or (2.71828)2 ˜ 7.38905

The same rules apply

lna×b=ln a+ln blnab=lna-lnblnab=blna

Binary Logarithms have a base of 2 and are widely used in engineering.

Changing the base of Logarithms

y=lnx?ey=x??????logey=logx?yloge=logx?y=logxloge?lnx=logxlogeand similarly????????logx=lnxln10

Summary: The logarithm in the new base equals the logarithm (in the old base) divided by the logarythm(in the old base) of the new base.

i.e..??lognx=logmxlogmn

Changing the log base can be useful for transforming a function into a convenient form for calculus as shown by the...