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Mastering Calculus: Key Concepts and Formulas Explained

School
J Percy Page School**We aren't endorsed by this school
Course
MATH 30
Subject
Mathematics
Date
Dec 12, 2024
Pages
19
Uploaded by DrIbexPerson1167
Math FormulasChapter 1: CalculusLimits of infinity1)=/ 2)=0?∞lim→?2∞?∞lim→1?Ex:?∞lim→4?3+2?2+1?2−2?-4?3?3+2?2?3+1?3?2?3−2??3-4+2?+1?31?−2?2-4+2∞+1∞31∞−2∞2-4+0+00−0- =4
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Differential Calculus: Constant ruley=(c) = 0⇒???Ex:Y=10-= 0????Differential Calculus: Power rule=n⇒?????−1Ex:Y=3–2+4x+1?3?2-????= 9?2− 4? + 4Differential Calculus: Sum & Difference rule⇒y=f(x)g(x)+−Ex:Y=4?4− 3?3+ 2? + 4-????= 16?3− 9?2+ 2Product Rule=U⇒????1+ ??1Ex:Y=(?2)(?4)-????= ??1+ ??1-(?2) · (4?3) + (?4) · (2?)-(4?5) + (2?5)-=6?5Quotient Rule=⇒?????1−??1?2Ex:Y=3?+1?
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-????=?·(3)−(3?+1)·(1)(?)2- =3?−3?+1?2- =1?2Tangent liney=mx+c⇒Ex:Find the equation of the tangent line to the curvea) y=at P(1,-2)?2− ? − 1-2x-1 at (1,-2)????=- m=2(1)-1- m=1-y=mx+c- (-2)=1(1)+c- (-2) – 1= (1)-1+c- -3=c- y=x-3Normal liney=x+c⇒−1?Ex:Find the equation of the normal line to the curvea) y=at P(1,2)?2− ? − 2-2x-1 at (1,2)????=- m=2(1)-1- m=1- y=x+c−1?
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- 2=(1)+c−11- 2+(1)=-1+(1)+c- 3=c- y=-x+3Chain Rule=x⇒???????????Ex:Find the derivative of the functiona) y=(?2+ 2?)4-x???=????????-4(2? + 2) · (?2+ 2?)- =8? + 8(?2+ 2?)Stationary points: Maximum point<0⇒?2???2Minimum point>0⇒?2???2Inflection point=0⇒?2???2Ex:Find the coordinates of the following curves and determinetheir nature a) y=?2− 4? + 1-= 2x-4=02x=4x=2????→→- y=(2–4(2)+1)2- y=4–8+1- y=-3- P=(2,-3)
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=(2x-4)=2 =Minimum point?2???2???b) y=2?3-= 6=0==?????2→6?2606→?20- x=0- y=2(0)3- y=0- P=(0,0)-=(6)=12x0 =Inflection point?2???2????2→ 12(0) =c) y=− ?2+ 4? + 1-=-2x+4=0-2x=-4=2????→→ ?- (2+4(2)+1)2- 4+8+1- y=13- P=(2,13)-=(-2x+4)=-2 =Maximum point?2???2???Differentiation in KinematicsV=/⇒?????????Ex:the position of a particle is given (S=-4+3t)?3?2a) Find the velocity function in terms of t- V==V=-8t+3????→3?2b) Find acceleration function in terms of t
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- A==A=6t-8????→Indefinite Integration:=(x+c)C is a constant1) ∫ ??⇒= (kx+c)K is a constant2) ∫ 𝑘??⇒= k3) ∫ 𝑘?(?)??∫ ?(?)??=+c4) ∫ ??????+1?+1Ex:a)∫ 3??- =3x+cb)∫ ?2??- =+c??+1?+1-+c?2+12+1- =+c?33c)∫ ?2− 4? − 2??-+c??+1?+1
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-+–2x +c?2+12+1−4?1+11+1-+–2x +c?33−4?22- =+ –2–2x + c?33?2Definite Integration:[f(x)(a)= lower limit⇒∫ ?(?)?? ]??→(b)= upper limit→f(b)- f(a)Ex:12∫ 2? ??-==2?1+11+1⎡⎢⎣⎤⎥⎦2?22⎡⎢⎣⎤⎥⎦?2- (2-(1)2)2- 4-1- =3Integration of kinematics:⇒? = ∫ ???? = ∫ ???S()=differentiation(V)differentiation (A)→→integrationintegration←←Ex:A particle moves to a straight line such that its acceleration is(a) m/when a=2t. If its initial velocity is 5m/s, find an?2expression for the s, the distance in meters traveled from thestart in t seconds.-==+c∫(2?) ?? 2?1+11+12?22
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- = V=+cv= 5m/st=0?2→→-V=+c?2- 5=(o+c)2- 5=c- V=+5?2- S=V=+5)dt∫(?2-+ 5t + c?2+12+1- S=+ 5t + C?33Area under the curve:A===f(b) – f(a)⇒??∫ ?(?)?? [?(?)]??Ex:Determine the area enclosed by y=2x+3 and the x-axis fromx=1, x=4- A=dx??∫[(𝑓)(?)]- A=dx14∫(2? + 3)- A=+3x + c2?1+11+1- A=+3x[?2]14- A= (4+3(4) – (1+3(1))2)2- A= (16+12) –(1+3)- 28-4
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A=24 unitsArea between two curves:A=– g(x)]⇒??∫[?(?)A= (upper function – lower function)??∫A= (right function – left function)??∫Ex:Find the area of the region bounded by the curves above y=+1 bounded below the curve is y=x by x=0 & x= 1?2- A= (upper function - lower function)??∫- A= (+1) – (x)01∫ ?2- A=(-x +1)01∫ ?2- A= [–+ x?2+12+1?1+11+1]01- A=–+ x[?33?22]01- A=–+ (1) ––+ (0)(1)33(1)22(0)33(0)22- A=–+– 0(1)33(1)22(1)1- A =–+263666- A=+−1666
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- A=square units56Volumes: Solids of revolutiondx⇒π??∫?2Ex:Find the volume of the solids formed when the following arerevolved through 2about the x-axis?πa) y=2xfor 0x3≤≤- V=()dxπ??∫ ?2- V=(2xdxπ03∫)2- V=(2xdxπ03∫)2- V=(4)π03∫?2→4?2+12+1- V=π03∫4?33- =–π[4(3)33][4(0)33]- =– 0π[4(27)3]- =π[1083]- V=36πDerivatives of trigonometric functions:(sin x) = cos x(1)⇒???(cos x) = -sin x(2)⇒???
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(tan x) = sex(3)⇒????2(cot x) = -csx(4)⇒????2(sec x) = sec x tan x(5)⇒???(csc x) = -csc x cot x(6)⇒???Ex:Differentiate each of the following functiona) f(x)=?2??? ?-= U+V=(-sin x) + (c0s x) (2x)?????1?1→?2- =-sin x + 2x cos x?2- =+−?2?𝑖? ??2? ??? ??- = x(-x sin x + 2 cos x)b) f(x)=??? ??𝑖? ?-==??????1−??1?2→(?𝑖? ?) (−?𝑖? ?) −(??? ?) (??? ?)(?𝑖? ?)2- =(−?𝑖?2?) − (???2?)(?𝑖? ?)2-(−1)(?𝑖? ?)2- =-cs?2?Integral of trigonometric functions:(1)dx = -cos x + c∫(?𝑖? ?)
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(2)dx = sin x + c∫(??? ?)(3)dx = tan x + c∫(???2?)Ex:integrate: a)+ x) dx∫(??? ?- =sin x ++ C?1+11+1- = sin x ++ C?22b)(+ sin x) dx3?2-– cos x + c3?2+12+1-- cos x + c3?33-– cos x + c= ?3Chapter 2: TrigonometryRadian & degree measure of angles: converting degreeinto radian⇒π180Ex:express each of these angles in radiansa) 45◦-==π18045180-=π4
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b) 60◦-==π18060180-=π3c) 270◦-==π180270180- =32πconverting radian into degree⇒180πEx:express each of these angles in degreesa)π6-=180π1806- =30◦b)56π-= 180180π·56- =150◦Arc length:S=⇒θ?Ex:ABC DEF is a regular hexagon inscribed in a circle of aradius 9 cm to find the length of the minor Arc (AB)?-== 60θ3606◦
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-=θ(60◦)·(π)180→60180π- =π3- S= r) x (9 cm)θ→ =(13π- =3 cmπArea of a sector:⇒12θ?2Ex:ABC DEF is a regular hexagon and has a radius of 9 cm,find the area of sector AB?-==60θ360◦6→◦- 60==◦π1806018013π- A=12θ?2- A=) ()(9(1213π)2- A=((81)1·12·3π)- A=(16π)(81)- A=13.5π??2Trigonometry functions at special angles (30 ,45 ,60 )◦◦◦
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θ30◦45◦60◦SIN122232COS322212TAN33123Quadrantal angles of angles (0 ,90 , 180 , 270 , 360 )◦◦◦◦◦θ0◦90◦180◦270◦360◦SIN010-10COS10-101TAN0∞0∞0Algebraic signs of trigonometry functions (QuadrantChart)
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Reference angles:Ar=AAr=180 – A◦
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Ar=A–180◦Ar=360 – A◦Trigonometry functions of right angle triangles:(1) sin x =??(2) cos x =hyp (r)??(3) tan x =opp (y)??(4) csc x =adj (x)??(5) sec x =??(6) cot x =??Verification of trigonometry identities:(1) Sin x = y(2) Cos x = x
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(3)Tan x =?𝑖??????(4) Csc x =1?𝑖??(5) Sec x =1????(6) Cot x =?????𝑖?? Fundamental Identities:1) Reciprocal identity(1) Csc x =1?𝑖??(2) Sec x =1????(3) Tan x =1????(4) Cot x =1???? 2)Pythagorean identity(1) Cox + six=1?2?2(2) Cox= 1 – six?2?2(3) Six = 1– cox?2?2(4) Sex= 1+ Tax?2?2(5) Csx= 1 + Cox?2?23)Quotient identity(1) Tan x=?𝑖?? ????
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(2) Cot x =?????𝑖??4)Co function relations(1) Sin x = cos(90– )θ(2) Cos x = sin (90– )θ(3) Tan x = cot (90– )θ(4) Cot x = tan (90– )θSum & Difference identity:- Sin (A+B) = SinA CosB + CosA SinB- Sin (A–B) = SinA CosB – CosA SinB- Cos (A+B)= CosA CosB – SinA SinB- Cos (A–B) = Cos A CosB + SinA SinB- Tan (A+B) =???? + ????1−???? ????-Tan (A–B) =???? − ????1+???? ????
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