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Appendices

Important Formulas

Differentiation Rules

1. ddx(cx)=c 10. ddx(ax)=lnaax 19. ddx(sin-1x)=11-x2 28. ddx(sechx)=-sechxtanhx
2. ddx(u±v)=u±v 11. ddx(lnx)=1x 20. ddx(cos-1x)=-11-x2 29. ddx(cschx)=-cschxcothx
3. ddx(uv)=uv+uv   12. ddx(logax)=1xlna 21. ddx(csc-1x)=-1|x|x2-1 30. ddx(cothx)=-csch2x
4. ddx(uv)=vu-uvv2 13. ddx(sinx)=cosx 22. ddx(sec-1x)=1|x|x2-1   31. ddx(cosh-1x)=1x2-1
5. ddx(u(v))=u(v)v 14. ddx(cosx)=-sinx 23. ddx(tan-1x)=11+x2 32. ddx(sinh-1x)=1x2+1
6. ddx(c)=0 15. ddx(cscx)=-cscxcotx   24. ddx(cot-1x)=-11+x2 33. ddx(sech-1x)=-1x1-x2
7. ddx(x)=1 16. ddx(secx)=secxtanx 25. ddx(coshx)=sinhx 34. ddx(csch-1x)=-1|x|1+x2
8. ddx(xn)=nxn-1 17. ddx(tanx)=sec2x 26. ddx(sinhx)=coshx 35. ddx(tanh-1x)=11-x2
9. ddx(ex)=ex 18. ddx(cotx)=-csc2x 27. ddx(tanhx)=sech2x 36. ddx(coth-1x)=11-x2

Integration Rules

11. cf(x)𝑑x=cf(x)𝑑x 11. tanxdx=-ln|cosx|+C 23. 1xx2-a2𝑑x=1asec-1(|x|a)+C
12. f(x)±g(x)dx= 12. secxdx=ln|secx+tanx|+C 24. coshxdx=sinhx+C
12. f(x)𝑑x±g(x)𝑑x 13. cscx=-ln|cscx+cotx|+C 25. sinhxdx=coshx+C
13. 0𝑑x=C 14. cotxdx=ln|sinx|+C 26. tanhxdx=ln(coshx)+C
14. 1𝑑x=x+C 15. sec2xdx=tanx+C 27. cothxdx=ln|sinhx|+C
1

5. xn𝑑x=1n+1xn+1+C,

n-1

16. csc2xdx=-cotx+C 28. 1x2-a2𝑑x=ln|x+x2-a2|+C
16. ex𝑑x=ex+C 17. secxtanxdx=secx+C 29. 1x2+a2𝑑x=ln|x+x2+a2|+C
17. ax𝑑x=1lnaax+C 18. cscxcotxdx=-cscx+C 30. 1a2-x2𝑑x=12aln|a+xa-x|+C
18. 1x𝑑x=ln|x|+C 19. cos2xdx=12x+14sin(2x)+C 31. 1xa2-x2𝑑x=1aln(xa+a2-x2)+C
19. cosxdx=sinx+C 20. sin2xdx=12x-14sin(2x)+C 32. 1xx2+a2𝑑x=1aln|xa+x2+a2|+C
10. sinxdx=-cosx+C 21. 1x2+a2𝑑x=1atan-1(xa)+C 33. x2+a2𝑑x=
22. 1a2-x2𝑑x=sin-1(x|a|)+C 33. x2x2+a2+a22ln(x+x2+a2)+C

The Unit Circle

x

y

0

0

(1,0)

30

π/6

(32,12)

45

π/4

(22,22)

60

π/3

(12,32)

90

π/2

(0,1)

120

2π/3

(-12,32)

135

3π/4

(-22,22)

150

5π/6

(-32,12)

180

π

(-1,0)

210

7π/6

(-32,-12)

225

5π/4

(-22,-22)

240

4π/3

(-12,-32)

270

3π/2

(0,-1)

300

5π/3

(12,-32)

315

7π/4

(22,-22)

330

11π/6

(32,-12)

Definitions of the Trigonometric Functions

Unit Circle Definition

x

y

(x,y)

y

x

θ
sinθ =y cosθ =x cscθ =1y secθ =1x tanθ =yx cotθ =xy

Right Triangle Definition

Adjacent

Opposite

Hypotenuse

θ
sinθ =OH cscθ =HO cosθ =AH secθ =HA tanθ =OA cotθ =AO

Common Trigonometric Identities

Pythagorean Identities

sin2x+cos2x=1
tan2x+1=sec2x
1+cot2x=csc2x

Cofunction Identities

sin(π2-x) =cosx csc(π2-x) =secx
cos(π2-x) =sinx sec(π2-x) =cscx
tan(π2-x) =cotx cot(π2-x) =tanx

Double Angle Formulas

sin2x =2sinxcosx
cos2x =cos2x-sin2x
=2cos2x-1
=1-2sin2x
tan2x =2tanx1-tan2x

Sum to Product Formulas

sinx+siny =2sin(x+y2)cos(x-y2)
sinx-siny =2sin(x-y2)cos(x+y2)
cosx+cosy =2cos(x+y2)cos(x-y2)
cosx-cosy =2sin(x+y2)sin(y-x2)

Power–Reducing Formulas

sin2x =1-cos2x2
cos2x =1+cos2x2
tan2x =1-cos2x1+cos2x

Even/Odd Identities

sin(-x) =-sinx
cos(-x) =cosx
tan(-x) =-tanx
csc(-x) =-cscx
sec(-x) =secx
cot(-x) =-cotx

Product to Sum Formulas

sinxsiny =12(cos(x-y)-cos(x+y))
cosxcosy =12(cos(x-y)+cos(x+y))
sinxcosy =12(sin(x+y)+sin(x-y))

Angle Sum/Difference Formulas

sin(x±y) =sinxcosy±cosxsiny
cos(x±y) =cosxcosysinxsiny
tan(x±y) =tanx±tany1tanxtany

Areas and Volumes

Triangles h=asinθ Area=12bh Law of Cosines: c2=a2+b2-2abcosθ

b

θ

a

c

h
Right Circular Cone Volume=13πr2h Surface Area= πrr2+h2+πr2

h

r
Parallelograms Area = bh

b

h
Right Circular Cylinder Volume=πr2h Surface Area= 2πrh+2πr2

h

r
Trapezoids Area = 12(a+b)h

b

a

h
Sphere Volume=43πr3 Surface Area=4πr2

r
Circles Area=πr2 Circumference=2πr

r
General Cone Area of Base=A Volume=13Ah

h

A
Sectors of Circles θ in radians Area=12θr2 s=rθ

r

s

θ
General Right Cylinder Area of Base=A Volume=Ah

h

A

Algebra

Factors and Zeros of Polynomials

Let p(x)=anxn+an-1xn-1++a1x+a0 be a polynomial. If p(a)=0, then a is a zero of the polynomial and a solution of the equation p(x)=0. Furthermore, (x-a) is a factor of the polynomial.

Fundamental Theorem of Algebra

An nth degree polynomial has n (not necessarily distinct) zeros. Although all of these zeros may be imaginary, a real polynomial of odd degree must have at least one real zero.

Quadratic Formula

If p(x)=ax2+bx+c, then the zeros of p are x=-b±b2-4ac2a

Special Factoring

x2-a2 =(x-a)(x+a) x3±a3 =(x±a)(x2ax+a2) x4-a4 =(x2-a2)(x2+a2)

Binomial Theorem

(x+y)2 =x2+2xy+y2 (x+y)3 =x3+3x2y+3xy2+y3
(x+y)4 =x4+4x3y+6x2y2+4xy3+y4 (x+y)n =k=0n(nk)xn-kyk

Rational Zero Theorem

If p(x)=anxn+an-1xn-1++a1x+a0 has integer coefficients, then every rational zero of p is of the form x=r/s, where r is a factor of a0 and s is a factor of an.

Factoring by Grouping

acx3+adx2+bcx+bd=ax2(cx+d)+b(cx+d)=(ax2+b)(cx+d)

Arithmetic Operations

ab+ac=a(b+c) ab+cd=ad+bcbd a+bc=ac+bc
(ab)(cd)=(ab)(dc)=adbc (ab)c=abc a(bc)=acb
a(bc)=abc a-bc-d=b-ad-c ab+aca=b+c

Exponents and Radicals

a0=1,a0 (ab)x =axbx axay =ax+y a =a1/2 axay =ax-y an =a1/n
(ab)x=axbx amn =am/n a-x =1ax abn =anbn (ax)y =axy abn =anbn

Additional Formulas

Summation Formulas

i=1nc =cn i=1ni =n(n+1)2 i=1ni2 =n(n+1)(2n+1)6 i=1ni3 =(n(n+1)2)2

Trapezoidal Rule

abf(x)𝑑xΔx2[f(x1)+2f(x2)+2f(x3)++2f(xn)+f(xn+1)]
with Error(b-a)312n2[max|f′′(x)|]

Simpson’s Rule

abf(x)𝑑xΔx3[f(x1)+4f(x2)+2f(x3)+4f(x4)++2f(xn-1)+4f(xn)+f(xn+1)]
with Error(b-a)5180n4[max|f(4)(x)|]


Arc Length

L=ab1+f(x)2𝑑x


Work Done by a Variable Force

W=abF(x)𝑑x

Force Exerted by a Fluid

F=abwd(y)(y)𝑑y

Taylor Series Expansion for f(x)

pn(x)=f(c)+f(c)(x-c)+f′′(c)2!(x-c)2+f′′′(c)3!(x-c)3++f(n)(c)n!(x-c)n+

Standard Form of Conic Sections

Parabola                Ellipse Hyperbola
Vertical axis Horizontal axis Foci and vertices Foci and vertices
on x-axis on y-axis
y=x24p x=y24p x2a2+y2b2=1 x2a2-y2b2=1 y2b2-x2a2=1

Summary of Tests for Series


Test
Series

Condition(s) of Convergence

Condition(s) of Divergence

Comment

nth-Term

Test for

Divergence

n=1an limnan0

cannot show convergence.


Alternating

Series

n=1(-1)nbn limnbn=0

bn must be positive and decreasing


Geometric

Series

n=0arn |r|<1 |r|1 Sum =a1-r

Telescoping

Series

n=1bn-bn+m limnbn=L

Sum =

(n=1mbn)-L


p-Series
n=11(an+b)p p>1 p1

p-Series For

Logarithms

n=11(an+b)(logn)p p>1 p1

logarithm’s base doesn’t affect convergence.


Integral

Test

n=1an

1a(n)𝑑n

convergesd

1a(n)𝑑n

diverges

an=a(n) must be positive and decreasing


Direct

Comparison

n=1an

n=0bn

converges and

0anbn

n=0bn

diverges and

0bnan


Limit

Comparison

n=1an

n=0bn

converges and

limnan/bn<

n=0bn

diverges and

limnan/bn>0or =

an,bn>0

Ratio Test
n=1an

limn|an+1an|<1

limn|an+1an|>1or =

limit of 1 is indeterminate


Root Test
n=1an

limn|an|1/n<1

limn|an|1/n>1or =

limit of 1 is indeterminate



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