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Calculus 2 > Midterm1 > Flashcards

Midterm1 Flashcards

1
Q

Integration by parts formula

A

uv- integral v du

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2
Q

LIATE

A

Logarithmic,Inverse Trig, Algebraic, Trig, Exponential

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3
Q

if integral ( p(x)sin(ax))

A

let u = p(x) du=the rest

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4
Q

sqrt( a^2 - x^2)

A

sub with x=asin(theta), identity: cos^2(theta)= 1- sin^2(theta)

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5
Q

sqrt(a^2 +x^2)

A

sub with x=atan(theta) sec^2(theta) = 1+tan^2(theta)

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6
Q

sqrt(x^2 - a^2)

A

sub with x=asec(theta) tan^2(theta)=sec^2(theta)-1

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7
Q

Integrating rational functions

A

if degree P(x)>=Q(x) use long division

otherwise use partial fractions

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8
Q

when doing partial fractions, denominator has (x-2)^2

A

do a variable over each power up to 2

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9
Q

when doing partial fractions, denominator when factored has irreducible quadrativ

A

make it BX+C/irreducible.

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10
Q

ODE

A

ordinary diff eq- equation containing a fcn of one variable and one or more derivatives of that fcn

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11
Q

order of an ODE

A

order of the highest derivative

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12
Q

Eullers Method equation

A

yn = yn-1+ h * F(xn-1,Yn-1)

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13
Q

I(x) in diffeq

A

e^(integral(P(x))dx)

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14
Q

standard form of linear 1st order ODE

A

dy/dx + yP(x) = Q(x)

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15
Q

solve a linear 1st order ODE

A

y* I(x) = integral(Q(x) I(x) dx)

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16
Q

d/dx sinx

A

cosx

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17
Q

d/dx cosx

A

-sinx

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18
Q

d/dx tanx

A

sec^2x

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19
Q

d/dx cotx

A

-csc^2x

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20
Q

d/dx secx

A

secxtanx

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21
Q

d/dx cscx

A

-cscxcotx

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22
Q

d/dx sin-1(x)

A

1/sqrt(1-x^2)

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23
Q

d/dx tan-1(x)

A

1/(1+x^2)

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24
Q

d/dx cost-1(x)

A

-1/sqrt(1-x^2)

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25
Q

d/dx cot-1(x)

A

-1/1+x^2

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26
Q

integral 1/ax+b

A

(1/a)ln(ax+b) +C

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27
Q

integral tanx

A

-lnI(cosx)+c

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28
Q

integral secx

A

ln(secx+tanx)+c

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29
Q

range of arcsin

A

pi/2 to -pi/2

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30
Q

range of arccos

A

0 to pi

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31
Q

cycloid

A

x=r(theta - sin(theta))

y=r(1 - cos(theta))

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32
Q

equation for dy/dx parametrics

A

=(dy/dt) / (dx/dt) if dx/dt is not 0

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33
Q

area under parametric curve

A 
A = integral a-b ( y(t) * x'(t) dt)
wher
where f(t) =x
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34
Q

if dx/dt =0and dy/dt=0 what is dy/dx

A

undefined

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35
Q

if dx/dt =0 and dy/dt != 0 what is dy/dx

A

vertical ( infinity)

36
Q

Length of parametric curve a->b

A

integral a->b [(dx/dt)^2 + (dy/dt)^2] dt

37
Q

converting x and y to polar

A

x= rcos(theta)

y =rsin(theta)

38
Q

r^2 =

A

x^2 + y^2

39
Q

tan(theta)

A

y/x

40
Q

dy/dx of a polar curve

A

dy/dx = (dy/dtheta)/(dx/dtheta) = [(dr/dtheta)(sintheta) + rcos(theta)] / [ (dr/dtheta)cost(theta) - rsin(theta)]

41
Q

Area of polar curve

A

A= integral a->b [ (1/2)r^2] dtheta

42
Q

Arc length of polar curve

A

integral a->b [ sqrt( r^2 + (dr/dtheta)^2 ] dtheta

43
Q

Improper integrals type 1

A

infinite integrals

44
Q

Improper integrals type 2

A

discts integrals

45
Q

integral a->infinity [ f(x)] dx

A

lim t->infinity[ integral a->t ( f(x) ) dx]

46
Q

integral (- infinity -> b) [f(x)] dx

A

lim t-> - infinity ( integral t->b ( f(x) dx) )

47
Q

integral -infinity -> infinity

A

integral -infinity -> a + integral a->infinity

48
Q

p series, where convergent

A

integral 1->infinity 1/x^p
convergent for p> 1./
divergent p<=1

49
Q

integral a->b DISCTS AT b

A

integral a->b = lim t->b- [ integral a->t ]

50
Q

integral a->b DISCTS AT a

A

= limit t-> a+ [integral t->b]

51
Q

linear electricity problem

A

L(dI/dt) + RI = E(t)
L is inductance
R is resistance
E(t) is voltage

52
Q

if c is in [a,b], f(x) discts at c then integral a->b

A

= integral a->c + integral b-> c

53
Q

comparison thm for improper integrals

A

supp f,g are cts with
f(x)>=g(x) >=0 for all x>=a
if integral a->infinity f(x) is convergent then so is a->infinity of g(x)

if integral a->infinity g(x) divergent so is a->infinity f(x)

54
Q

{An} has limit L

A

lim n->infinity an =L or an->L as n->infinity

{An} converges to L

55
Q

if {An} limit DNE or is infinite

A

diverges

56
Q

Squeeze thm for sequences

A

if {an}{bn}{cn} aninfinity an = lim n->infinity cn =L then lim n->infinity bn=L

if lim abs[an] as n->infinity =0 theb lim n-> an=0

57
Q

if {an} converges to L and fis a fcn that is cts at L then

A

lim n->infinity f(an) =f(L)

58
Q

integral cotx

A

ln(sinx)

59
Q

Second derivative for parametrics

A

dy/dx = (d/dx)(dy/dx) = [ d/dt(dy/dx) ] / [dx/dt]

60
Q

concave up

A

second deriv >=0

61
Q

concave down

A

second deriv <= 0

62
Q

formula for a circle (h,k) and radius r

A

(x-h)^2 +(y-k)^2=r^2

63
Q

arctan(infinity)

A

pi/2

64
Q

Absolutely convergent

A

if the series of absolute values of an is convergent

65
Q

Conditionally convergent

A

if its convergent but not absolutely

66
Q

if a series is absolutely convergent

A

then it is convergent

67
Q

ratio test

A

if lim abs[(an+1)/(an)]= L1 or infinite, then divergent

if =1 inconclusive

68
Q

root test

A

if limn->infinity (root(abs(an))) = L 1 or infinity then divergent

if =1 then inconclusive

69
Q

given a power series sigma(cn(x-a)^n) there are only three possibilities:

A

the series converges only when x=a

the series converges for all x

there isa positive number R such that the series converges if abs(x-a) < R and diverges if abs(x-a)>R

70
Q

(1/(1-x)) = 1 +x +x^2 +x^3 + … =

A

sigma(x^n) for abs(x)<1

71
Q

if f hasa power series representation at a then its represented by

A

sigma[ (f(^n)(a)/n!)(x-a)^n) ] for abs(x-a)<R

72
Q

Maclaurian series

A

when taylor series is centered at 0. equation is:

sigma [ f(^n)(0)/n! * x^n ]

73
Q

power series of sinx

A

sigma [ (-1)^n (x^2n+1)/(2n+1)! for all x

74
Q

power series of cosx

A

sigma [ (-1)^n (x^2n)/(2n!) ]

75
Q

geometric series

A

sigma [ar^n-1] = a/(1-r)

76
Q

if a series is convergent then

A

lim n->infinity = 0

77
Q

if limit n-> infinity an dne or is non zero

A

then series is divergent

78
Q

integral test

A

if integral 1->infinity f(x) is convergent the series is convergent.

if integral divergent, series is divergent

79
Q

limit comparison test

A

limn->infinity (an/bn) =c if c>0 then either both converge or both diverge

80
Q

alternating series test

A

if decreasing and limn->infinity =0 then series converges`

81
Q

lim n->infinity (1+(1/n))^n

A

e

82
Q

taylor series 1/(1-x)

A

sigma x^n

83
Q

taylor series e^x

A

sigma (x^n)/(n!)

84
Q

taylor series sinx

A

sigma (-1)^n * (x^2n+1)/(2n+1)!

85
Q

taylor series cosx

A

sigma (-1)^n * (x^2n)/(2n)!

86
Q

estimating series using integral

A

Rn needs to be less than some decimal number. so Rn by thm >= integral n->infinity an. the answer to that < the error. solve for n

Calculus 2 (1 decks)

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