CALC MIDTERM 2

theorems and equations and general stuff to remember

Xi =

a+ (i dx)

i =

n(n+1)/2

i²

n(2n²+3n+1)/6

definition of a definite integral

integral [a,b] f(x)dx= lim n→ infinity summation [i=1,n) f(xi)dx

comparison theorem

p>1 converges, p<=1 diverge

a. if FOI is greater than simple and simple diverges, then interest diverges

b. if FOI is less than simple and simple converges than FOI converges

cos theta= (change in coordinates formula)

x/r

x= (charge in coordinates formula)

cos theta r

y = (charge in coordinates formula)

sin theta r

disk area

pi*r²

washer area

pi r²big - pi r²small

cylinder area

2pi r* h

L’Hopitals rule

if some f/g has 0/0 or infinity/infinity from then lim f/g can be replaced by lim f’/g’ (if f’/g’ exists)

Improper integrals

converge if all limits exist. diverge if any limit DNE

theorem: Limits are Nice

lim n→infinity [Kan*bn+cn^1/2 ] = (k lim _n→infinity an) _*(k lim _n→infinity bn) +(lim_n→infinity cn)^1/2

Theorem a sequence converges to zero if and only if abs value an →0

an→ 0 ←→ l an l → 0

continuous →

integrable

FTC 2

if f is continuous and F is an antiderivative of f F’ = f, then [a,b] integral f(x)dx= F(b)-F(a)

net change theorem

integral of a rate of change is the net change

Why is FTC 1 limited

only works when upper limit of integration is a single, independent variable and lower limit is a constant

Evil theorem

lim x→ n 1/n-n^±, sign of terms will determine if answer is + or - 0.

sequence {an} has limit L and we write

lim n→infinity = L or an→L as n→ infinity

squeeze theorem

an</= bn<= cn for n>= n0 and lim an = lim can = L, then lim bn= L. as n → infinity

MTC Monotone convergence theorem

a sequence that is monotone (inc/dec) and bonded converges. lim n→infinity an converges to L (least upper bound)

Theorem for geometric series

a series with first term a and ration r converges to a/1-r when l r l < 1 and diverges otherwise

divergence test

if lim k→ infinity ak doesnt equal zero then the series diverges, all else divergence test fails

what does an equal in terms of partial sums

(Sn) - (Sn-1)

ratio of integers

is expressed as a fraction where both the numerator and denominator are whole numbers, useful for finding a and r values

harmonic series

1/n diverges because S2^n 1 + n/2

Integral test

f is continuous, positive and decreasing. the series Summation an is convergent if and only if the improper integral an is convergent.

p-series test

1/n^p is convergent if p>1 and divergent if p <= 1

comparison test

i) if summation bn is convergent and an</=bn for all n then summation an is also convergent

ii) if summation bn is divergent and an>/= bn for all n then summation an is also divergent