Math 20B term 2

explain proper vs improper integrals: Improper integrals: when can u use P power rule to find convergence or not? if not in above notation, find integral of infinite or discontinuous at certain pt: What if f is discontinuous at infinity, discontinuity, or other pts (not exact end pt bounds)? Convergent vs divergent.: If u use u-sub to solve integral, do u change bounds of the improper bounds? Explain how Distillation helps with figuring out if hard-to-solve-integrals are convergent or not? Which things can get crossed out to better simplify the integral?

Proper: [a,b] has finite length, and f is continuous over [a,b], otherwise its improper. power rule can be used when integral of 1/x^P. Set infinite bound to t. Let lim t approaches infinite of integral. Solve integral, then solve limit as t approaches. same for discontinuity (put coming from left or right). for other pts: Split up integral into 2 integrals. Convergent: improper integral is finite, otherwise its divergent. Yes, bounds change. Distillation helps compare original integral with simplified integral. Remove constants and x's that don't matter can get crossed out, only keep powerful ones.

if z= a + bi what is the complex conjugate? What is the magnitude of z? What does i^2= Dividing complex numbers when i in denominator. e^(bi) = (in trig polar form) e^(-bi) = (in trig polar form) how to solve integral by using complex exponentials, and ans has to be in trig: sin(2theta) double angle identity =

z= a - bi z= square root (a^2 + b^2) -1 multiply by conjugate. cosb + isinb cosb -isinb use formulas for cos(ax) and sin(ax) that turn into exponentials. simplify and put back in trig to take integral of easier function. = 2sin(theta)cos(theta) .

Lim n->infinity (fraction)^n = . S3 vs a3: When given an infinite Sum in cr^n notation, how do you find Sum? Trick to see if its convergent: What if its not in that notation, but they give u an? how do u find Sum when given Sn Greek Symbol Sum from n to a certain number on top?

0. S3: sum of a1+a2+a3 . a3: individual term. Ex: S3-S2=a3. Find c and r. C is initial term. R is 2nd term/1st term. Then use S = c /(1-r) . Convergent when | r |<1 . If | r |> or equal to 1, its divergent. If not in that notation: solve few of the sum in order, split up using theorems, cross out opposite terms (a1+a2+a3..). Plug top # into given Sn.

Series/sum Vs sequence: Test For Divergence: when given the Sum of an, If limit n->infinity an does not=0, then the Sum from 1 to infinity of an is_____. If the limit of an is a finite #, the series is _____. if the limit of an = 0, does this conclude the sum converges? what does Sn in a series mean? Given: Sum of 1 to infinity ai; how do u find if this is divergent or not?

Series/sum: what u get from adding up all terms of the sequence. Sequence: an order list of #s. Divergent. Convergent at that finite #. No, its inconclusive. Sn is the pattern the sequence follows in 'n' terms. Find what Sn = , then take lim n->infinity of Sn. Taking them limit gives ans if convergent or not.

List the steps for integrating using trig substitution: And when do u know when to use trig sub vs not? integrals using sine sub. when to use it and let x=? When do u use secant sub? let x= When do you use tan sub? let x= if there are bounds set, do you change them?

Steps: 1) find a, plug in appropriate x, solve for dx. 2) plug in x and dx into integral. 3) use trig identities and simplify. 4) integrate and put "theta" in terms of "x". Use: # on numerator. (no x). Not use: x on numerator Sine sub: When: if you see sqr(a^2 - x^2). let x= asine theta Sec: when you see: sqr(x^2 - a^2). let x= asec theta Tan: when you see: sqr(x^2+a^2). let x= atan theta yes, change bounds in terms of theta.

Integrating rational functions: what should u first look for: List the steps for integrating partial fractions if above doesn't work: and how do you know when to do partial fractions first? how do you know if you have to polynomial long divide first? what to do if bottom is a quadratic?

denominator can be u-sub. inverse trig derivatives. partial fractions: 1. if it's improper, meaning degree of numerator is bigger, use polynomial long division. 2. factor out denominator into linear and inreducable. 3. write partial fraction expansions with A and B as constants, and equal it to original equation. 4. find A and B. 5. plug in and take integral. polynomial long division if improper rational f (power of numerator is greater). list the quadratic and 2 partial fractions. the top should be a linear function.

If u have a complex integral of e^x.. how can you simplify this to find if convergent? Determining closed form of sequence: Determining recursive of sequence: Limit law for lim n->infinity (an)^p = Limit law for lim n->infinity f(an) = 3 tools for finding limit: when you ln(f), and u are finding the limit, what do u do with your final ans?

e^x can be 1, and use comparison test. U might need to change the coefficient for it to be true. Closed: pattern term and order. Recursive: initial term & relationship between 2 consecutive items. = (lim n->infinity an)^p . = f(lim n->infinity an) when f is continuous. 1) l'hopitals when infinity/infinity or 0/0. 2) squeeze theorem. 3) multiply by conjugate when radicals. final answer: e^(ans)

trig integrals of to the powers: what method should be used if powers are odd? What do u do if the powers are even? What if ur multiplying tan and sec?

odd: breaking up trig with whichever power is odd. use trig identities to simplify. use u-sub to cross out a single trig identity. even: use reduction formulas of cos^2 or sin^2. ex: cos^= (1+cos2x)/2 Use trig identify: tan^2 + 1 = sec^2.