Final Exam: Strategy for Limits

& strategy for limits if a substitute directly & if = = ↳ & symptotelimit found indeterminate trig identities . L'hopitals . 3 4 : form : y -y point slope = , m(x - x ) , y hole : factor from top and bottom HA : if degree on top is less than bottom · bottom the arcsinx arrange a if continuous domain : limf(x) exists . X99 C imf(x)=fl . linearization : quotient ((x) f(a) + F(a)(x= from rule (f(x)(g(x)) ) rule product proof : tanx (fg)) fg fg · f(x)(g(x)) ) f(x)(g(x)))) f(x)g(x) (g(x)) 2(f(x)g(x) f(x)g(x) co+X + - = - - ) (x) : X : - - (g(x))2 ) - - if f has sec CeXCO - f(x) AsinBx + D g(x) point values 1 . are F'(() 0 or - f() DNE - - Critical point 5 Compare all the values and . choose the largest and smalles a - rectangular cylinderVerse a A DE is separable if it = 3 if . given by solving h(y) = 0 and xy3 separation 1. find , 1 , , = 0 O solutions : 6 separate variables dy/ax = g(x)h(y) -> integrate bothc ides and soIve . fory . 3 UnTyay (g(x)dx exponential decay: = =k(y) k(y)Yk doubling y(t=zy h y = : : net rate : rate in-rate P(t) Poekt = v= rate Po-starting out where G = D+1 SeCX n = (if(x) cosx CSCX : - = - 03 = = + = = /x=- an = = " Jaafa if fiseren , x (DNE) = 2in(ab) : eb , (g(dx value average - ina = e sphere cone : : SA=4rer 108 not I = 0 4 differentiable : corners . 2 3 . . . cusps VA Jumps . discontinuities 6 are = sa horizontal unt are 111- 1 "11 on X -> all slopes in column are same -> all relies slopes in row are same ony relies on both - no pattern (Ha+) polynomials roots inverse 2 . vertical lines 5 - = continuous functions When defined : Verrah Ver 0 = = A j irgisodd g(PNE) FTCI : = (DNE) 1 Slope fields : equilibria where slopes stable . 01 A A eln(ap) 00 0 00 + = A = = concave down: overestimate g(x))dx - - = concave Fra /Exa Ma sinx - = linear approxi up : underestimate between Fandy= area ↓ : (f q)(x) X integral Sea logbM (DNE verran 0 relies - min(x) = cylinder 0 = k = = determinate forms : fory exponential growth : amn (ab)m amym x . of variables any equilibrium solven(y) ap if = . F(x) + for a X1b then LHR Stuff : = . 0 -> antiderivative = indefinite f(x)1g(x) if S am -n (am)n = exty y = if 0 + r = IN find the global extremum aman = am am Unitys dy (g(x)dx separable : being exponent be written in form such limit rules : C - . State the conclusion with units can = if 4 . are - a more <(b a arctan( ((f q)")'(x) Which arevarying algea is rules : a (a) quantity are you optimizing ? , = ]x]b =natixtality general antiderivative t varying possible - hyp constant ? Which how are the quantities related? g(x) n(y) solutions O variable , and determine its domain . . logp (Mk) quantities · SA = 2LW + 2LH+2WH V= L W H between - b geometry to write the quantity optimi zed as a function of a single box Ms(n) O varying? What = argrace assign variable namesto anything thats .calculatethe correspondinag n 4 and f'(l) F(b) F(a) that on a particular antiderivatives solves an initial condition and finds the particular value of E= arcsin(P) what variables are - . determine the critical points 3 : 0 - max min=-1 identify the involved computef'(x) off (sin(1 x) + z optimization I f(a) and f (b) = = F'(x) , anx-value c with D : average value 1. calculate the end . f(x) T Bim-period global extremum 2 AcosBx + D A : amplitude it is continuous where = Sucta+ isSe [A BJ then a root , afunctionfasamaxandseed , the interval by velocity and t = b t=a axis is positive below axis is negative (f(x)dx=(x)dx=F(b) -Fa= mean value theorem C if f is differentiable on the general exists interval [a , b] then there xd = [a b) on a and +-axis between FTCI : differentiable is up bounded f A and B 1 + extreme value theorem interval and f I' must have arctanxx - roots at b When concave . negative rolle's theorem cax . underest definiteintegralhetareaofthegraph andthe as a x logs of overest -> area y = decreasing When . - area above =InA- elnx When . Overest : curve InABIT secx 1 ) In , , When . increasing decreasing underest When increasing When concave down overest displacement=net area In (AB) = InA-InB arcsin f(x)g(x) f(x)g(x) =f(x)g(x) y'(x)f(x) (g(x))E so : 2g(x)sexitanx - = = properties denominator underest . = a)OY - = range : (M from 24 uals the deriv the left LHS = ( x) common divideiprocal MS : ((nx) in ((nx) y arctanX : a b , =n A RHS : (e) = a (x) enx-(nx)= na domain : zero 1 = 01708 - A =cos(arctanx) proof In domain : [1 , 1] i 100 . = 72 range Go : differentiability check th When flas is defined derivedseif a : tanx domain : , HA= O , if degrees are the same then use coefficients VA : makes = 0 1 Jarctanx) Secaxarctanx F(x- Sapproxusing atabriv L . Sech(arctanx)Carctanx) domain : conjugates 2 . indet Forms proof a(tan(arctanx)) = -=COST & factor 1. arctan : trig In1 0 = In o S = f(x) = rational functions trig onometric exponentsal logarithmic finding domains : or Ndx=arctan who radicals variables in the denominator : domain is all real numbers dx=arctanx , polynomial a undefined function with denominator: set & fraction in the a the bottom equal to Zero a function with a variable inside the radical : Set the terms inside) lim Inx = - D X+0 lim X- 0 a function using natural Set the terms in the cotX = 0 70 and solve log : paran theses to