Final Exam: Strategy for Limits

Strategy for limits: directly substitute asymptote limit found indeterminate factor 2. combagales 3. trig identities 4. 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On any closed, bourded Interval [a,b] when it is continuous D: average value Optimization period global extremuan 1. calculate the end point values F(a) and f(b) 2. compute f'(x) 3. determine the critical points OFF: F'(L) Gor FOHE 4. calculate the corresponding Critical values F(c) for ea each Critical paint 5. compare all the values and Choose the largest and smallest LWH Venth 1. idcntify the involved quamities skal varieties are content! which assign variable namesto angthing thals verging how are the quantities nisid whatquantity are you optimizing? which out varjina a. use the constraint, algeam, and geometry to write the quantity being optimised as a function of a single variabic, and detemine in dansa 3-ir possible, Finst the global extremum 4. state the conclusion with waits expment vales DEis spreable if it can be written in fome 000 given by solving kly): and if (am) (ob seperation of variabtes determinate foran: ८ 001001 00 110000 200gs (ONE) exponential growth exponential decay: when Stope fields: -18 mean value theorem if f is differentiabit on the interval [a, b], thee there exish anx-venec with accesuchen governantitat indefinite integral that P'(C)=F(b)-f(a) b-a 5000 log_b(ll * l ^ a) = k * log_b(M) 656 (r ^ - 1)(x) + p' * (q) limit ruies Cot 1000 10000 (DE) 200 sparre V = 1/3 * Rr * r ^ 2 * n 3. VA unstaple equilibris comert stopes art horizontal Stare retics an voll slepes in coluen av Sme relles any all slopes in rous are same relies on Dem no pattern if Fivame average value i/(b - alpha) * integrate f(x) dx from a to b linear approx. Concave up underestimate concave doun evarestianate Continuous functions when defined polynοιητίας f(x) rational Functions trigonometric Exponential Inverse trig rogarithmic finding domains In 1=0 In 0=undefined int^ x 1/(1 + x ^ 3) dx=arctan x+C 5. venical Wes lim Inx=-00 lim cot x= 00 290 a polynomial w/o radicals or variables in the denominator: domain is all real nawkers a function with a fraction in the denominator: set the bottoon equa 20 a function with a variable imid the radical set the forms inud 10 a Function using nahuni log ser the terms in the και το 20 and solve limit definition of derivative f(4) = lim f(4th)-f(4) hto h lim 274th-4 h slim 274th-4 274th +4 h lim 4/4th)-16 heon(avuth+u) lim 16-16 lim th #Yuth 14 consider f(x)=x+ax²+ D what conditions will sand bhave f'(x)=2x+dax=2x(x²τα) Critical point bath P(x) and f'(x) ose defined for allree) numbers, so critical points occur at pornts where FOO aso then 0=x²ra has no real solutions sof has one critical point: x=0 if ao then(x)=ax, so has onecritical Daini ifaco then solutions, sof h Fhos three critical points at x=0,- ond has two distinct real assume aro, we classify using first derivative TRIG REFERENCE radians o 1/6 1/3 42 Sin COS 0 tan 0 test intemal test point F(c) sign of f behavior asing, the initial condisin to save про п(274 lim 14) (100,0) 1- decreasing A (0,00) + increasing thus if a 20, regardless the valm of b, f has a single critical point at 0, at which fhasa local min. 4 १० 24th +4 what fanctions are continuous on their if a differentiable function f(x) nas 5 distinct rootsye then f(x) must have of least a distinct roots domains? C P(x)= Sa if x<1 3 if x>1 true statements F(a), if F(x)=sin(at)dt F(x)=x*sin(a)(x) axsin (G) F'(৯) ৯(১) son(a(a*))=16sing x300 004X ((1+))= lim In(1+x)= 00 x400 the n'h derivative of an exponential function is an cxponential Function, for any positive whole asumbern, lim the ratio between the rate of change of an exponent is! function at s point and the valve of the function at that point is constant exponential function are differentiable everywhere a Function that's concave up and decreasing: e nonlincar function w/ instant rate same al average rate (0,5) x(x-5) FF F(x)in In(f(x)): arcsinx-Inax 1 aresinansini in(ar) ((x)=2x Vi-x Find the value'e guarantent to exut by the MYT 4 a fanction that dominain ( (4-8)= **(1+x)=e=ean antidestination of fix) cxe satisfpeg (1): J'Le'de 2 f'(x): 2x³+3ay 6x²+20 f(x) if aso lao still has no inflection points , then for all xáo, so even though f(0)=0,F(x) then 1"(x) 6x raaxo, sof(x) has no inflection paints (ax+5) dx dx related rates: Verh (35) a2o, then and for F(x)>0 for and x or equivalentiy x Mr tous ifaco, regardless the vafar of b, Hel hoo exactly tuo inflection ponis, at x at 1 note that f(x) is differentiabic on [0,4] where F(x(2x+1)-2.2 Vaxri the MVT states that there exists sone XFC with 0xCC4 such that f())-f(0) computing 4)-40)√2-√2011 V-VI 4-0 4-0 4 h thes we solve the following equation forc, where we requise OLCCH 4=acts (since Occarimрlію 21170) C2 population growth 9:50 wmen P:100 dt 50-100 (600-100) 50 60000K-10000% 50-50000円 1000 a function two different locations dp 50, d- 1000 P(600-P),(0)-100 1 Show that the areas of the rectangle is the length of the rectangleis 2x and the ay since height is ay. Shown y Viw-x ty 16, at the point area=length height =dx-2y=ax(27/16-) graph of V graph of inx Sinx+ cosxl lim sinx=1 temp. ina muse dotm is 76°F tutude temps 15. the temp in the dormat lam is 65 I-K(T-35) T(0)=16 (1) 65 1-35 1-35 grapinofe since temp in decrewing In/T-35-R+C76-(0) don't forget C or DNE or abs vaive reread your answers o eulers method lim A76-35 SoT (+35 lim +400 if you are giveno TANGENT LINE, you can Only Fied (x) and f(x) rate of change of area of square: 2(f(x)f(x)) perimeter 4f(x) on an Interval at can achieve its global maximum if f has a local extremum at xxp, then f has a critical point at