critical number where f '(x) = 0 or f '(x) is undef (where denom. of f '(x) = 0)


 Clarence Horton
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1 Decoding AB Clculus Voculry solute mx/min x f(x) (sometimes do sign digrm line lso) Edpts C.N. ccelertion rte of chnge in velocity or x''(t) = v'(t) = (t) AROC Slope of secnt line, f () f () verge vlue 1 f(x)dx verge velocity s () s () 1 or v(t)dt concve down f ''(x) < 0 mens find ppoi s then test 2 nd deriv # line concve up f ''(x) > 0 mens find ppoi s then test 2 nd deriv # line continuity 3 steps: 1) f () =? 2) f(x) = f(x) x x + 3) x f(x) = f() criticl numer where f '(x) = 0 or f '(x) is undef (where denom. of f '(x) = 0) cross section Are of cross section
2 decresing f '(x) < 0, ut in Relted Rtes mens d(something)/dt definite integrl n n k=1 (f(c kx ) x) = f(x)dx defn of derivtive f(x+h) f(x) = f (x) h 0 h derivtive Slope of tngent line t specific point point f(+h) f() f () = or f () = h 0 h f(x) f() x x, IROC differentile continuous differentile 1) f () =? 1) continuous 2) f(x) = f(x) x x + 2) f (x) = f (x) x x + 3) x f(x) = f() eqution tng. line y f() = f '() (x ) f '() IROC, slope of tngent line t x =, f(+h) f() h 0 h, f(x) f() x x f(2) the yvlue of the function f t x = 2
3 FTC f(x)dx = F() F() given point Use for 1 of 4 possiilities: 1) FTC 2) Eq of tngent/norml line 3) Sketch grph 4) Diff Eq horiz. symptote (HAY) Horiz. Asymptote must e in the form of y = # HA = x ± f(x) horizontil tngent f '(x)= 0 incresing f '(x) > 0, ut in Relted Rtes mens d(something)/dt integrte ccumulte, re, chnge, how mny IVT sys IF 1) f(x) continuous on [, ] AND 2) k is ny yvlue such tht f() < k < f() THEN there is t lest one # c in [, ] such tht f(c) = k interpret must include the 1) rel sitution 2) independent vrile mening nd the units 3) dependent vrile mening nd the unit
4 IROC Slope of tngent specific point sin (x) x 0 x sin (x) grph is the mountin, = 1 x 0 x f(x) x (Algericlly) To find it pproching one xvlue: 1) try direct sustitution 2) try fctor, cncel 3) tke the it of the expression tht is left (Grphiclly) x f(x) = f(x) must pproch the sme yvlue + x IF it exists f(x) x ± Write f(x)s frction: 1) If the highest power of x ppers in the denomintor (ottom hevy), f(x) = 0 x ± 2) If the highest power of x ppers in the numertor (top hevy), x ± f(x) = ±. Plug in very lrge or smll numers nd determine the sign of the nswer 3) If the highest power of x ppers oth the numertor nd denomintor (powers equl), f(x) = coefficient of numertor s highest power x ± coefficent of denomintor s highest power (Algericlly) (Grphiclly) log < poly < exponentil Wht is the yvlue the function is pproching out to ±?
5 MVT the xvlue on function where the slope of the secnt line f(x) must e: 1) continuous [, ] 2) differentile on (, ), THEN 3) there exists c such tht f '(c) = f () f () mx/min vlue the yvlue of the function where the mx/min occurs moving to right (left) v(t) > 0 (v(t) < 0) optimiztion solute mx/min, frthest to the left/right, lest mount prllel sme slope (derive t point) prticle t rest v(t) = 0 prticle moves right v(t) > 0 mens find x'(t) = 0 nd test # line prticle moves left v(t) < 0 mens find x'(t) = 0 nd test # line point of inflection point where the concvity chnges [f ''(x) = 0 or where undefined (s long s point on f ), test # line 2 nd deriv, signs must chnge] rte derivtive reltive mx f '(x) = 0 or undefined, test # line, f '(x) chnges from + to
6 reltive min f '(x) = 0 or undefined, test # line, f '(x) chnges from  to + Rolle s Theorem If 1) f is continuous [, ] 2) f is differentile over the intervl (,) 3) f() = f() then there exists t lest 1 # c in (, ) where f '(x) = 0. slope genertor f(x), derivtive function tht will find the slope t ny point speed velocity tngent tke the derivtive totl distnce v(t) trpezoid rule A = h 2 ( ), where the ses re prllel sides which re found y the heights of f(x) t tht xvlue nd the height of the trpezoid is Δx, which is the distnce etween the ses vries directly (or directly proportionl) y = kx vries inversely (or inversely proportionl) y = k x velocity rte of chnge in position, x'(t) = v(t) speed with direction
7 verticl symptote VAX, eqution for VA must e of the form x = 1) fctor 2) cross out common fctors 3) set denomintor = 0, eqution for VA is x = tht # verticl tngent is the xvlue where the denomintor of f '(x) = 0, IF it exists volume (disk) π r 2 volume (wsher) π R 2 r 2
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