Important Facts You Need To Know/Review:

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Dylan Clarke
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1 Importt Fcts You Need To Kow/Review: Clculus: If fuctio is cotiuous o itervl I, the its grph is coected o I If f is cotiuous, d lim g Emple: lim eists, the lim lim f g f g d lim cos cos lim 3 si lim, t lim, cos lim 4 If f is cotiuous o,, d zero i the itervl, f d Itermedite Vlue Theorem f hve opposite sigs, the f hs t lest oe Emple: For f, f d itervl f, so f hs t lest oe zero i the, 5 f lim, or f h f h f h f y f lim y y Differece Quotiets 6 All the rules of differetitio: product rule, quotiet rule, chi rule, 7 The lieriztio of the fuctio f t c is L f c f c c Approimtio 8 If f is cotiuous o, t lest oe zero i,, d differetile o Rolle s Theorem Tget Lie,, d if f f, the f hs Emple: For f, f f, so f hs t lest oe zero i,

2 9 If f is cotiuous o, f, with fc, d differetile o,, the there is t lest oe vlue c i f Me Vlue Theorem Emple: For f, there is t lest oe vlue c i f f fc, with If f is positive o itervl I, the f icreses o I If f is egtive o itervl I, the f decreses o I If f is zero o itervl I, the f is costt o I 3 If f is cotiuous t c d differetile o ope itervl cotiig c, ut possily ot t c, the if f chges from egtive to positive cross c from left to right, the f hs locl miimum t c; if f chges from positive to egtive cross c from left to right, the f hs locl mimum t c; if f hs the sme sig o ope itervl cotiig c, ut possily ot t c, the f hs either mimum or miimum t c First Derivtive Test 4 If fc, the fc implies tht f hs locl miimum t c, while fc implies tht f hs locl mimum t c Secod Derivtive Test 5 Curve sketchig usig the first d secod derivtives 6 Relted rtes prolems 7 If f is defied o,, the lim i f d f c Riem Sums i i

3 8 If f is itegrle o,, the ouded y the grph of f d the -is f d is the sum of the siged res of the regios 9 If f is itegrle o,, the f d f d If f eists, the f d If f is itegrle o,, d c is y umer i, c f d f d f d c, the If f, g, d h re itegrle o,, d g f h o, g d f d h d, d if g f h o, g d f d h d d cos d d 4 4 Emple: Sice cos, 9 lower oud, the,, which mes tht cos d upper oud the

4 3 If f is cotiuous o,, the F f tdt is differetile o, F f A Fudmetl Theorem of Clculus Emple: For F sit dt, F si with 4 If f is cotiuous o, f d F F d F is y tiderivtive of f o,, the A Fudmetl Theorem of Clculus cos si si Emple: d 5 All the sic tiderivtive rules 6 If f is cotiuous o,, the there is t lest oe vlue c i, with f c f d Me Vlue Theorem for Itegrls Emple: Sice cos is cotiuous o, with cos, d cos d, there must e vlue c i c 7 If u is cotiuous o, u d f is cotiuous o the rge of u, the f uu d f udu Sustitutio for Itegrls u

5 8 If f is itegrle o the symmetric itervl, f d f d f d, while if f is odd f d Odd d Eve Fuctios Preclculus: eve fuctio eve fuctio odd fuctio Specil Products/Fctoriztios: eve power odd power eve power odd power odd fuctio odd fuctio odd fuctio eve fuctio eve fuctio eve fuctio odd fuctio eve fuctio odd fuctio odd fuctio eve fuctio eve fuctio eve fuctio eve fuctio Differece of Squres: Squre of Sum: Squre of Differece: Sum of Cues: Differece of Cues: Biomil Powers: The coefficiets i the epsio of c e determied from Pscl s Trigle: 3 3, the if f is eve The sic methods of fctorig: GCF, triomils,

6 3 Complete the Squre: 4 Qudrtic Formul: The solutios of c c c c c, with re give y 4c 5 Asolute Vlue Properties: i) ; ; ii) iii) iv) v) vi), vii) 6 Epoet Properties: i) m m m m ii), m iii) m iv) v), vi), vii), 7 Rdicl d Rtiol Epoet Properties: i), d re rel umers ii), d re rel umers d iii) ; is odd ; is eve m m m iv), is rel umer

7 8 Distce, Midpoit, d Slope Formuls for the two poits, y d, 9 Specil Equtios for Lies: Polyomil Properties: Distce Formul: y y Midpoit Formul:, y y y y Slope Formul:, Poit-Slope Form or Formul: y y m Slope-Itercept Form or Formul: y m y Itercept-Itercept Form or Formul: y y y y Two-Poit Form or Formul: y : If pc if d oly if c is fctor of p p is polyomil of degree, the it hs zeros, coutig repeted zeros Logrithm Properties: log MN log M log N ii) log M log M log N i) N iii) log p M plog M iv) log M log M v) log log vi) log Coic Sectios: i) Circle of rdius r cetered t hk, : i) Ellipse cetered t, ii) Hyperol cetered t hk : h y k h y k r h y k hk, : or y k h

8 3 Trig Idetities: i) ii) si si cos iii) si cos cos cos si iv) cos cos v) si cos vi) sec t vii) csc cot viii) cos cos cos si si i) si si cos cos si ) si si i) cos cos 4 The Priciple of Mthemticl Iductio: Emple: Prove tht for positive iteger, 3 4 The Bse Step: Show tht the equtio is true for the strtig vlue: For, the left side of the equtio is, d the right side of the equtio is So the equtio is true for The Iductio Step: Assume tht the equtio is true for k, d show tht it is true for k : kk Assume tht 3 4 k Addig k to oth sides leds to kk 3 4 k k k k k k k k k k Therefore, the equtio is lso true for k So 3 4, for ll positive itegers, y Mthemticl Iductio

Limit of a function:

Limit of a function: - Limit of fuctio: We sy tht f ( ) eists d is equl with (rel) umer L if f( ) gets s close s we wt to L if is close eough to (This defiitio c e geerlized for L y syig tht f( ) ecomes s lrge (or s lrge egtive