The type of limit that is used to find TANGENTS and VELOCITIES gives rise to the central idea in DIFFERENTIAL CALCULUS, the DERIVATIVE.

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1 NOTES : LIMITS AND DERIVATIVES Name: Date: Period: Iitial: LESSON.1 THE TANGENT AND VELOCITY PROBLEMS Pre-Calculus Mathematics Limit Process Calculus The type of it that is used to fid TANGENTS ad VELOCITIES gives rise to the cetral idea i DIFFERENTIAL CALCULUS, the DERIVATIVE. Taget : Lati word tages meas touchig. Taget lie should have the same directio as the curve at the poit of cotact. (Defiitio of taget i Euclidea Geometry is iadequate.) Eample: The Taget Lie Problem P (c, f(c)) Q c, f ( c ) Slope of PQ = = f ( c ) f ( c) c c f ( c ) f ( c) Slope of a Taget Lie at poit c, f ( c ) : f ( c ) f ( c) 0 Slope of the taget lie is the it of the slopes of the secat lies. Symbolically: m QP PQ m Practice Problem 1: Fid the equatio of the taget lie usig the it process to the graph of the fuctio at a give poit c, f ( c ).

2 f ( ) Limit process 0 f ( c ) f ( c) a. 1, b., c., f ( ) Rates of Chage i.e.: populatio growth, rates, productio rates, water flow rates, velocity, ad acceleratio. Positio Fuctio Average Velocity Istataeous Velocity The fuctio(s) that gives the positio (relative to the origi) of a object as a fuctio of time t. chage i dis ta ce s chage i time t s( t t) s( t) v( t) s'( t) t 0 t Eample: The positio of a free-fallig object is represeted by 1 where: s() t gt v0t s0 g = the acceleratio due to gravity ft / s 9.8 m/ s v 0 = iitial velocity s = iitial height 0 Note: Velocity is speed with directios Note: Istataeous velocity fuctio is the derivative of the positio fuctio Practice Problems:. Fid the average velocity give:. Give: s t t t ( ) 16 16

3 s t t t ( ) a. 1, a. Whe does the object hit the groud? b. 1,1.5 b. What is the object s velocity at impact? c. 1, Page 87 # t (mi) Heartbeats Estimate the heart rate betwee: a) 6,4 b)8,4 c) 40,4 d) 4,44 e) What are your coclusios? 5. Fid a equatio of the taget lie to 4 f ( ) at 1

4 LESSON. THE LIMIT OF A FUNCTION Iformal Descriptio of a Limit The Eistece of a Limit Ifiite Limits Types of Limits Methods of fidig its If f( ) becomes arbitrarily close to a sigle umber L as approaches c from either side, the it of f( ), as approaches c, is L. f ( ) L c Note: The eistece or o-eistece of f( ) at = c has o bearig o the eistece of the it of f( ) as approaches c. Let f be a fuctio ad let c ad L be real umbers. The it of f( ) as approaches c is L iff f ( ) L ad f ( ) L c c f( ) or f( ) c We say: c the it of f( ) as approaches c is ifiity (or egative ifiity) or f( ) becomes ifiite as approaches c or f( ) icreases w/o boud as approaches c A real umber L or Does ot eist (DNE) Direct substitutio Whe Direct Substitutio fails: Hole (or jump): 0 (idertermiate form) 0 - factor - cleaup fractios - multiply by the cojugate - swap trig idetities - simplify logs - last resort: plug i # with tables ad graphs Vertical Asymptote: # 0 Eample: Give f ( ), fid the it as approaches 5. Eample: Techically the it does ot eist because is ot a real umber. However, the aswer is either or. It s a particular way of epressig the it. Eample: Eamples: a) 5 5 5

5 - check both sides: possible or or DNE b) 5 5 Practice Problems: Fid the its h0 ( h) ( h) 5 ( 5) h 9. f ( ), where f ( ) 4 6, 4, 10. 1, 1 f ( ), where f ( ) 1 1, 1

6 11. csc 1. ta 1. l l si(4 ) 4 0

7 LESSON. CALCULATING LIMITS USING THE LIMIT LAWS Limit Laws Let b ad c be real umbers, let be a positive iteger, ad let f ad g be fuctios with the followig its. f ( ) L ad g( ) K c 1. Scalar Multiple bf ( ) c bl. Sum or Differece ( ) ( ) c c f g L K. Product f ( ) g( ) c LK Eample: Eample: Eample: 4. Quotiet 5. Power f ( ) L, provided K 0 c g( ) K ( ) f L c b b c c 6. Horizotal Lie 7. Lie: y 8. Polyomial: 9. y c y c c c 10. Root Law Practice Problems: Fid the its. Eample: Eample: Eample: Eample: Eample: c where is a positive iteger (if is eve, we assume c 0.) f ( ) f ( ) c c where is a positive iteger (if is eve, we assume f( ) 0.) c Eample: Eample:

8 . si cos The Squeeze Theorem If h( ) f ( ) g( ) for all i a ope iterval cotaiig c, ecept possibly at c itself, ad if h( ) L g( ) the f( ) c c c eists ad is equal to L. Eample: c a b a f ( ) b a Fid: f( ) c Practice Problems: Fid the its. 5. Suppose Fid 4 f( ) 0 ad 4 f( ) f( ) 5, si si 9. Suppose f( ). 1 f 6 ( ) 6, fid cos 9

9 LESSON.5 CONTINUITY Defiitio of Cotiuity Cotiuity at a Poit: A fuctio f is cotiuous at c if the followig three coditios are met. 1. f() c is defied.. f( ) c eists. f ( ) f ( c) c Note: Fuctio f is cotiuous at c meas that there is o iterruptio i the graph of f at c (o holes, jumps, or gaps). Cotiuity o a A fuctio is cotiuous o a ope iterval ab, if it is cotiuous at each poit i ope iterval the iterval. A fuctio that is cotiuous o the etire real lie, is everywhere cotiuous. Discussio Most fuctios where we do t divide by zero are cotiuous o their domais Check whe dividig by zero Polyomials are cotiuous everywhere Ratioal fuctios are cotiuous o their domais Iverse fuctio of ay cotiuous oe-to-oe fuctio is also cotiuous Cotiuity of composite fuctio: f ( g( )) f g( ) a a Discotiuity: if a fuctio f is defied o a iterval (ecept possibly at c) ad f is ot cotiuous at c Removable Discotiuity ad No-Removable Discotiuity Eample 1: f( ) 56 4 Eample : Describe ay discotiuities 5 7 f ( ) Practice Problems: Describe ay discotiuities

10 1. 1 f( ) f ( ) For what value(s) of the costat c is the fuctio,? f( ) cotiuous o c 1 f( ) c 1 4. For what value(s) of the costat c is the fuctio,? f( ) cotiuous o c f( ) c Itermediate Value Theorem ab, ad k is ay umber betwee f( a ) ad f() b,, that f () c k. If f is cotiuous o the closed iterval the there is at least oe umber c i ab such Note: I other words, if takes o all values betwee a ad b, f( ) must take o all values betwee f( a ) ad f() b. Practice Problems: Use the Itermediate Value Theorem to show that f( ) has at least oe zero o ab,. 5. f ( ) 1, 0,1 6.,, 0 f 5 ( ) 4 7. f ( ) 5cos 4, 0,

11 LESSON.6 LIMITS AT INFINITY; HORIZONTAL ASYMPTOTES Horizotal Asymptote i. < m y =0 ii. = m a y b iii. > m o horizotal asymptotes m Note: Horizotal asymptotes are i the form of y... Slat Asymptote Whe > m by 1 the slat asymptote is the quotiet of the ratioal epressio. (If > m by or more, the fuctio has either horizotal or slat asymptotes) Note: Slat asymptotes are i the form y = m + b Limits at Ifiity f ( ) L or f ( ) L Everythig for larger the M fits iside the give widow aroud L. Defiitio of a Horizotal Asymptote Limits at Ifiity The lie y L is a horizotal asymptote of the graph f is f ( ) L or f ( ) L. If r is a positive ratioal umber ad c is ay real c umber, the 0. r r Furthermore, if is defied whe 0 c 0. r, the Remember: Give f ( ) L ad c g( ) K c, the f ( ) g( ) L K c f ( ) g( ) c I other words: cos tat 0 r cos tat 0 r LK or Practice Problems: Evaluate the followig its ad fid their HA.

12 Note: whe you ecouter a idetermiate form, the suggested method is to DIVIDE the NUMERATOR ad the DENOMINATOR by the HIGHEST POWER of i the DENOMINATOR. Whe asked to fid all horizotal asymptotes, check both its as ad Practice Problems: Evaluate the followig its. 7. arcta( ) 8. e

13 Fuctios with two horizotal asymptotes ad oratioal fuctios. a. 1 Note: for 0 Eample 1: Fid the followig its for 0 b. 1 Graph: y 1 Fuctios with Ifiite Limits at Ifiity a. 4 1 Note: Eample : Fid the followig its f( ) b. 4 1 Graph: y 4 1 Be careful!!! 0 ( 1) Istead:

14 LESSON.7 DERIVATIVES AND RATES OF CHANGE Defiitio of Taget Lie with Slope m Vertical Taget Lie The Derivative of a Fuctio Defiitio of the Derivative of a Fuctio Rates of Chage Positio Fuctio Average Velocity If f is defied o a ope iterval cotaiig c, ad if the it y f ( c ) f ( c) m c, f ( c ) 0 0 eists, the the lie passig through with slope m is the taget lie to the graph of f at the poit c, f ( c ). If f is cotiuous at c ad f ( c ) f ( c) or 0 f ( c ) f ( c), the vertical lie, 0 c, passig through c, f ( c ) is a Vertical Taget Lie to the Eample: Eample: graph of f. Fudametal operatio of calculus: Differetiatio is the process of fidig the derivative of a fuctio. A fuctio is differetiable at if its derivative eists at ad differetiable o a ope iterval ab, if it is differetiable at every poit i the iterval. The derivative of f at is give by Notatio for Derivatives: f ( ) f ( ) dy d f '( ) provided f '( ),, y ', f ( ), ad D y 0 d d the it eists. For all for which this it eists, f ' is a fuctio of. Note: The Derivative fuctio is the Slope fuctio of the origial fuctio. i.e.: populatio growth, rates, productio rates, water flow rates, velocity, ad acceleratio. The fuctio(s) that gives the positio Eample: The positio of a free-fallig (relative to the origi) of a object as a object is represeted by fuctio of time t. 1 s() t gt v0t s0 where: g = the acceleratio due to gravity ft / s 9.8 m/ s v 0 = iitial velocity s = iitial height 0 Displacemet chage i dis ta ce s Note: Velocity is speed with directios Time chage i time t Istataeous s( t t) s( t) Velocity v( t) s '( t) t 0 t Note: Istataeous velocity fuctio is the derivative of the positio fuctio Practice Problems

15 1. Fid the average rate of chage of f ( ) 9,,4. Fid the istataeous rate of chage of f ( ) 4. Fid the equatio of the taget lie to f ( ) Fid the equatio of the taget lie to f ( 7 5. Give f ( ) cos, fid f ' 6 6. Fid the equatio of the taget lie to 5 f ( Note: cosh1 sih 0 ad 1 h0 h h0 h

16 LESSON.8 THE DERIVATIVE AS A FUNCTION Defiitio of the Derivative of a Fuctio Where might a fuctio ot have a derivative (ot differetiable)? Higher-Order Derivatives Free-Fallig Object Problems The derivative of f at is give by f ( ) f ( ) f '( ) provided 0 the it eists. For all for which this it eists, f ' is a fuctio of. - Taget lie is vertical (udefied slope) - Corers (like absolute values) b/c of more tha oe taget a corer - Not cotiuous (jump) f () fuctio d dy f '( ) f ( ) y ' d d First derivative d d y f ''( ) f ( ) y '' d d Secod Derivative d d y f '''( ) f ( ) y ''' d d Third Derivative 4 4 (4) d d y 4 f ( ) 4 f ( ) y Fourth Derivative 4 d d ( ) ( ) d ( ) d y f f y d d th Derivative Positio Fuctio st () Velocity Fuctio v( t) s'( t) Notatio for Derivatives: dy d f '( ),, y ', f ( ), ad D y d d Note: The Derivative fuctio is the Slope fuctio of the origial fuctio. Graphs: Acceleratio Fuctio a( t) v'( t) s''( t) Eample 1: Prove f ( ) 4 is ot differetiable at. Practice Problems: Fid f '( ) usig the defiitio.

17 1. f ( ). 5 f( ). f( ) 4 4. f( ) 4 Practice Problems: Give the followig graphs, fid the derivative of each fuctio.

18 5. 6. Practice Problems: Give the followig graphs, sketch the derivative

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