Name: _________________________ 1.4 1 .4 T h e P re ci se D ef in it io n o f a L im it 1.4 The Precise Definition of a Limit Ticket in the Door In order to be prepared for class you must watch the...

Complete various problems showing the work through each step (explanations).


Name: _________________________ 1.4 1 .4 T h e P re ci se D ef in it io n o f a L im it 1.4 The Precise Definition of a Limit Ticket in the Door In order to be prepared for class you must watch the module and complete the following activity. Remember you must submit your work as a PDF file. Example 1: Use the given graph of ??(??) = √?? to find a ?? such that if |?? − 4| < then="" �√??="" −="" 2�="">< 0.4="" example="" 2:="" use="" the="" given="" graph="" of="" (??)="??2" to="" find="" a="" such="" that="" if="" |??="" −="" 2|="">< then="" |??2="" −="" 4|="">< 0.3 example 3: prove the statement using the ??, ?? definition of a limit. lim x→1 9 + 3x 4 = 3 ?? ?? = ??2 ?? 1.4 the precise definition of a limit name: _________________________ 2.3 2. 3 tr ig on om et ri c f un ct io ns unit 2.3 trigonometric functions ticket in the door in order to be prepared for class you must watch the module and complete the following activity. remember you must submit your work as a pdf file. pre-calculus review: rewrite cos (x+h) using the sum/differences identities example 1. as we did in the video, using the limiting definition to prove ???? ???? [cos x] = -sin x. example 2: use the quotient rule to find the derivative of f(x) = 2-5cos x 3x2 + 5 example 3: given ??′′(??) = −4??????(??) and ??′(0) = −3 and ??(0) = 3. find f �π 3 � unit 2.3 trigonometric functions name: _________________________ 2.4 un it 2 – le ss on 2 .4 c om po si tio n of f un ct io ns p ar t 1 unit 2 – lesson 2.4 composition of functions part 1 ticket in the door in order to be prepared for class you must watch the module and complete the following activity. remember you must submit your work as a pdf file. example #1 consider y = √3??2 − ?? a. what is the “inner” function? _______ what is the derivative of this function? ___________ b. what is the “outer” function? _______ what is the derivative of this function? ____________ c. find ?? ???? √3??2 − ?? example #2 evaluate ?? ???? ??????(5??3 + 12??) example #3 evaluate the integral. �(2 + x)5 ???? let u = _______ ________ du = dx example #4 evaluate the integral. �2x sin(4 + x2)???? let u = _______ ________ du = dx unit 2 – lesson 2.4 composition of functions part 1 name: _________________________ 2.7 un it 2 – le ss on 2 .7 d er iv at iv es o f h yp er bo lic f un ct io ns unit 2 – lesson 2.7 derivatives of hyperbolic functions ticket in the door in order to be prepared for class you must watch the module and complete the following activity. remember you must submit your work as a pdf file. pre-calculus review: given the definitions of the hyperbolic sine, cosine and tangent functions are ??????ℎ ?? = ?? ??−??−?? 2 ??????ℎ ?? = ?? ??+??−?? 2 ??????ℎ ?? = ??????ℎ ?? ??????ℎ ?? find the numerical value of each expression (a) ??????ℎ 0 (b) ??????ℎ 1 (c) ??????ℎ (???? 3) example 1: find the first derivative of ??(??) = sinh−1(??2 − 1). example 2: find the first derivative of ??(??) = ln[cosh(5??2 − 2??)] unit 2 – lesson 2.7 derivatives of hyperbolic functions name: ________________ section 3.1: related rates ticket-in-the-door in order to be prepared for class, you must complete the following activity and turn it in online. guidelines for solving related rates problems 1. make a ___________ and ___________ the quantities. 2. read the problem and identify all quantities as: ___________ , ___________ and ___________with the appropriate information. 3. write an ___________ involving the variables whose ___________ of ___________ either are given or are to be determined. 4. using the _________ _________, implicitly _________ both sides of the equation with respect to time, _________. 5. _________ into the resulting equation all known values for the variables and their rates of change. then _________ for the required rate of change. example: the spreading circle an oil rig springs a leak, and the oil spreads in a circular patch around the rig. if the radius of the oil patch increases at a rate of 30 m / hr, how fast is the area are of the patch increasing when the patch has a radius of 100 meters? draw and label a sketch know given find 0.3="" example="" 3:="" prove="" the="" statement="" using="" the="" ,="" definition="" of="" a="" limit.="" lim="" x→1="" 9="" +="" 3x="" 4="3" =="" 2="" 1.4="" the="" precise="" definition="" of="" a="" limit="" name:="" _________________________="" 2.3="" 2.="" 3="" tr="" ig="" on="" om="" et="" ri="" c="" f="" un="" ct="" io="" ns="" unit="" 2.3="" trigonometric="" functions="" ticket="" in="" the="" door="" in="" order="" to="" be="" prepared="" for="" class="" you="" must="" watch="" the="" module="" and="" complete="" the="" following="" activity.="" remember="" you="" must="" submit="" your="" work="" as="" a="" pdf="" file.="" pre-calculus="" review:="" rewrite="" cos="" (x+h)="" using="" the="" sum/differences="" identities="" example="" 1.="" as="" we="" did="" in="" the="" video,="" using="" the="" limiting="" definition="" to="" prove="" [cos="" x]="-sin" x.="" example="" 2:="" use="" the="" quotient="" rule="" to="" find="" the="" derivative="" of="" f(x)="2-5cos" x="" 3x2="" +="" 5="" example="" 3:="" given="" ′′(??)="−4??????(??)" and="" ′(0)="−3" and="" (0)="3." find="" f="" �π="" 3="" �="" unit="" 2.3="" trigonometric="" functions="" name:="" _________________________="" 2.4="" un="" it="" 2="" –="" le="" ss="" on="" 2="" .4="" c="" om="" po="" si="" tio="" n="" of="" f="" un="" ct="" io="" ns="" p="" ar="" t="" 1="" unit="" 2="" –="" lesson="" 2.4="" composition="" of="" functions="" part="" 1="" ticket="" in="" the="" door="" in="" order="" to="" be="" prepared="" for="" class="" you="" must="" watch="" the="" module="" and="" complete="" the="" following="" activity.="" remember="" you="" must="" submit="" your="" work="" as="" a="" pdf="" file.="" example="" #1="" consider="" y="√3??2" −="" a.="" what="" is="" the="" “inner”="" function?="" _______="" what="" is="" the="" derivative="" of="" this="" function?="" ___________="" b.="" what="" is="" the="" “outer”="" function?="" _______="" what="" is="" the="" derivative="" of="" this="" function?="" ____________="" c.="" find="" √3??2="" −="" example="" #2="" evaluate="" (5??3="" +="" 12??)="" example="" #3="" evaluate="" the="" integral.="" �(2="" +="" x)5="" let="" u="_______" ________="" du="dx" example="" #4="" evaluate="" the="" integral.="" �2x="" sin(4="" +="" x2)????="" let="" u="_______" ________="" du="dx" unit="" 2="" –="" lesson="" 2.4="" composition="" of="" functions="" part="" 1="" name:="" _________________________="" 2.7="" un="" it="" 2="" –="" le="" ss="" on="" 2="" .7="" d="" er="" iv="" at="" iv="" es="" o="" f="" h="" yp="" er="" bo="" lic="" f="" un="" ct="" io="" ns="" unit="" 2="" –="" lesson="" 2.7="" derivatives="" of="" hyperbolic="" functions="" ticket="" in="" the="" door="" in="" order="" to="" be="" prepared="" for="" class="" you="" must="" watch="" the="" module="" and="" complete="" the="" following="" activity.="" remember="" you="" must="" submit="" your="" work="" as="" a="" pdf="" file.="" pre-calculus="" review:="" given="" the="" definitions="" of="" the="" hyperbolic="" sine,="" cosine="" and="" tangent="" functions="" are="" ℎ="" =="" −??−??="" 2="" ℎ="" =="" +??−??="" 2="" ℎ="" =="" ℎ="" ℎ="" find="" the="" numerical="" value="" of="" each="" expression="" (a)="" ℎ="" 0="" (b)="" ℎ="" 1="" (c)="" ℎ="" (????="" 3)="" example="" 1:="" find="" the="" first="" derivative="" of="" (??)="sinh−1(??2" −="" 1).="" example="" 2:="" find="" the="" first="" derivative="" of="" (??)="ln[cosh(5??2" −="" 2??)]="" unit="" 2="" –="" lesson="" 2.7="" derivatives="" of="" hyperbolic="" functions="" name:="" ________________="" section="" 3.1:="" related="" rates="" ticket-in-the-door="" in="" order="" to="" be="" prepared="" for="" class,="" you="" must="" complete="" the="" following="" activity="" and="" turn="" it="" in="" online.="" guidelines="" for="" solving="" related="" rates="" problems="" 1.="" make="" a="" ___________="" and="" ___________="" the="" quantities.="" 2.="" read="" the="" problem="" and="" identify="" all="" quantities="" as:="" ___________="" ,="" ___________="" and="" ___________with="" the="" appropriate="" information.="" 3.="" write="" an="" ___________="" involving="" the="" variables="" whose="" ___________="" of="" ___________="" either="" are="" given="" or="" are="" to="" be="" determined.="" 4.="" using="" the="" _________="" _________,="" implicitly="" _________="" both="" sides="" of="" the="" equation="" with="" respect="" to="" time,="" _________.="" 5.="" _________="" into="" the="" resulting="" equation="" all="" known="" values="" for="" the="" variables="" and="" their="" rates="" of="" change.="" then="" _________="" for="" the="" required="" rate="" of="" change.="" example:="" the="" spreading="" circle="" an="" oil="" rig="" springs="" a="" leak,="" and="" the="" oil="" spreads="" in="" a="" circular="" patch="" around="" the="" rig.="" if="" the="" radius="" of="" the="" oil="" patch="" increases="" at="" a="" rate="" of="" 30="" m="" hr,="" how="" fast="" is="" the="" area="" are="" of="" the="" patch="" increasing="" when="" the="" patch="" has="" a="" radius="" of="" 100="" meters?="" draw="" and="" label="" a="" sketch="" know="" given="">