Math ~ Final Exam Review Guide* *This is only a guide, for your benefit, and it in no way replaces class notes, homework, or studying

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1 Math ~ Fial Exam Review Guide* *This is oly a guide, for your eefit, ad it i o way replaces class otes, homework, or studyig Geeral Tips for Studyig: 1. Review this guide, class otes, ad the text 2. Review commets o quizzes ad exams ad rework ALL quiz ad exam prolems 3. Review (assumig you have completed it) ALL homework assiged for the sectios that will covered o the exam: , N.1 N.5, 1.1, , , Complete ALL suggested prolems for the fial exam review give i class 5. Start studyig early eough to ask questios! *Pay careful attetio to what I have emphasized/de emphasized (or completely elimiated) Sectios or specific topics ot listed here will NOT e covered o the fial exam If the directios ask you to use a certai method, you receive NO CREDIT for usig aother method. I've olded ad uderlied formulas you eed to memorize with: Memorize Pay careful attetio to what I have oted i class as commo mistakes o past exams ad quizzes Keep a eye out for otes that say: this WILL e o the exam Chapter 0: Prelimiaries *There will ot e direct questios from Chapter 0 sice it is all review, ut you eed this material to complete the future material, so it is VITAL that you uderstad this etire chapter. 0.5 Fuctios ad Their Graphs: Be ale to fid the domai of a fuctio Be ale to test whether fuctios are eve (y axis symmetry) or odd (origi symmetry) Eve: f x= f x Odd: f x= f x 0.6 Operatios o Fuctios: Be familiar with the asic paret fuctios, ad kow what their graphs look like (liear, squarig, cuic, square root, reciprocal, asolute value). Be ale to graph piecewise fuctios. Useful shifts: Vertical shift c uits up: h x= f x c Vertical shift c uits dow: h x= f x c Horizotal shift c uits right: hx= f x c Horizotal shift c uits left: hx= f xc Reflectio i x axis: hx= f x Reflectio i y axis: hx= f x Kow how to arithmetically comie fuctios f g x= f x g x f g x= f x g x fg x= f x g x f f x g x= g x Uderstad fuctio compositio: f gx= f g x 0.7 Trigoometric Fuctios: Kow the asics aout the sie ad cosie fuctios (graphs, values at 0, period, etc.) Kow the 4 other trig fuctios from pg 45 (ta, cot, sec, csc) i terms of si ad cos

2 Chapter N: Itroductio to Polyomial Calculus *Because all of Chapter N is either review or covered i detail later i the course, it will ot e set apart from the rest of the exam (prolems will occur where we covered them i the ook). N.1 Straight Lies: Kow how to fid the slope of a (overtical) lie through two poits: m= y 2 y 1 x 2 x 1 Slope Itercept form: y = mx + Poit Slope form: y y 1 = m(x x 1 ) Use slope to determie whether lies are parallel or perpedicular N.2 Slope of a Curve: BIG IDEA: slope of a curve depeds o where you are o the curve! Uderstad how to use ad memorize the defiitio of the derivative: f ' x=lim h 0 f xh f x h Rememer! The value of the derivative at x is the slope of the curve or graph f x at x, f x N.3 Derivative of a Polyomial: Use the followig rules to fid the derivatives of polyomials: For f x= x, f ' x=x 1 cf x'=cf ' x f x g x'= f ' xg ' x N.4 Atiderivatives of Polyomials: Use the followig rules (the aove i reverse) to fid atiderivatives of polyomials F = x dx= x1 1 C af x dx=a f xdx f xgxdx= f xdx gxdx **Uderstad positio, velocity, ad acceleratio (see N.5 as well). You WILL see a applicatio questio of this type o the fial (see 2.6 ad 3.9). N.5 Defiite Itegrals: Use the 2 d Fudametal Theorem of Calculus to fid defiite itegrals a f x dx=f F a where F is ay atiderivative of f ( C=0 for coveiece) Chapter 1: Limits 1.1 Itroductio to Limits: Uderstad the ituitive meaig of a limit Recall that a limit oly exists if the right ad left had limits match (Thrm A pg 59) Kow how to evaluate asic limits limits that ivolve simply puggig i the value of x, or simplifyig (ie: factorig ad cacelig zero deomiators) so pluggig i works 1.3 Limit Theorems: Kow ad uderstad all of the limit rules i the Mai Limit Theorem (Thrm A pg 68) Rememer! You are ALWAYS required to simplify complex limits to smaller limits that you kow the limit of y applyig this theorem. Kow that the Sustitutio Theorem from this sectio allows you to evaluate polyomials ad ratioal fuctios with ozero deomiators y pluggig i

3 1.4 Limits Ivolvig Trigoometric Fuctios: six Memorize: lim x 0 x =1 Big Idea: Simplify the trig fuctios ito small pieces you kow the limit of y usig the limit rules from 1.3. Key Tools: Write ay trig fuctios i terms of si ad cos Multiply y 1 as the ecessary x x to get pieces that look like six x 1.5 Limits at Ifiity; Ifiite Limits: 1 Memorize: lim x x =0 k ad lim x 1 x k =0 (this is the key tool here) Evaluate limits at ifiity y rewritig fuctios with pieces like aove (to do this, you divide each term y the largest x k i the deomiator) Geeral Limit Iformatio There are 3 cases for x ± (you must show the work to get the aswer i every case!): degree umerator > degree deomiator: limit goes to ± degree umerator = degree deomiator: limit is ratio of leadig coefficiets degree umerator < degree deomiator: limit is zero (Note these 3 cases relate to horizotal asymptotes) There are 3 cases for x c (you must show the work to get the aswer i every case!): The first step is always plug i c (i your head if you wat) If you get a fiite umer ack, you're doe (limits i 1.1, 1.3, 1.4) 0 If you get, you eed to simplify more to fid the limit. Simplify util you 0 ca plug i c ad get a fiite umer (1.1, 1.3, 1.4) umer If you get, the limit will go to or or it does ot exist. To 0 figure out which, you must evaluate the right ad left had limits. (Note that this case relates to vertical asymptotes) 1.6 Cotiuity of Fuctios: Kow ad uderstad how to use the defiitio of cotiuity at a poit f x= f c This defiitio requires 3 thigs (memorize): 1. lim f x must exist 2. f c must exist 3. lim x c x c lim x c f x= f c Chapter 2: The Derivative 2.2 The Derivative: Uderstad how to use ad memorize the defiitio of the derivative: f xh f x f ' x=lim There WILL e a questio of this type o the exam! h 0 h Uderstad that differetiaility implies cotiuity (ut ot vice versa) Rememer! Fuctios are ot differetiale at corers ad vertical tagets 2.3 Rules for Fidig Derivatives:

4 Kow how to use the rules from this sectio to fid derivatives (memorize) Costat Rule D x k=0 Idetity Fuctio Rule D x x=1 Power Rule D x x =x 1 Costat Multiple Rule D x [k f x]=k D x f x Sum/Differece Rule D x [ f x±g x]=d x f x±d x g x Product Rule D x [ f x g x]= f x D x g xg x D x f x OR f g' x= f x g ' xg x f ' x Quotiet Rule D x f x g x = g x D x f x f x D x g x g 2 x OR f g x f ' x f x g ' x g ' x= g 2 x 2.4 Derivatives of Trigoometric Fuctios: Memorize: D x si x=cos x ad D x cos x= si x Be ale to use the aove with the rules from 2.3 to fid derivatives of trig fuctios 2.5 The Chai Rule: Uderstad whe ad how to apply the chai rule (memorize): f g' x= f ' g x g ' x Rememer! You must use the chai rule ay time that fuctio compositio is ivolved 2.6 Higher Order Derivatives: Be ale to take higher order derivatives, icludig those ivolved i applied questios ivolvig positio, velocity, ad acceleratio (see N.4, 3.9). Kow how positio, velocity ad acceleratio are related. This WILL e o the fial! 2.7 Implicit Differetiatio: Steps: 1. Take the derivative of oth sides (usig dy/dx otatio!) of the equatio. 2. Move all terms ivolvig dy/dx to oe side ad solve for dy/dx Rememer! y is a fuctio of x (y(x)), so you must use the chai rule to differetiate ay terms ivolvig y. 2.8 Related Rates: There WILL e 1 related rates questio o the exam. It will e similar to questios give i the homework, o the take home quiz, ad exam 2. Complex formulas will e give if eeded (ut you eed to kow similar triagles ad the Pythagorea theorem). Steps: Draw a picture that is valid for all time>0. Lael the values you kow, ad chose variales for ad lael the values you do't kow, ut eed. State what is give ad what you wat to fid. This iformatio will e i the form of derivatives with respect to t. Relate the variales y writig a equatio that is true for ALL time>0. Do NOT plug i what you kow!! Implicitly differetiate with respect to time. Fially, plug i what you kow ad solve for what you wat to fid! Note: you may eed to elimiate a extra variale y usig the Pythagorea theorem or similar triagles.

5 Chapter 3: Applicatios of the Derivative 3.1 Maxima ad Miima: Uderstad the Critical Poit Thrm: If f(c) is a extrema, the c must e a critical poit. Either c is a edpoit of I a statioary poit of f ( f ' c=0 ) a sigular poit of f ( f ' c does ot exist) Note that this does NOT mea that all critical poits are extrema 3.2 Mootoicity ad Cocavity (memorize): Uderstad the Mootoicity Thrm f ' x0 the f is icreasig f ' x0 the f is decreasig Uderstad the Cocavity Thrm f ' ' x0 the f is cocave up f ' ' x0 the f is cocave dow Kow what iflectio poits are ad how to fid them 3.3 Local Extrema: Kow the differece etwee local ad gloal extrema Kow ad uderstad whe ad how to apply the first derivative test (memorize) f ' x0 o the left of c ad f ' x0 o the right of c, meas c is a local max f ' x0 o the left of c ad f ' x0 o the right of c, meas c is a local mi If f ' x has the sig same o oth sides of c, c is ot a local extreme value Kow ad uderstad whe ad how to apply the secod derivative test If f ' ' c0 the f c is a local max If f ' ' c0 the f c is a local mi Note that if f ' ' c=0 the test is icoclusive 3.5 Graphig Fuctios Usig Calculus: Steps: Precalculus: Fid the domai of the fuctio Test for symmetry (is the fuctio eve or odd) Fid the itercepts (oth x ad y) Calculus: Fid the critical poits Fid out where the fuctio is icreasig ad decreasig Fid local miima ad maxima Fid out where the graph is cocave up ad cocave dow Fid ay iflectio poits Fid the asymptotes *There WILL e a questio aout the material from sectios 3.1, 3.2, 3.3, ad 3.5 o the fial* 3.8 Atiderivatives: F is a atiderivative of f if F ' x= f x Rules (memorize): Power Rule: F = x r dx= x r1 kf xdx=k f xdx C (r is ay ratioal umer except 1) r1

6 f x±g xdx= f xdx± g xdx si xdx= cosxc ad cosxdx=si xc Geeral Power Rule: F = [ g x] r g ' xdx= [ g x]r1 C (udoig chai rule) r1 Rememer! Do't forget the C o the atiderivatives!! 3.9 Itroductio to Differetial Equatios: Be ale to use Separatio of Variales to solve first order, separale DEs Steps: separate the 2 variales, itegrate oth sides (do't forget the +C!), use algera to solve for the depedet variale (ie: y for y(x)) There WILL e a questio o the fial relatig positio, velocity, ad acceleratio. See HW #17 for a example of how this relatioship is applied for itegratio. Chapter 4: The Defiite Itegral 4.1 Itroductio to Area: Uderstad the asics of sigma otatio, e ale to use it to work with sums If you have to simplify a sum, you are required to show the work required for usig liearity. Do NOT memorize the special sum rules, ut kow how to use them! i =1 i =1 c=c a i ± i = i =1 a i ± i i=1 i=1 ca i =c i=1 a i 4.2 The Defiite Itegral: Uderstad itegratio i terms of the Riema sum: Sum up the areas of rectagles uder a curve, ad take the limit as the umer of rectagles goes to ifiity. Thus, the defiite itegral is the area uder the curve. You will e give appropriate formulas if asked to solve a questio from this sectio. Rememer that area is siged area here (area elow the curve is egative, ut area is always positive, so we chage the sig. This does NOT apply for area etwee curves!) a f xdx= Aup A dow Kow how to use the followig properties (memorize): a f xdx= a f xdx a c f x dx= a f x dx c f x dx 4.3 The 1 st Fudametal Thrm of Calculus: It has the word fudametal i the title: it WILL e o the fial! Uderstad the 1 st Fudametal Thrm of Calculus (memorize) d dx x a f t dt= f x The locatio of the x matters. If x is a fuctio, the chai rule is required! Rememer! The est part aout this theorem is that there is NO actual itegratio ivolved i fidig the aswer! Uderstad liearity of itegratio a kf xdx=k a f xdx a f x±g xdx= a f xdx± a g xdx

7 4.4 The 2 d Fudametal Thrm of Calculus ad the Method of Sustitutio: It has the word fudametal i the title: oth the 2 d Fud. Thrm ad sustitutio for defiite ad idefiite itegrals WILL e o the fial! Recall the 2 d Fud. Thrm from Chapter N (memorize): a f x dx=f F a Use sustitutio to evaluate idefiite itegrals ivolvig fuctio compositio f g xg ' xdx= f udu=f uc=f g xc let u=g x so du=g ' x dx Rememer! Defie u=g(x) ased o the fuctio compositio (it is the iside fuctio), ad the FIND du from this u (do't just read it off from the itegral...it may ot match exactly!). You may eed to multiply or divide y a costat to get the du term to fit your itegral. Use sustitutio to evaluate defiite itegrals ivolvig fuctio compositio g a f g x g' x dx= g a f udu=[f u] g ga = F g F ga Rememer! Use the same methods for sustitutio as for idefiite itegrals, ut chage your limits of itegratio to e i terms of u=g(x). Chapter 5: Applicatios of the Itegral 5.1 The Area of a Plae Regio: Be ale to fid the area of a plae regio Steps: sketch the regio slice ito thi pieces, lael ad typical piece (choose dx or dy) approximate the area of the typical piece as if it were a rectagle Rememer! If it's the area etwee 2 curves, use f(x) g(x) as the legth of the rectagle where f(x)> g(x) Rememer! The sig of the aswer automatically works out if you sutract correctly (you do't correct for eig elow the x axis)! fid the limits of itegratio (you eed the itersectio poits of the 2 graphs) itegrate: Area = a legth * width (width is dx or dy) (memorize) 5.2 Volumes of Solids: Disks, Washers: Be ale to use the disk ad washer methods to fid the volume of solids Disk Method sketch graph decide o dx or dy thickess fid limits of itegratio determie the area for a typical slice (slices are always circular, A= r 2 ) itegrate: Volume = a Area(x) dx (or A(y) ad dy) (memorize) Washer Method 2 2 same as aove, ut A=[r outer r ier ] sice the slice is a washer (memorize) Rememer! Measure the radius from the axis of revolutio! Rememer! The slice looks perpedicular to the axis of revolutio for these methods (ie: revolve aout x axis, use dx slices, revolve aout y axis, use dy slices) Rememer! The itegratio limits are i terms of the variale of thickess (x for dx or y for dy) ad oly rage i the origial area (ot the revolved area).

8 5.3 Volumes of Solids: Shells: Be ale to use the shell method to fid the volume of solid Rememer! Shell ad washer methods are est for similar prolems (those with ope areas i the middle of the revolved shape). Shell Method sketch graph decide o dx or dy thickess fid limits of itegratio determie the radius ad height for a typical shell (oth deped o your variale...there are ifiitely may shells iside the volume) Itegrate: Volume = 2 a radius*height dx (or dy) (memorize) Rememer! The shell looks parallel to the axis of revolutio (ie: revolve aout x axis, use dy slices (shell o side), revolve aout y axis, use dx slices (shell upright)) Rememer! The itegratio limits are i terms of the variale of thickess (i x for dx or y for dy) ad oly rage i the origial area (ot the revolved area). 5.4 Legth of a Plae Curve: Be ale to use the 3 arc legth equatios to fid the arc legth of various curves (do NOT memorize, ut kow whe to use each formula). L= a dx 2 dt dy 2 dt dt for a t (parametric equatios, x=f(t), y=g(t) ) L= 1 a dy 2 dx dx for a x (ot parametric, y=f(x) ) d L= c dx 2 dy 1dy for c y d (ot parametric, x=g(y) ) 5.5 Work ad Fluid Force: Be ale to solve asic force questios ivolvig sprigs (see HW for what to focus o). Rememer! Work = Force * Distace Take variale force ito accout y summig over small distaces W= a F xdx Hooke's Law: F(x)=kx, where k is the sprig costat *There WILL e oe questio from 5.4 OR 5.5 (worth a smaller amout of poits tha the other questios)*

Math 21B-B - Homework Set 2

Math 21B-B - Homework Set 2 Math B-B - Homework Set Sectio 5.:. a) lim P k= c k c k ) x k, where P is a partitio of [, 5. x x ) dx b) lim P k= 4 ck x k, where P is a partitio of [,. 4 x dx c) lim P k= ta c k ) x k, where P is a partitio