B Veitch. Calculus I Study Guide


 Deborah Gibson
 1 years ago
 Views:
Transcription
1 Clculus I Stuy Guie This stuy guie is in no wy exhustive. As stte in clss, ny type of question from clss, quizzes, exms, n homeworks re fir gme. There s no informtion here bout the wor problems. 1. Some Algebr Review () Fctoring n Solving i. Qurtic Formul: x 2 + bx + c = 0 ii. Difference of Two Squres x = b ± b 2 4c 2 u 2 2 = (u )(u + ) Creful! Sometimes you hve to rewrite the eqution, ex. x = 4 x 2 = (2 x)(2 + x) iii. Fctor Trinomils: x 2 + bx + c. Some exmples (b) Exponentil Properties x x + 16, 6x 2 + x 12, 2x 2 6x 10 i. n m = n+m ii. ( n ) m = nm iii. (b) n = n b n n iv. m = n m = 1 m n v. n = 1 n vi. 1 = n n vii. m/n = ( m ) 1/n = ( 1/n ) m (c) Properties of Ricls i. n = 1/n ii. n m = m/n 2. Limits () Nottion i. Generl Limit Nottion: lim f(x) = L ii. Left Hn Limit: lim f(x) = L iii. Right Hn Limit: lim f(x) = L + iv. If lim f(x) = lim f(x) = L then lim f(x) = L + 1
2 Clculus I Stuy Guie (b) Limits t ±. Assume ll polynomils re in escening orer. i. lim x x r = 0, r > 0 ii. If n > m, then lim x x m +... bx n +... = 0 iii. x n +... lim x bx n +... = b iv. If m > n, then lim x x m +... bx n +... = ± (c) Evlution Techniques i. If f(x) is continuous, then lim f(x) = f() ii. Fctor n Cncel lim x2 + 4 = = 13 x 3 If you evlute rtionl function n get 0, then try to fctor. 0 x 2 9 lim x 3 x 3 = lim (x 3)(x + 3) (x + 3) = lim = 6 x 3 x 3 x 3 1 iii. Rtionlizing Numertors / Denomintors Try this technique if you hve ricls. Multiply top n bottom by the conjugte. x 4 x 4 x + 4 lim x 4 x 16 = lim x 4 x 16 x 16 = lim x + 4 x 4 (x 16)( x + 4) = lim 1 x 4 x + 4 iv. Combine by Using Common Denomintors Try this when you nee to combine frctions within frctions = lim lim 2 x x = lim 2 x x 2 2x (x )(x + 4)( + 4) = lim (x + 4)( + 4) (x + 4)( + 4) = lim 2( + 4) 2(x + 4) (x )(x + 4)( + 4) 2(x ) (x )(x + 4)( + 4) = lim 2 (x + 4)( + 4) = 2 ( + 4)( + 4) () Piecewise Functions  Know how to grph n evlute Piecewise Functions. These re goo ones to test your unerstning of lefthn, righthn, n generl limits. They re lso use to test your unerstning of continuity. (e) Definition of Continuity A function f is continuous t x = if the following three conitions re stisfie: i. f() must exist ii. lim f(x) must exist iii. lim f(x) = f() (f) Asymptotes 2
3 Clculus I Stuy Guie i. Verticl Asymptotes: The line x = is verticl symptote if ny of the following occur: lim f(x) = ± or lim From lgebr, you cn fin verticl symptotes by f(x) = ± + A. Reucing your rtionl function (no common fctors in numertor n enomintor). B. Fining the x vlues tht mke the enomintor 0. ii. Horizontl Asymptotes The line y = b is horizontl symptote if either or both occur: lim f(x) = b or lim x f(x) = b x Refer to 2(b) of this guie for shortcuts on evluting these limits. iii. Slnt Asymptotes: Given the function f(x) = P (x), slnt symptote occurs when the egree of the numertor Q(x) is 1 greter thn the egree of the enomintor. Ex. f(x) = x2 4 x 1 You fin the slnt symptote by oing long ivision. 3
4 Clculus I Stuy Guie 3. Derivtives () Limit Definition of Derivtive (b) Tngent Lines f (x) = lim h 0 f(x + h) f(x) h or f () = lim f(x) f() x When ske to fin the eqution of tngent line on f t x =, you nee two things: A point n slope. i. You re usully given the x vlue. If they on t tell you the y vlue, you must plug the x vlue into f(x) to get the y vlue. Now you hve point (, f()) ii. To fin the slope m, you fin m = f (). iii. The eqution of the tngent line is y f() = f ()(x ). You my nee to write this in slopeintercept form. (c) Derivtive Formuls i. x (c) = 0 ii. x (f ± g) = f ± g iii. x (x) = 1 iv. x (kx) = k v. Power Rule: x (xn ) = nx n 1 vi. Chin Rule: x f(g(x)) = f (g(x)) g (x) vii. Prouct Rule: (f g) = f g + fg ( ) f viii. Quotient Rule: = f g fg g g 2 ix. sin(x) = cos(x) x x. cos(x) = sin(x) x xi. x tn(x) = sec2 (x) xii. sec(x) = sec(x) tn(x) x xiii. csc(x) = csc(x) cot(x) x xiv. x cot(x) = csc2 (x) () Criticl Points x = c is vlue of f(x) if f (c) = 0 or f (c) oes not exist. (e) Incresing / Decresing i. If f (x) > 0 on n intervl I, then f(x) is incresing. ii. If f (x) < 0 on n intervl I, then f(x) is ecresing. (f) Concve Up / Concve Down i. If f (x) > 0 on n intervl I, then f(x) is concve up. ii. If f (x) < 0 on n intervl I, then f(x) is concve own. (g) Inflection Points x = c is n inflection point of f if 4
5 Clculus I Stuy Guie i. The point t x = c must exist. ii. Concvity chnges t x = c (h) First Derivtive Test to fin Locl (Reltive) Extrem i. Fin ll criticl vlue of f(x) where f (c) = 0 ii. Locl Mx t x = c if f (x) chnges from (+) to ( ) t x = c. iii. Locl Min t x = c if f (x) chnges from ( ) to (+) t x = c. iv. If f (x) oes not chnge signs, it s still n importnt point to plot. It my be plce where the slope is 0, corner, n symptote, verticl tngent line, etc. v. Mke sure you write your reltive mx n mins s points (c, f(c)) (i) Secon Derivtive Test to fin Locl (Reltive) Extrem i. Fin ll criticl vlue of f(x) where f (c) = 0 ii. Locl Mx t x = c if f (c) < 0 iii. Locl Min t x = c if f (x) > 0 iv. Mke sure you write your reltive mx n mins s points (c, f(c)) (j) Absolute Extrem i. (c, f(c)) is n bsolute mximum of f(x) if f(c) f(x) for ll x in the omin. ii. (c, f(c)) is n bsolute minimum of f(x) if f(c) f(x) for ll x in the omin. (k) Fining Absolute Extrem of continuous f(x) over [, b] i. Fin ll criticl vlues of f(x) on [, b]. ii. Evlute f(x) t ll criticl vlues. iii. Evlute f(x) t the enpoints, f() n f(b). iv. The bsolute mx is the lrgest function vlue n the bsolute min is the smllest function vlue. (l) Differentils / Lineriztion i. Lineriztion of f(x) t x = L(x) = f() + f ()(x ) ii. Differentils A. x  is the true chnge in x B. ifferentil x  is our inepenent vrible tht represents the chnge in x. We let x = x. C. y  is the true chnge in y D. ifferentil y  is the estimte chnge in y E. Formul: y x = f (x) or y = f (x) x 5
6 Clculus I Stuy Guie 4. Summry of Curve Sketching Alwys strt by noting the omin of f(x) () x n y intercepts i. xintercepts occur when f(x) = 0 ii. yintercept occurs when x = 0 (b) Fin ny verticl, horizontl symptotes, or slnt symptotes. i. Verticl Asymptote: Fin ll xvlues where lim f(x) = ±. Usully when the enomintor is 0 n the numertor is not 0. Rtionl function MUST be reuce. ii. Horizontl Aymptotes: Fin lim f(x) n lim f(x). There re shortcuts bse on the x x egree of the numertor n enomintor. (c) Fin f (x) iii. Slnt Asymptotes: Occurs when the egree of the numertor is one lrger thn the enomintor. You must o long ivision to etermine the symptotes. i. Fin the criticl vlues, ll xvlues where f (x) = 0 or when f (x) oes not exist. ii. Plot the criticl vlues on number line. iii. Fin incresing / ecresing intervls using number line iv. Use The First Derivtive Test to fin locl mximums / minimums (if ny exist). () Fin f (x) Remember to write them s points. A. Locl Mx t x = c if f (x) chnges from (+) to ( ) t x = c. B. Locl Min t x = c if f (x) chnges from ( ) to (+) t x = c. C. Note: If f (x) oes not chnge signs, it s still n importnt point. It my be plce where the slope is 0, corner, n symptote, verticl tngent line, etc. i. Fin ll xvlues where f (x) = 0 or when f (x) oes not exist. ii. Plot these xvlues on number line. iii. Fin intervls of concvity using the number line iv. Fin points of inflection (e) Sketch A. Must be plce where concvity chnges B. The point must exist (i.e, cn t be n symptote, iscontinuity) i. Drw every symptote ii. Plot ll intercepts iii. Plot ll criticl points (even if they re not reltive extrem). They were criticl points for reson. iv. Plot ll inflection points. v. Connect points on the grph by using informtion bout incresing/ecresing n its concvity. 6
7 Clculus I Stuy Guie 5. Integrls () Definitions i. Antieritive: An ntierivtive of f(x) is function F (x), where F (x) = f(x). ii. Generl Antierivtive: The generl ntierivtive of f(x) is F (x) + C, where F (x) = f(x). Also known s the Inefinite Integrl f(x) x = F (x) + C iii. Definite Integrl: (b) Common Integrls b f(x) x = F (b) F () i. ii. iii. iv. k x = kx + C x n x = xn+1 + C, n 1 n + 1 sin(x) x = cos(x) + C cos(x) x = sin(x) + C v. vi. vii. viii. sec 2 (x) x = tn(x) + C sec(x) tn(x) x = sec(x) + C csc 2 (x) x = cot(x) + C csc(x) cot(x) = csc(x) + C (c) u Substitution Given b i. Let u = g(x) f(g(x))g (x) x, ii. Then u = g (x) x iii. If there re bouns, you must chnge them using u = g(b) n u = g() b f(g(x))g (x) x = g(b) g() f(u) u () Every integrl must be written into the proper form in orer to use the formuls. For exmple, (e) Fining Are uner or between Curves x x = x 1/2 x 3 3 5x 4 x = 5 x 4 x 8 5 x = 8x 9/5 x x 9 7
8 Clculus I Stuy Guie 6. Approximtion Integrtion Techniques Given the integrl x i = + i x, then b f(x) x n n (for Simpson s Rule n must be even), with x = b n n Lefthn b Righthn Mipoint b b f(x) x = x [f(x 0 ) + f(x 1 ) + f(x 2 ) + f(x 3 ) f(x n 1 )] f(x) x = x [f(x 1 ) + f(x 2 ) + f(x 3 ) + f(x 4 ) f(x n )] f(x) x = x [f(x 1) + f(x 2) + f(x 3) + f(x 4) f(x n)], where x i is mipoint in [x i 1, x i ] 8
Final Exam Review. Exam 1 Material
Lessons 24: Limits Limit Solving Strtegy for Finl Exm Review Exm 1 Mteril For piecewise functions, you lwys nee to look t the left n right its! If f(x) is not piecewise function, plug c into f(x), i.e.,
More informationf a L Most reasonable functions are continuous, as seen in the following theorem:
Limits Suppose f : R R. To sy lim f(x) = L x mens tht s x gets closer n closer to, then f(x) gets closer n closer to L. This suggests tht the grph of f looks like one of the following three pictures: f
More informationMATH 144: Business Calculus Final Review
MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives
More informationDefinition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim
Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)
More informationAP Calculus AB First Semester Final Review
P Clculus B This review is esigne to give the stuent BSIC outline of wht nees to e reviewe for the P Clculus B First Semester Finl m. It is up to the iniviul stuent to etermine how much etr work is require
More informationOverview of Calculus I
Overview of Clculus I Prof. Jim Swift Northern Arizon University There re three key concepts in clculus: The limit, the derivtive, nd the integrl. You need to understnd the definitions of these three things,
More information2. Limits. Reading: 2.2, 2.3, 4.4 Definitions.
TOPICS COVERED IN MATH 162 This is summry of the topics we cover in 162. You my use it s outline for clss, or s review. Note: the topics re not necessrily liste in the orer presente in clss. 1. Review
More informationMAT137 Calculus! Lecture 20
officil website http://uoft.me/mat137 MAT137 Clculus! Lecture 20 Tody: 4.6 Concvity 4.7 Asypmtotes Net: 4.8 Curve Sketching 4.5 More Optimiztion Problems MVT Applictions Emple 1 Let f () = 3 27 20. 1 Find
More information( ) as a fraction. Determine location of the highest
AB Clculus Exm Review Sheet  Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if
More information( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f
More informationAB Calculus Review Sheet
AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is
More informationAntiderivatives/Indefinite Integrals of Basic Functions
Antiderivtives/Indefinite Integrls of Bsic Functions Power Rule: In prticulr, this mens tht x n+ x n n + + C, dx = ln x + C, if n if n = x 0 dx = dx = dx = x + C nd x (lthough you won t use the second
More informationOverview of Calculus
Overview of Clculus June 6, 2016 1 Limits Clculus begins with the notion of limit. In symbols, lim f(x) = L x c In wors, however close you emn tht the function f evlute t x, f(x), to be to the limit L
More information1.1 Functions. 0.1 Lines. 1.2 Linear Functions. 1.3 Rates of change. 0.2 Fractions. 0.3 Rules of exponents. 1.4 Applications of Functions to Economics
0.1 Lines Definition. Here re two forms of the eqution of line: y = mx + b y = m(x x 0 ) + y 0 ( m = slope, b = yintercept, (x 0, y 0 ) = some given point ) slopeintercept pointslope There re two importnt
More informationMain topics for the First Midterm
Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 23, Sections 4.14.8, nd Sections 5.15.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the
More information( ) Same as above but m = f x = f x  symmetric to yaxis. find where f ( x) Relative: Find where f ( x) x a + lim exists ( lim f exists.
AP Clculus Finl Review Sheet solutions When you see the words This is wht you think of doing Find the zeros Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor Find
More informationMORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)
MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give
More informationChapter 7 Notes, Stewart 8e. 7.1 Integration by Parts Trigonometric Integrals Evaluating sin m x cos n (x) dx...
Contents 7.1 Integrtion by Prts................................... 2 7.2 Trigonometric Integrls.................................. 8 7.2.1 Evluting sin m x cos n (x)......................... 8 7.2.2 Evluting
More informationsec x over the interval (, ). x ) dx dx x 14. Use a graphing utility to generate some representative integral curves of the function Curve on 5
Curve on Clcultor eperience Fin n ownlo (or type in) progrm on your clcultor tht will fin the re uner curve using given number of rectngles. Mke sure tht the progrm fins LRAM, RRAM, n MRAM. (You nee to
More information( ) where f ( x ) is a. AB/BC Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB/ Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 Find the intersection of f ( x) nd g( x). A3 Show tht f ( x) is even. A4 Show tht
More informationcritical number where f '(x) = 0 or f '(x) is undef (where denom. of f '(x) = 0)
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
More informationThe graphs of Rational Functions
Lecture 4 5A: The its of Rtionl Functions s x nd s x + The grphs of Rtionl Functions The grphs of rtionl functions hve severl differences compred to power functions. One of the differences is the behvior
More informationFinal Exam  Review MATH Spring 2017
Finl Exm  Review MATH 5  Spring 7 Chpter, 3, nd Sections 5.5.5, 5.7 Finl Exm: Tuesdy 5/9, :37:pm The following is list of importnt concepts from the sections which were not covered by Midterm Exm or.
More information1 Techniques of Integration
November 8, 8 MAT86 Week Justin Ko Techniques of Integrtion. Integrtion By Substitution (Chnge of Vribles) We cn think of integrtion by substitution s the counterprt of the chin rule for differentition.
More informationMath 142: Final Exam Formulas to Know
Mth 4: Finl Exm Formuls to Know This ocument tells you every formul/strtegy tht you shoul know in orer to o well on your finl. Stuy it well! The helpful rules/formuls from the vrious review sheets my be
More informationIntroduction and Review
Chpter 6A Notes Pge of Introuction n Review Derivtives y = f(x) y x = f (x) Evlute erivtive t x = : y = x x= f f(+h) f() () = lim h h Geometric Interprettion: see figure slope of the line tngent to f t
More information5.4, 6.1, 6.2 Handout. As we ve discussed, the integral is in some way the opposite of taking a derivative. The exact relationship
5.4, 6.1, 6.2 Hnout As we ve iscusse, the integrl is in some wy the opposite of tking erivtive. The exct reltionship is given by the Funmentl Theorem of Clculus: The Funmentl Theorem of Clculus: If f is
More informationx ) dx dx x sec x over the interval (, ).
Curve on 6 For , () Evlute the integrl, n (b) check your nswer by ifferentiting. ( ). ( ). ( ).. 6. sin cos 7. sec csccot 8. sec (sec tn ) 9. sin csc. Evlute the integrl sin by multiplying the numertor
More informationKeys to Success. 1. MC Calculator Usually only 5 out of 17 questions actually require calculators.
Keys to Success Aout the Test:. MC Clcultor Usully only 5 out of 7 questions ctully require clcultors.. FreeResponse Tips. You get ooklets write ll work in the nswer ooklet (it is white on the insie)
More informationTopics for final
Topics for 161.01 finl.1 The tngent nd velocity problems. Estimting limits from tbles. Instntneous velocity is limit of verge velocity. Slope of tngent line is limit of slope of secnt lines.. The limit
More informationUnit 5. Integration techniques
18.01 EXERCISES Unit 5. Integrtion techniques 5A. Inverse trigonometric functions; Hyperbolic functions 5A1 Evlute ) tn 1 3 b) sin 1 ( 3/) c) If θ = tn 1 5, then evlute sin θ, cos θ, cot θ, csc θ, nd
More informationSpecial notes. ftp://ftp.math.gatech.edu/pub/users/heil/1501. Chapter 1
MATH 1501 QUICK REVIEW FOR FINAL EXAM FALL 2001 C. Heil Below is quick list of some of the highlights from the sections of the text tht we hve covere. You shoul be unerstn n be ble to use or pply ech item
More informationMUSTKNOW MATERIAL FOR CALCULUS
MUSTKNOW MATERIAL FOR CALCULUS MISCELLANEOUS: intervl nottion: (, b), [, b], (, b], (, ), etc. Rewrite ricls s frctionl exponents: 3 x = x 1/3, x3 = x 3/2 etc. An impliction If A then B is equivlent to
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 200910 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationMath 107H Topics for the first exam. csc 2 x dx = cot x + C csc x cotx dx = csc x + C tan x dx = ln secx + C cot x dx = ln sinx + C e x dx = e x + C
Integrtion Mth 07H Topics for the first exm Bsic list: x n dx = xn+ + C (provided n ) n + sin(kx) dx = cos(kx) + C k sec x dx = tnx + C sec x tnx dx = sec x + C /x dx = ln x + C cos(kx) dx = sin(kx) +
More informationPractice Final. Name: Problem 1. Show all of your work, label your answers clearly, and do not use a calculator.
Nme: MATH 2250 Clculus Eric Perkerson Dte: December 11, 2015 Prctice Finl Show ll of your work, lbel your nswers clerly, nd do not use clcultor. Problem 1 Compute the following limits, showing pproprite
More information0.1 Chapters 1: Limits and continuity
1 REVIEW SHEET FOR CALCULUS 140 Some of the topics hve smple problems from previous finls indicted next to the hedings. 0.1 Chpters 1: Limits nd continuity Theorem 0.1.1 Sndwich Theorem(F 96 # 20, F 97
More informationf ) AVERAGE RATE OF CHANGE p. 87 DEFINITION OF DERIVATIVE p. 99
AVERAGE RATE OF CHANGE p. 87 The verge rte of chnge of fnction over n intervl is the mont of chnge ivie by the length of the intervl. DEFINITION OF DERIVATIVE p. 99 f ( h) f () f () lim h0 h Averge rte
More informationThe Fundamental Theorem of Calculus Part 2, The Evaluation Part
AP Clculus AB 6.4 Funmentl Theorem of Clculus The Funmentl Theorem of Clculus hs two prts. These two prts tie together the concept of integrtion n ifferentition n is regre by some to by the most importnt
More informationx dx does exist, what does the answer look like? What does the answer to
Review Guie or MAT Finl Em Prt II. Mony Decemer th 8:.m. 9:5.m. (or the 8:3.m. clss) :.m. :5.m. (or the :3.m. clss) Prt is worth 5% o your Finl Em gre. NO CALCULATORS re llowe on this portion o the Finl
More informationPrecalculus Spring 2017
Preclculus Spring 2017 Exm 3 Summry (Section 4.1 through 5.2, nd 9.4) Section P.5 Find domins of lgebric expressions Simplify rtionl expressions Add, subtrct, multiply, & divide rtionl expressions Simplify
More informationSummary Information and Formulae MTH109 College Algebra
Generl Formuls Summry Informtion nd Formule MTH109 College Algebr Temperture: F = 9 5 C + 32 nd C = 5 ( 9 F 32 ) F = degrees Fhrenheit C = degrees Celsius Simple Interest: I = Pr t I = Interest erned (chrged)
More informationBefore we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!!
Nme: Algebr II Honors PreChpter Homework Before we cn begin Ch on Rdicls, we need to be fmilir with perfect squres, cubes, etc Try nd do s mny s you cn without clcultor!!! n The nth root of n n Be ble
More informationInstantaneous Rate of Change of at a :
AP Clculus AB Formuls & Justiictions Averge Rte o Chnge o on [, ]:.r.c. = ( ) ( ) (lger slope o Deinition o the Derivtive: y ) (slope o secnt line) ( h) ( ) ( ) ( ) '( ) lim lim h0 h 0 3 ( ) ( ) '( ) lim
More informationFirst midterm topics Second midterm topics End of quarter topics. Math 3B Review. Steve. 18 March 2009
Mth 3B Review Steve 18 Mrch 2009 About the finl Fridy Mrch 20, 3pm6pm, Lkretz 110 No notes, no book, no clcultor Ten questions Five review questions (Chpters 6,7,8) Five new questions (Chpters 9,10) No
More informationMATH SS124 Sec 39 Concepts summary with examples
This note is mde for students in MTH124 Section 39 to review most(not ll) topics I think we covered in this semester, nd there s exmples fter these concepts, go over this note nd try to solve those exmples
More informationMath 1431 Section M TH 4:00 PM 6:00 PM Susan Wheeler Office Hours: Wed 6:00 7:00 PM Online ***NOTE LABS ARE MON AND WED
Mth 43 Section 4839 M TH 4: PM 6: PM Susn Wheeler swheeler@mth.uh.edu Office Hours: Wed 6: 7: PM Online ***NOTE LABS ARE MON AND WED t :3 PM to 3: pm ONLINE Approimting the re under curve given the type
More informationapproaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below
. Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.
More informationA. Limits  L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C.
A. Limits  L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( x) lim where lim f x x! c g x ( ) = or lim f ( x) = limg( x) = ". ( ) x! c limg( x) = 0 x! c x! c
More informationThe First Half of Calculus in 10 (or 13) pages
Limits n Continuity Rtes of chnge n its: The First Hlf of Clculus in 10 (or 13) pges Limit of function f t point = the vlue the function shoul tke t the point = the vlue tht the points ner tell you f shoul
More information38 Riemann sums and existence of the definite integral.
38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the xxis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These
More informationStuff You Need to Know From Calculus
Stuff You Need to Know From Clculus For the first time in the semester, the stuff we re doing is finlly going to look like clculus (with vector slnt, of course). This mens tht in order to succeed, you
More informationMath 3B Final Review
Mth 3B Finl Review Written by Victori Kl vtkl@mth.ucsb.edu SH 6432u Office Hours: R 9:4510:45m SH 1607 Mth Lb Hours: TR 12pm Lst updted: 12/06/14 This is continution of the midterm review. Prctice problems
More informationM 106 Integral Calculus and Applications
M 6 Integrl Clculus n Applictions Contents The Inefinite Integrls.................................................... Antierivtives n Inefinite Integrls.. Antierivtives.............................................................
More informationTopics Covered AP Calculus AB
Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.
More informationCalculus II: Integrations and Series
Clculus II: Integrtions nd Series August 7, 200 Integrls Suppose we hve generl function y = f(x) For simplicity, let f(x) > 0 nd f(x) continuous Denote F (x) = re under the grph of f in the intervl [,x]
More informationMA Exam 2 Study Guide, Fall u n du (or the integral of linear combinations
LESSON 0 Chpter 7.2 Trigonometric Integrls. Bsic trig integrls you should know. sin = cos + C cos = sin + C sec 2 = tn + C sec tn = sec + C csc 2 = cot + C csc cot = csc + C MA 6200 Em 2 Study Guide, Fll
More informationx 2 1 dx x 3 dx = ln(x) + 2e u du = 2e u + C = 2e x + C 2x dx = arcsin x + 1 x 1 x du = 2 u + C (t + 2) 50 dt x 2 4 dx
. Compute the following indefinite integrls: ) sin(5 + )d b) c) d e d d) + d Solutions: ) After substituting u 5 +, we get: sin(5 + )d sin(u)du cos(u) + C cos(5 + ) + C b) We hve: d d ln() + + C c) Substitute
More informationReview of basic calculus
Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below
More informationImproper Integrals. Type I Improper Integrals How do we evaluate an integral such as
Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph
More informationMath 118: Honours Calculus II Winter, 2005 List of Theorems. L(P, f) U(Q, f). f exists for each ǫ > 0 there exists a partition P of [a, b] such that
Mth 118: Honours Clculus II Winter, 2005 List of Theorems Lemm 5.1 (Prtition Refinement): If P nd Q re prtitions of [, b] such tht Q P, then L(P, f) L(Q, f) U(Q, f) U(P, f). Lemm 5.2 (Upper Sums Bound
More informationMath 113 Exam 2 Practice
Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number
More informationWe divide the interval [a, b] into subintervals of equal length x = b a n
Arc Length Given curve C defined by function f(x), we wnt to find the length of this curve between nd b. We do this by using process similr to wht we did in defining the Riemnn Sum of definite integrl:
More informationMATH , Calculus 2, Fall 2018
MATH 362, 363 Clculus 2, Fll 28 The FUNdmentl Theorem of Clculus Sections 5.4 nd 5.5 This worksheet focuses on the most importnt theorem in clculus. In fct, the Fundmentl Theorem of Clculus (FTC is rgubly
More informationMathematics 19A; Fall 2001; V. Ginzburg Practice Final Solutions
Mthemtics 9A; Fll 200; V Ginzburg Prctice Finl Solutions For ech of the ten questions below, stte whether the ssertion is true or flse ) Let fx) be continuous t x Then x fx) f) Answer: T b) Let f be differentible
More informationCalculus III Review Sheet
Clculus III Review Sheet 1 Definitions 1.1 Functions A function is f is incresing on n intervl if x y implies f(x) f(y), nd decresing if x y implies f(x) f(y). It is clled monotonic if it is either incresing
More information1 The fundamental theorems of calculus.
The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl new nme for ntiderivtive. Differentiting integrls. Theorem Suppose f is continuous
More informationChapter 5 1. = on [ 1, 2] 1. Let gx ( ) e x. . The derivative of g is g ( x) e 1
Chpter 5. Let g ( e. on [, ]. The derivtive of g is g ( e ( Write the slope intercept form of the eqution of the tngent line to the grph of g t. (b Determine the coordinte of ech criticl vlue of g. Show
More informationMath& 152 Section Integration by Parts
Mth& 5 Section 7.  Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible
More information( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f
More information1 The fundamental theorems of calculus.
The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl new nme for ntiderivtive. Differentiting integrls. Tody we provide the connection
More informationHow can we approximate the area of a region in the plane? What is an interpretation of the area under the graph of a velocity function?
Mth 125 Summry Here re some thoughts I ws hving while considering wht to put on the first midterm. The core of your studying should be the ssigned homework problems: mke sure you relly understnd those
More informationExam 1 Study Guide. Differentiation and Antidifferentiation Rules from Calculus I
Exm Stuy Guie Mth 26  Clulus II, Fll 205 The following is list of importnt onepts from eh setion tht will be teste on exm. This is not omplete list of the mteril tht you shoul know for the ourse, but
More informationDERIVATIVES NOTES HARRIS MATH CAMP Introduction
f DERIVATIVES NOTES HARRIS MATH CAMP 208. Introduction Reding: Section 2. The derivtive of function t point is the slope of the tngent line to the function t tht point. Wht does this men, nd how do we
More informationReview on Integration (Secs ) Review: Sec Origins of Calculus. Riemann Sums. New functions from old ones.
Mth 20B Integrl Clculus Lecture Review on Integrtion (Secs. 5.  5.3) Remrks on the course. Slide Review: Sec. 5.5.3 Origins of Clculus. Riemnn Sums. New functions from old ones. A mthemticl description
More informationPolynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230
Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given
More informationIf we have a function f(x) which is welldefined for some a x b, its integral over those two values is defined as
Y. D. Chong (26) MH28: Complex Methos for the Sciences 2. Integrls If we hve function f(x) which is wellefine for some x, its integrl over those two vlues is efine s N ( ) f(x) = lim x f(x n ) where x
More informationLesson 1: Quadratic Equations
Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring
More informationf(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral
Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one
More informationExam 3 Review. Lesson 19: Concavity, Inflection Points, and the Second Derivative Test. Lesson 20: Absolute Extrema on an Interval
Exam 3 Review Lessons 1718: Relative Extrema, Critical Numbers, an First Derivative Test (from exam 2 review neee for curve sketching) Critical Numbers: where the erivative of a function is zero or unefine.
More informationLoudoun Valley High School Calculus Summertime Fun Packet
Loudoun Vlley High School Clculus Summertime Fun Pcket We HIGHLY recommend tht you go through this pcket nd mke sure tht you know how to do everything in it. Prctice the problems tht you do NOT remember!
More informationIntegral points on the rational curve
Integrl points on the rtionl curve y x bx c x ;, b, c integers. Konstntine Zeltor Mthemtics University of Wisconsin  Mrinette 750 W. Byshore Street Mrinette, WI 5443453 Also: Konstntine Zeltor P.O. Box
More informationREVIEW Chapter 1 The Real Number System
Mth 7 REVIEW Chpter The Rel Number System In clss work: Solve ll exercises. (Sections. &. Definition A set is collection of objects (elements. The Set of Nturl Numbers N N = {,,,, 5, } The Set of Whole
More informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More informationIdentify graphs of linear inequalities on a number line.
COMPETENCY 1.0 KNOWLEDGE OF ALGEBRA SKILL 1.1 Identify grphs of liner inequlities on number line.  When grphing firstdegree eqution, solve for the vrible. The grph of this solution will be single point
More informationMath 113 Exam 1Review
Mth 113 Exm 1Review September 26, 2016 Exm 1 covers 6.17.3 in the textbook. It is dvisble to lso review the mteril from 5.3 nd 5.5 s this will be helpful in solving some of the problems. 6.1 Are Between
More informationReversing the Chain Rule. As we have seen from the Second Fundamental Theorem ( 4.3), the easiest way to evaluate an integral b
Mth 32 Substitution Method Stewrt 4.5 Reversing the Chin Rule. As we hve seen from the Second Fundmentl Theorem ( 4.3), the esiest wy to evlute n integrl b f(x) dx is to find n ntiderivtive, the indefinite
More informationSection Areas and Distances. Example 1: Suppose a car travels at a constant 50 miles per hour for 2 hours. What is the total distance traveled?
Section 5.  Ares nd Distnces Exmple : Suppose cr trvels t constnt 5 miles per hour for 2 hours. Wht is the totl distnce trveled? Exmple 2: Suppose cr trvels 75 miles per hour for the first hour, 7 miles
More information1 Functions Defined in Terms of Integrals
November 5, 8 MAT86 Week 3 Justin Ko Functions Defined in Terms of Integrls Integrls llow us to define new functions in terms of the bsic functions introduced in Week. Given continuous function f(), consider
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More informationMath Bootcamp 2012 Calculus Refresher
Mth Bootcmp 0 Clculus Refresher Exponents For ny rel number x, the powers of x re : x 0 =, x = x, x = x x, etc. Powers re lso clled exponents. Remrk: 0 0 is indeterminte. Frctionl exponents re lso clled
More informationAP Calculus Multiple Choice: BC Edition Solutions
AP Clculus Multiple Choice: BC Edition Solutions J. Slon Mrch 8, 04 ) 0 dx ( x) is A) B) C) D) E) Divergent This function inside the integrl hs verticl symptotes t x =, nd the integrl bounds contin this
More informationINTRODUCTION TO INTEGRATION
INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide
More informationLogarithmic Functions
Logrithmic Functions Definition: Let > 0,. Then log is the number to which you rise to get. Logrithms re in essence eponents. Their domins re powers of the bse nd their rnges re the eponents needed to
More informationChapter 1: Fundamentals
Chpter 1: Fundmentls 1.1 Rel Numbers Types of Rel Numbers: Nturl Numbers: {1, 2, 3,...}; These re the counting numbers. Integers: {... 3, 2, 1, 0, 1, 2, 3,...}; These re ll the nturl numbers, their negtives,
More informationIf u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then f(g(x))g (x) dx = f(u) du
Integrtion by Substitution: The Fundmentl Theorem of Clculus demonstrted the importnce of being ble to find ntiderivtives. We now introduce some methods for finding ntiderivtives: If u = g(x) is differentible
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationDisclaimer: This Final Exam Study Guide is meant to help you start studying. It is not necessarily a complete list of everything you need to know.
Disclimer: This is ment to help you strt studying. It is not necessrily complete list of everything you need to know. The MTH 33 finl exm minly consists of stndrd response questions where students must
More informationBasic Derivative Properties
Bsic Derivtive Properties Let s strt this section by remining ourselves tht the erivtive is the slope of function Wht is the slope of constnt function? c FACT 2 Let f () =c, where c is constnt Then f 0
More information