# B Veitch. Calculus I Study Guide

Save this PDF as:

Size: px
Start display at page:

Download "B Veitch. Calculus I Study Guide"

## 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 left-hn, right-hn, 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 slope-intercept 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. x-intercepts occur when f(x) = 0 ii. y-intercept occurs when x = 0 (b) Fin ny verticl, horizontl symptotes, or slnt symptotes. i. Verticl Asymptote: Fin ll x-vlues 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 x-vlues 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 x-vlues where f (x) = 0 or when f (x) oes not exist. ii. Plot these x-vlues 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 Left-hn b Right-hn 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 2-4: 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.,

### f 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

### MATH 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

### Definition 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)

### AP 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

### Overview 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,

### 2. 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

### MAT137 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

### ( ) 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

### ( ) 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

### AB 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

### Anti-derivatives/Indefinite Integrals of Basic Functions

Anti-derivtives/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

### Overview 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

### 1.1 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 = y-intercept, (x 0, y 0 ) = some given point ) slope-intercept point-slope There re two importnt

### Main topics for the First Midterm

Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 2-3, Sections 4.1-4.8, nd Sections 5.1-5.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the

### ( ) Same as above but m = f x = f x - symmetric to y-axis. 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 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

### Chapter 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

### sec 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

### ( ) 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

### critical 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

### The 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

### Final 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, :3-7:pm The following is list of importnt concepts from the sections which were not covered by Midterm Exm or.

### 1 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.

### Math 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

### Introduction 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

### 5.4, 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

### x ) 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

### Keys 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.. Free-Response Tips. You get ooklets write ll work in the nswer ooklet (it is white on the insie)

### Topics 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

### Unit 5. Integration techniques

18.01 EXERCISES Unit 5. Integrtion techniques 5A. Inverse trigonometric functions; Hyperbolic functions 5A-1 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

### Special 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

### MUST-KNOW MATERIAL FOR CALCULUS

MUST-KNOW 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

### Review of Calculus, cont d

Jim Lmbers MAT 460 Fll Semester 2009-10 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

### Math 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) +

### Practice 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

### 0.1 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

### f ) 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

### The 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

### x 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

### Precalculus 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

### Summary 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)

### Before 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 Pre-Chpter 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

### Instantaneous 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

### First 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, 3pm-6pm, Lkretz 110 No notes, no book, no clcultor Ten questions Five review questions (Chpters 6,7,8) Five new questions (Chpters 9,10) No

### MATH 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

### Math 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

### approaches 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.

### A. 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

### The 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

### 38 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 x-xis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These

### Stuff 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

### Math 3B Final Review

Mth 3B Finl Review Written by Victori Kl vtkl@mth.ucsb.edu SH 6432u Office Hours: R 9:45-10:45m SH 1607 Mth Lb Hours: TR 1-2pm Lst updted: 12/06/14 This is continution of the midterm review. Prctice problems

### M 106 Integral Calculus and Applications

M 6 Integrl Clculus n Applictions Contents The Inefinite Integrls.................................................... Antierivtives n Inefinite Integrls.. Antierivtives.............................................................

### Topics 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.

### Calculus 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]

### MA 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

### x 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

### Review 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

### Improper 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

### Math 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

### Math 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

### We 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:

### MATH , Calculus 2, Fall 2018

MATH 36-2, 36-3 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

### Mathematics 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

### Calculus I-II Review Sheet

Clculus I-II 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

### 1 The fundamental theorems of calculus.

The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl- new nme for nti-derivtive. Differentiting integrls. Theorem Suppose f is continuous

### Chapter 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

### Math& 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

### ( ) 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

### 1 The fundamental theorems of calculus.

The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl- new nme for nti-derivtive. Differentiting integrls. Tody we provide the connection

### How 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

### Exam 1 Study Guide. Differentiation and Anti-differentiation 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

### DERIVATIVES 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

### Review 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

### Polynomial 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

### If we have a function f(x) which is well-defined 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 well-efine for some x, its integrl over those two vlues is efine s N ( ) f(x) = lim x f(x n ) where x

### Lesson 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

### f(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

### Exam 3 Review. Lesson 19: Concavity, Inflection Points, and the Second Derivative Test. Lesson 20: Absolute Extrema on an Interval

Exam 3 Review Lessons 17-18: 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.

### Loudoun 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!

### Integral 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 5443-453 Also: Konstntine Zeltor P.O. Box

### REVIEW 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

### ARITHMETIC 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

### Identify 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 first-degree eqution, solve for the vrible. The grph of this solution will be single point

### Math 113 Exam 1-Review

Mth 113 Exm 1-Review September 26, 2016 Exm 1 covers 6.1-7.3 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

### Reversing 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

### Section 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

### 1 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

### Improper 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

### Math 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

### AP 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

### INTRODUCTION 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

### Logarithmic 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

### Chapter 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,

### If 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 nti-derivtives. We now introduce some methods for finding ntiderivtives: If u = g(x) is differentible

### Unit #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