Loudoun Valley High School Calculus Summertime Fun Packet

Save this PDF as:
Size: px
Start display at page:

Download "Loudoun Valley High School Calculus Summertime Fun Packet"

Transcription

1 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! Solutions will be posted on our websites. Nme:

2 Comple Frctions When simplifying comple frctions, multiply by frction equl to 1 which hs numertor nd denomintor composed of the common denomintor of ll the denomintors in the comple frction. Emple: ( ) ( ) ( ) ( ) 5( )( ) 1( ) Simplify ech of the following

3 Functions To evlute function for given vlue, simply plug the vlue into the function for. Recll: f g( ) f( g( )) OR f[ g( )] red f of g of mens to plug the inside function (in this cse g ) in for in the outside function (in this cse, f). Emple: Given f ( ) 1 nd g( ) find fg. f( g( )) f( ) ( ) 1 ( 8 16) ( ( )) 16 f g Let f ( ) 1 nd g( ) 1. Find ech. 6. f () 7. g( ) 8. f( t1) 9. f g( ) 10. ( ) g f m 11. f ( h) f( ) h Let f ( ) sin Find ech ectly. 1. f 1. f Let f ( ), g( ) 5, nd h( ) 1. Find ech. 1. h f( ) 15. f g( 1) 16. g h ( )

4 f( h) f( ) Find for the given function f. h 17. f( ) f( ) 5 Intercepts nd Points of Intersection To find the intercepts, let y 0 in your eqution nd solve. To find the y intercepts, let 0 in your eqution nd solve. Emple: y int. ( Let y 0) 0 0 ( )( 1) 1or i ntercepts ( 1,0) nd (,0) yint. ( Let 0) y 0 (0) y y intercept (0, ) Find the nd y intercepts for ech. 19. y 5 0. y 1. y 16. y

5 Use substitution or elimintion method to solve the system of equtions. Emple: y169 0 y 90 Elimintion Method ( )( 5) 0 nd 5 Plug = nd 5 into one originl y y y 0 16 y y 0 y Points of Intersection (5,), (5, ) nd (,0) Substitution Method Solve one eqution for one vrible (1st eqution solved for y) y ( 169) 9 0 Plug wht y is equl to into second eqution. The rest is the sme s ( ) previous emple ( )( 5) 0 or 5 Find the point(s) of intersection of the grphs for the given equtions.. y 8 y 7. y 6 y 5. y y Intervl Nottion 6. Complete the tble with the pproprite nottion or grph. Solution Intervl Nottion Grph 1, 7 8 5

6 Solve ech eqution. Stte your nswer in BOTH intervl nottion nd grphiclly Domin nd Rnge Find the domin nd rnge of ech function. Write your nswer in INTERVAL nottion. 0. f 1. f( ). f( ) sin. ( ) 5 f( ) 1 Mth Mni Crossword 6

7 Inverses To find the inverse of function, simply switch the nd the y nd solve for the new y vlue. Emple: f( ) 1 Rewrite f() s y y= 1 Switchndy = y 1 Solve for your new y y 1 Cube both sides y1 Simplify y 1 Solve for y 1 f ( ) 1 Rewrite in inverse nottion Find the inverse for ech function.. f ( ) 1 5. f( ) if 0 Fourteen mtchsticks form two identicl equilterl tringles nd two identicl squres. Move five mtchsticks to form three identicl equilterl tringles nd three identicl squres. 7

8 Also, recll tht to PROVE one function is n inverse of nother function, you need to show tht: f( g( )) g( f( )) Emple: If: 9 f( ) nd g( ) 9 show f() nd g() re inverses of ech other f( g( )) 9 g( f( )) f( g( )) g( f( )) therefore they re inverses of ech other. Prove f nd g re inverses of ech other. 6. f( ) g( ) 7. f( ) 9, 0 g( ) 9 8 Euclid

9 Eqution of line Slope intercept form: y m b Verticl line: c (slope is undefined) Point slope form: y y1 m( 1) Horizontl line: yc (slope is 0) 8. Use slope intercept form to find the eqution of the line hving slope of nd y intercept of Determine the eqution of line pssing through the point 5, with n undefined slope. 0. Determine the eqution of line pssing through the point, with slope of Use point slope form to find the eqution of the line pssing through the point 0, 5 with slope of /.. Find the eqution of line pssing through the point, 8 nd prllel to the line y 1.. Find the eqution of line perpendiculr to the y is pssing through the point, 7.. Find the eqution of line pssing through the points, 6 nd 1,. 5. Find the eqution of line with n intercept, 0 nd y intercept 0,. 9

10 Rdin nd Degree Mesure Use degrees rdins to convert bck nd forth between degrees nd rdins Convert to degrees:. 5 6 b. 5 c..6 rdins 7. Convert to rdins:. 5 b. 17 c. 7 Angles in Stndrd Position 8. Sketch the ngle in stndrd position b. 0 c. 5 d. 1.8 rdins 10 Riddle Me This Why re mn hole covers round?

11 Reference Tringles 9. Sketch the ngle in stndrd position. Drw the reference tringle nd lbel the sides, if possible.. b. 5 c. d. 0 Qudrntls You cn determine the sine or cosine of qudrntl ngle by using the chrts. SIN COS TAN 1 0 ND Emple: sin 90 1 cos 0 0 ND 50..) sin180 b.) cos 70 c.) sin( 90 ) d.) tn e.) cos60 f.) cos( ) 11

12 Grphing Trig Functions f = sin f = cos ysin nd ycos hve period of π nd n mplitude of 1. Use the prent grphs bove to help you sketch grph of the functions below. For f AsinB C D, A mplitude, = period, B opposite of C phse shift nd D verticl shift. Grph one complete period of the function. 51. f ( ) 5sin 5. f ( ) sin 5. f( ) cos 5. f ( ) cos Trigonometric Equtions: Solve ech of the equtions for 0,. Isolte the vrible, sketch reference tringle, find ll the solutions within the given domin, 0,. Remember to double the domin when solving for double ngle. Use trig identities, if needed, to rewrite the trig functions. (See formul sheet t the end of the pcket.) sin 56. cos 1

13 57. 1 cos 58. sin sin 60. cos 1cos cos 0 6. sin cos cos 0 1

14 Inverse Trigonometric Functions: Recll: Inverse Trig Functions cn be written in one of wys: rcsin sin 1 Inverse trig functions re defined only in the qudrnts s indicted below due to their restricted domins. cos 0 sin 0 cos 0 tn 0 sin 0 tn 0 Emple: Epress the vlue of y in rdins. 1 y rctn Drw reference tringle. 1 This mens the reference ngle is 0 or. So, y = so tht it flls in the intervl from y 6 6 Answer: y For ech of the following, epress the vlue for y in rdins. 6. y rcsin 6. rccos 1 y 65. y rctn( 1) 1

15 Emple: Find the vlue without clcultor. 5 cosrctn 6 61 Drw the reference tringle in the correct qudrnt first. 5 θ Find the missing side using Pythgoren Theorem. 6 Find the rtio of the cosine of the reference tringle. cos 6 61 For ech of the following give the vlue without clcultor. 66. tn rccos secsin sin rctn sin sin

16 Circles nd Ellipses h y k r Minor Ais h yk b 1 b CENTER (h, k) Mjor Ais FOCUS (h - c, k) c FOCUS (h + c, k) For circle centered t the origin, the eqution is y r, where r is the rdius of the circle. y For n ellipse centered t the origin, the eqution is 1, where is the distnce from the center to the ellipse b long the is nd b is the distnce from the center to the ellipse long the y is. If the lrger number is under the y term, the ellipse is elongted long the y is. For our purposes in Clculus, you will not need to locte the foci. Grph the circles nd ellipses below: 70. y y y y

17 Logrithms There re TWO bses tht re used for logrithms Common Log (Bse 10) y log sme s 10 y Nturl Log (Bse e) y y ln sme s e y log rgument nswer bse Eponentil Form Logrithmic Form y y log Inverse Properties of Logrithmic Epnsion Eponentil Equivlent 1. log 1 0 nd ln1 0 comes from the fcts tht 0 1 nd 0 e 1. log 1 nd ln e 1 comes from the fcts tht 1 1 nd e e log. M ln M M nd e M comes from rewriting ech s logrithmic function log M log M nd ln M e ln M r. log r r nd ln e r comes from rewriting logrithm in eponentil form r r r r nd e e (remember the understood e in the ln) M N M N log MN log M log N comes from the eponent rule log M log M log N N comes from the eponent rule M N M N 7. r log M rlog M proved by using log M rewritten eponentilly, ech side rised to the r power then simplified 8. if log M log N then M=N comes from the eponent rule if M N then M=N 17

18 Epnd or condense: log 1 log 1 log log Solve: 76. log log log 77. log log 9 18

19 Formul Sheet Reciprocl Identities: 1 csc sin 1 sec cos 1 cot tn Quotient Identities: sin tn cos cos cot sin Pythgoren Identities: sin cos 1 tn 1 sec 1cot csc Double Angle Identities: sin sin cos tn tn 1 tn cos cos sin 1 sin cos 1 Logrithms: y log is equivlent to y Product property: log mn log m log n b b b m Quotient property: log log m log n b b b p Power property: log m plog m Property of equlity: If log m log b b b b n n, then m = n Chnge of bse formul: Derivtive of Function: log log n log b b n Slope of tngent line to curve or the derivtive: lim Slope intercept form: y m b Point slope form: y y1 m( 1) Stndrd form: A By C 0 19

TO: Next Year s AP Calculus Students

TO: Next Year s AP Calculus Students TO: Net Yer s AP Clculus Students As you probbly know, the students who tke AP Clculus AB nd pss the Advnced Plcement Test will plce out of one semester of college Clculus; those who tke AP Clculus BC

More information

AP Calculus AB Summer Packet

AP Calculus AB Summer Packet AP Clculus AB Summer Pcket Nme: Welcome to AP Clculus AB! Congrtultions! You hve mde it to one of the most dvnced mth course in high school! It s quite n ccomplishment nd you should e proud of yourself

More information

AP Calculus AB Summer Packet

AP Calculus AB Summer Packet AP Clculus AB Summer Pcket Nme: Welcome to AP Clculus AB! Congrtultions! You hve mde it to one of the most dvnced mth course in high school! It s quite n ccomplishment nd you should e proud of yourself

More information

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!!!

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

More information

Sections 1.3, 7.1, and 9.2: Properties of Exponents and Radical Notation

Sections 1.3, 7.1, and 9.2: Properties of Exponents and Radical Notation Sections., 7., nd 9.: Properties of Eponents nd Rdicl Nottion Let p nd q be rtionl numbers. For ll rel numbers nd b for which the epressions re rel numbers, the following properties hold. i = + p q p q

More information

Math 1102: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 230 Homework 2 solutions

Math 1102: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 230 Homework 2 solutions Mth 1102: Clculus I (Mth/Sci mjors) MWF 3pm, Fulton Hll 230 Homework 2 solutions Plese write netly, nd show ll work. Cution: An nswer with no work is wrong! Do the following problems from Chpter III: 6,

More information

Unit 1 Exponentials and Logarithms

Unit 1 Exponentials and Logarithms HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 1 Unit 1 Eponentils nd Logrithms (2) Eponentil Functions (3) The number e (4) Logrithms (5) Specil Logrithms (7) Chnge of Bse Formul (8) Logrithmic Functions (10)

More information

REVIEW SHEET FOR PRE-CALCULUS MIDTERM

REVIEW SHEET FOR PRE-CALCULUS MIDTERM . If A, nd B 8, REVIEW SHEET FOR PRE-CALCULUS MIDTERM. For the following figure, wht is the eqution of the line?, write n eqution of the line tht psses through these points.. Given the following lines,

More information

Chapter 1 - Functions and Variables

Chapter 1 - Functions and Variables Business Clculus 1 Chpter 1 - Functions nd Vribles This Acdemic Review is brought to you free of chrge by preptests4u.com. Any sle or trde of this review is strictly prohibited. Business Clculus 1 Ch 1:

More information

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector

More information

approaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below

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.

More information

Higher Maths. Self Check Booklet. visit for a wealth of free online maths resources at all levels from S1 to S6

Higher Maths. Self Check Booklet. visit   for a wealth of free online maths resources at all levels from S1 to S6 Higher Mths Self Check Booklet visit www.ntionl5mths.co.uk for welth of free online mths resources t ll levels from S to S6 How To Use This Booklet You could use this booklet on your own, but it my be

More information

Prerequisite Knowledge Required from O Level Add Math. d n a = c and b = d

Prerequisite Knowledge Required from O Level Add Math. d n a = c and b = d Prerequisite Knowledge Required from O Level Add Mth ) Surds, Indices & Logrithms Rules for Surds. b= b =. 3. 4. b = b = ( ) = = = 5. + b n = c+ d n = c nd b = d Cution: + +, - Rtionlising the Denomintor

More information

Lesson 1: Quadratic Equations

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

More information

Summer Work Packet for MPH Math Classes

Summer Work Packet for MPH Math Classes Summer Work Pcket for MPH Mth Clsses Students going into Pre-clculus AC Sept. 018 Nme: This pcket is designed to help students sty current with their mth skills. Ech mth clss expects certin level of number

More information

Math 113 Exam 2 Practice

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

More information

The discriminant of a quadratic function, including the conditions for real and repeated roots. Completing the square. ax 2 + bx + c = a x+

The discriminant of a quadratic function, including the conditions for real and repeated roots. Completing the square. ax 2 + bx + c = a x+ .1 Understnd nd use the lws of indices for ll rtionl eponents.. Use nd mnipulte surds, including rtionlising the denomintor..3 Work with qudrtic nd their grphs. The discriminnt of qudrtic function, including

More information

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac

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

More information

Calculus AB. For a function f(x), the derivative would be f '(

Calculus AB. For a function f(x), the derivative would be f '( lculus AB Derivtive Formuls Derivtive Nottion: For function f(), the derivtive would e f '( ) Leiniz's Nottion: For the derivtive of y in terms of, we write d For the second derivtive using Leiniz's Nottion:

More information

Linear Inequalities: Each of the following carries five marks each: 1. Solve the system of equations graphically.

Linear Inequalities: Each of the following carries five marks each: 1. Solve the system of equations graphically. Liner Inequlities: Ech of the following crries five mrks ech:. Solve the system of equtions grphiclly. x + 2y 8, 2x + y 8, x 0, y 0 Solution: Considerx + 2y 8.. () Drw the grph for x + 2y = 8 by line.it

More information

Anti-derivatives/Indefinite Integrals of Basic Functions

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

More information

1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE

1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check

More information

Logarithmic Functions

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

More information

k ) and directrix x = h p is A focal chord is a line segment which passes through the focus of a parabola and has endpoints on the parabola.

k ) and directrix x = h p is A focal chord is a line segment which passes through the focus of a parabola and has endpoints on the parabola. Stndrd Eqution of Prol with vertex ( h, k ) nd directrix y = k p is ( x h) p ( y k ) = 4. Verticl xis of symmetry Stndrd Eqution of Prol with vertex ( h, k ) nd directrix x = h p is ( y k ) p( x h) = 4.

More information

Chapter 1: Logarithmic functions and indices

Chapter 1: Logarithmic functions and indices Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4

More information

A LEVEL TOPIC REVIEW. factor and remainder theorems

A LEVEL TOPIC REVIEW. factor and remainder theorems A LEVEL TOPIC REVIEW unit C fctor nd reminder theorems. Use the Fctor Theorem to show tht: ) ( ) is fctor of +. ( mrks) ( + ) is fctor of ( ) is fctor of + 7+. ( mrks) +. ( mrks). Use lgebric division

More information

Chapter 7 Notes, Stewart 8e. 7.1 Integration by Parts Trigonometric Integrals Evaluating sin m x cos n (x) dx...

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

More information

We divide the interval [a, b] into subintervals of equal length x = b a n

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:

More information

AB Calculus Review Sheet

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

More information

Quotient Rule: am a n = am n (a 0) Negative Exponents: a n = 1 (a 0) an Power Rules: (a m ) n = a m n (ab) m = a m b m

Quotient Rule: am a n = am n (a 0) Negative Exponents: a n = 1 (a 0) an Power Rules: (a m ) n = a m n (ab) m = a m b m Formuls nd Concepts MAT 099: Intermedite Algebr repring for Tests: The formuls nd concepts here m not be inclusive. You should first tke our prctice test with no notes or help to see wht mteril ou re comfortble

More information

Obj: SWBAT Recall the many important types and properties of functions

Obj: SWBAT Recall the many important types and properties of functions Obj: SWBAT Recll the mny importnt types nd properties of functions Functions Domin nd Rnge Function Nottion Trnsformtion of Functions Combintions/Composition of Functions One-to-One nd Inverse Functions

More information

I. Equations of a Circle a. At the origin center= r= b. Standard from: center= r=

I. Equations of a Circle a. At the origin center= r= b. Standard from: center= r= 11.: Circle & Ellipse I cn Write the eqution of circle given specific informtion Grph circle in coordinte plne. Grph n ellipse nd determine ll criticl informtion. Write the eqution of n ellipse from rel

More information

( ) as a fraction. Determine location of the highest

( ) 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

Identify graphs of linear inequalities on a number line.

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

More information

Edexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks

Edexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks Edexcel GCE Core Mthemtics (C) Required Knowledge Informtion Sheet C Formule Given in Mthemticl Formule nd Sttisticl Tles Booklet Cosine Rule o = + c c cosine (A) Binomil Series o ( + ) n = n + n 1 n 1

More information

( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).

( ) 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 information

Precalculus Spring 2017

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

More information

Chapter 8: Methods of Integration

Chapter 8: Methods of Integration Chpter 8: Methods of Integrtion Bsic Integrls 8. Note: We hve the following list of Bsic Integrls p p+ + c, for p sec tn + c p + ln + c sec tn sec + c e e + c tn ln sec + c ln + c sec ln sec + tn + c ln

More information

Sect 10.2 Trigonometric Ratios

Sect 10.2 Trigonometric Ratios 86 Sect 0. Trigonometric Rtios Objective : Understnding djcent, Hypotenuse, nd Opposite sides of n cute ngle in right tringle. In right tringle, the otenuse is lwys the longest side; it is the side opposite

More information

R(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of

R(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of Higher Mthemtics Ojective Test Prctice ook The digrm shows sketch of prt of the grph of f ( ). The digrm shows sketch of the cuic f ( ). R(, 8) f ( ) f ( ) P(, ) Q(, ) S(, ) Wht re the domin nd rnge of

More information

MTH 4-16a Trigonometry

MTH 4-16a Trigonometry MTH 4-16 Trigonometry Level 4 [UNIT 5 REVISION SECTION ] I cn identify the opposite, djcent nd hypotenuse sides on right-ngled tringle. Identify the opposite, djcent nd hypotenuse in the following right-ngled

More information

Chapter 1: Fundamentals

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,

More information

MA Exam 2 Study Guide, Fall u n du (or the integral of linear combinations

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

More information

Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230

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

More information

A-Level Mathematics Transition Task (compulsory for all maths students and all further maths student)

A-Level Mathematics Transition Task (compulsory for all maths students and all further maths student) A-Level Mthemtics Trnsition Tsk (compulsory for ll mths students nd ll further mths student) Due: st Lesson of the yer. Length: - hours work (depending on prior knowledge) This trnsition tsk provides revision

More information

FUNCTIONS: Grade 11. or y = ax 2 +bx + c or y = a(x- x1)(x- x2) a y

FUNCTIONS: Grade 11. or y = ax 2 +bx + c or y = a(x- x1)(x- x2) a y FUNCTIONS: Grde 11 The prbol: ( p) q or = +b + c or = (- 1)(- ) The hperbol: p q The eponentil function: b p q Importnt fetures: -intercept : Let = 0 -intercept : Let = 0 Turning points (Where pplicble)

More information

Calculus AB Section I Part A A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION

Calculus AB Section I Part A A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION lculus Section I Prt LULTOR MY NOT US ON THIS PRT OF TH XMINTION In this test: Unless otherwise specified, the domin of function f is ssumed to e the set of ll rel numers for which f () is rel numer..

More information

Read section 3.3, 3.4 Announcements:

Read section 3.3, 3.4 Announcements: Dte: 3/1/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: 1. f x = 3x 6, find the inverse, f 1 x., Using your grphing clcultor, Grph 1. f x,f

More information

ES.182A Topic 32 Notes Jeremy Orloff

ES.182A Topic 32 Notes Jeremy Orloff ES.8A Topic 3 Notes Jerem Orloff 3 Polr coordintes nd double integrls 3. Polr Coordintes (, ) = (r cos(θ), r sin(θ)) r θ Stndrd,, r, θ tringle Polr coordintes re just stndrd trigonometric reltions. In

More information

MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)

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

More information

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

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

More information

3.1 Exponential Functions and Their Graphs

3.1 Exponential Functions and Their Graphs . Eponentil Functions nd Their Grphs Sllbus Objective: 9. The student will sketch the grph of eponentil, logistic, or logrithmic function. 9. The student will evlute eponentil or logrithmic epressions.

More information

Unit 5. Integration techniques

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

More information

4 7x =250; 5 3x =500; Read section 3.3, 3.4 Announcements: Bell Ringer: Use your calculator to solve

4 7x =250; 5 3x =500; Read section 3.3, 3.4 Announcements: Bell Ringer: Use your calculator to solve Dte: 3/14/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: Use your clcultor to solve 4 7x =250; 5 3x =500; HW Requests: Properties of Log Equtions

More information

15 - TRIGONOMETRY Page 1 ( Answers at the end of all questions )

15 - TRIGONOMETRY Page 1 ( Answers at the end of all questions ) - TRIGONOMETRY Pge P ( ) In tringle PQR, R =. If tn b c = 0, 0, then Q nd tn re the roots of the eqution = b c c = b b = c b = c [ AIEEE 00 ] ( ) In tringle ABC, let C =. If r is the inrdius nd R is the

More information

. Double-angle formulas. Your answer should involve trig functions of θ, and not of 2θ. cos(2θ) = sin(2θ) =.

. Double-angle formulas. Your answer should involve trig functions of θ, and not of 2θ. cos(2θ) = sin(2θ) =. Review of some needed Trig Identities for Integrtion Your nswers should be n ngle in RADIANS rccos( 1 2 ) = rccos( - 1 2 ) = rcsin( 1 2 ) = rcsin( - 1 2 ) = Cn you do similr problems? Review of Bsic Concepts

More information

/ 3, then (A) 3(a 2 m 2 + b 2 ) = 4c 2 (B) 3(a 2 + b 2 m 2 ) = 4c 2 (C) a 2 m 2 + b 2 = 4c 2 (D) a 2 + b 2 m 2 = 4c 2

/ 3, then (A) 3(a 2 m 2 + b 2 ) = 4c 2 (B) 3(a 2 + b 2 m 2 ) = 4c 2 (C) a 2 m 2 + b 2 = 4c 2 (D) a 2 + b 2 m 2 = 4c 2 SET I. If the locus of the point of intersection of perpendiculr tngents to the ellipse x circle with centre t (0, 0), then the rdius of the circle would e + / ( ) is. There re exctl two points on the

More information

x ) dx dx x sec x over the interval (, ).

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

More information

Topics Covered AP Calculus AB

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.

More information

6.2 CONCEPTS FOR ADVANCED MATHEMATICS, C2 (4752) AS

6.2 CONCEPTS FOR ADVANCED MATHEMATICS, C2 (4752) AS 6. CONCEPTS FOR ADVANCED MATHEMATICS, C (475) AS Objectives To introduce students to number of topics which re fundmentl to the dvnced study of mthemtics. Assessment Emintion (7 mrks) 1 hour 30 minutes.

More information

= = 6 radians 15. θ = s 30 feet (2π) = 13 radians. rad θ R, we have: 180 = 60

= = 6 radians 15. θ = s 30 feet (2π) = 13 radians. rad θ R, we have: 180 = 60 SECTION - 7 CHAPTER Section. A positive ngle is produced y counterclockwise rottion from the initil side to the terminl side, negtive ngle y clockwise rottion.. Answers will vry. 5. Answers will vry. 7.

More information

= = 6 radians 15. θ = s 30 feet (2π) = 13 radians. rad θ R, we have: 180 = 60

= = 6 radians 15. θ = s 30 feet (2π) = 13 radians. rad θ R, we have: 180 = 60 SECTION - 7 CHAPTER Section. A positive ngle is produced y counterclockwise rottion from the initil side to the terminl side, negtive ngle y clockwise rottion.. Answers will vry. 5. Answers will vry. 7.

More information

SUMMER ASSIGNMENT FOR Pre-AP FUNCTIONS/TRIGONOMETRY Due Tuesday After Labor Day!

SUMMER ASSIGNMENT FOR Pre-AP FUNCTIONS/TRIGONOMETRY Due Tuesday After Labor Day! SUMMER ASSIGNMENT FOR Pre-AP FUNCTIONS/TRIGONOMETRY Due Tuesdy After Lor Dy! This summer ssignment is designed to prepre you for Functions/Trigonometry. Nothing on the summer ssignment is new. Everything

More information

Section 13.1 Right Triangles

Section 13.1 Right Triangles Section 13.1 Right Tringles Ojectives: 1. To find vlues of trigonometric functions for cute ngles. 2. To solve tringles involving right ngles. Review - - 1. SOH sin = Reciprocl csc = 2. H cos = Reciprocl

More information

MCR 3U Exam Review. 1. Determine which of the following equations represent functions. Explain. Include a graph. 2. y x

MCR 3U Exam Review. 1. Determine which of the following equations represent functions. Explain. Include a graph. 2. y x MCR U MCR U Em Review Introduction to Functions. Determine which of the following equtions represent functions. Eplin. Include grph. ) b) c) d) 0. Stte the domin nd rnge for ech reltion in question.. If

More information

Chapter 5 1. = on [ 1, 2] 1. Let gx ( ) e x. . The derivative of g is g ( x) e 1

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

More information

Definition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim

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)

More information

HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time allowed Two hours (Plus 5 minutes reading time)

HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time allowed Two hours (Plus 5 minutes reading time) HIGHER SCHOOL CERTIFICATE EXAMINATION 998 MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time llowed Two hours (Plus 5 minutes reding time) DIRECTIONS TO CANDIDATES Attempt ALL questions ALL questions

More information

Summary Information and Formulae MTH109 College Algebra

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)

More information

Algebra & Functions (Maths ) opposite side

Algebra & Functions (Maths ) opposite side Instructor: Dr. R.A.G. Seel Trigonometr Algebr & Functions (Mths 0 0) 0th Prctice Assignment hpotenuse hpotenuse side opposite side sin = opposite hpotenuse tn = opposite. Find sin, cos nd tn in 9 sin

More information

CONIC SECTIONS. Chapter 11

CONIC SECTIONS. Chapter 11 CONIC SECTIONS Chpter. Overview.. Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig..). Fig.. Suppose we rotte the line m round

More information

Mathematics Extension 2

Mathematics Extension 2 00 HIGHER SCHOOL CERTIFICATE EXAMINATION Mthemtics Etension Generl Instructions Reding time 5 minutes Working time hours Write using blck or blue pen Bord-pproved clcultors my be used A tble of stndrd

More information

Chapter 3 Exponential and Logarithmic Functions Section 3.1

Chapter 3 Exponential and Logarithmic Functions Section 3.1 Chpter 3 Eponentil nd Logrithmic Functions Section 3. EXPONENTIAL FUNCTIONS AND THEIR GRAPHS Eponentil Functions Eponentil functions re non-lgebric functions. The re clled trnscendentl functions. The eponentil

More information

Practice Final. Name: Problem 1. Show all of your work, label your answers clearly, and do not use a calculator.

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

More information

. Double-angle formulas. Your answer should involve trig functions of θ, and not of 2θ. sin 2 (θ) =

. Double-angle formulas. Your answer should involve trig functions of θ, and not of 2θ. sin 2 (θ) = Review of some needed Trig. Identities for Integrtion. Your nswers should be n ngle in RADIANS. rccos( 1 ) = π rccos( - 1 ) = 2π 2 3 2 3 rcsin( 1 ) = π rcsin( - 1 ) = -π 2 6 2 6 Cn you do similr problems?

More information

Calculus AB Bible. (2nd most important book in the world) (Written and compiled by Doug Graham)

Calculus AB Bible. (2nd most important book in the world) (Written and compiled by Doug Graham) PG. Clculus AB Bile (nd most importnt ook in the world) (Written nd compiled y Doug Grhm) Topic Limits Continuity 6 Derivtive y Definition 7 8 Derivtive Formuls Relted Rtes Properties of Derivtives Applictions

More information

Warm-up for Honors Calculus

Warm-up for Honors Calculus Summer Work Assignment Wrm-up for Honors Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Honors Clculus in the fll of 018. Due Dte: The

More information

Are You Ready for PreCalculus? Summer Packet **Required for all PreCalculus CP and Honors students**

Are You Ready for PreCalculus? Summer Packet **Required for all PreCalculus CP and Honors students** Are You Redy for PreClculus? Summer Pcket **Required for ll PreClculus CP nd Honors students** Pge of The PreClculus course prepres students for Clculus nd college science courses. In order to ccomplish

More information

Year 12 Mathematics Extension 2 HSC Trial Examination 2014

Year 12 Mathematics Extension 2 HSC Trial Examination 2014 Yer Mthemtics Etension HSC Tril Emintion 04 Generl Instructions. Reding time 5 minutes Working time hours Write using blck or blue pen. Blck pen is preferred. Bord-pproved clcultors my be used A tble of

More information

Improper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.

Improper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral. Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:

More information

THE DISCRIMINANT & ITS APPLICATIONS

THE DISCRIMINANT & ITS APPLICATIONS THE DISCRIMINANT & ITS APPLICATIONS The discriminnt ( Δ ) is the epression tht is locted under the squre root sign in the qudrtic formul i.e. Δ b c. For emple: Given +, Δ () ( )() The discriminnt is used

More information

Mathematics Extension 1

Mathematics Extension 1 04 Bored of Studies Tril Emintions Mthemtics Etension Written by Crrotsticks & Trebl. Generl Instructions Totl Mrks 70 Reding time 5 minutes. Working time hours. Write using blck or blue pen. Blck pen

More information

ntegration (p3) Integration by Inspection When differentiating using function of a function or the chain rule: If y = f(u), where in turn u = f(x)

ntegration (p3) Integration by Inspection When differentiating using function of a function or the chain rule: If y = f(u), where in turn u = f(x) ntegrtion (p) Integrtion by Inspection When differentiting using function of function or the chin rule: If y f(u), where in turn u f( y y So, to differentite u where u +, we write ( + ) nd get ( + ) (.

More information

KEY CONCEPTS. satisfies the differential equation da. = 0. Note : If F (x) is any integral of f (x) then, x a

KEY CONCEPTS. satisfies the differential equation da. = 0. Note : If F (x) is any integral of f (x) then, x a KEY CONCEPTS THINGS TO REMEMBER :. The re ounded y the curve y = f(), the -is nd the ordintes t = & = is given y, A = f () d = y d.. If the re is elow the is then A is negtive. The convention is to consider

More information

7h1 Simplifying Rational Expressions. Goals:

7h1 Simplifying Rational Expressions. Goals: h Simplifying Rtionl Epressions Gols Fctoring epressions (common fctor, & -, no fctoring qudrtics) Stting restrictions Epnding rtionl epressions Simplifying (reducin rtionl epressions (Kürzen) Adding nd

More information

First Semester Review Calculus BC

First Semester Review Calculus BC First Semester Review lculus. Wht is the coordinte of the point of inflection on the grph of Multiple hoice: No lcultor y 3 3 5 4? 5 0 0 3 5 0. The grph of piecewise-liner function f, for 4, is shown below.

More information

MA Lesson 21 Notes

MA Lesson 21 Notes MA 000 Lesson 1 Notes ( 5) How would person solve n eqution with vrible in n eponent, such s 9? (We cnnot re-write this eqution esil with the sme bse.) A nottion ws developed so tht equtions such s this

More information

Main topics for the Second Midterm

Main topics for the Second Midterm Min topics for the Second Midterm The Midterm will cover Sections 5.4-5.9, Sections 6.1-6.3, nd Sections 7.1-7.7 (essentilly ll of the mteril covered in clss from the First Midterm). Be sure to know the

More information

Math 113 Exam 1-Review

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

More information

Space Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space.

Space Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space. Clculus 3 Li Vs Spce Curves Recll the prmetric equtions of curve in xy-plne nd compre them with prmetric equtions of curve in spce. Prmetric curve in plne x = x(t) y = y(t) Prmetric curve in spce x = x(t)

More information

MATHEMATICS PART A. 1. ABC is a triangle, right angled at A. The resultant of the forces acting along AB, AC

MATHEMATICS PART A. 1. ABC is a triangle, right angled at A. The resultant of the forces acting along AB, AC FIITJEE Solutions to AIEEE MATHEMATICS PART A. ABC is tringle, right ngled t A. The resultnt of the forces cting long AB, AC with mgnitudes AB nd respectively is the force long AD, where D is the AC foot

More information

Optimization Lecture 1 Review of Differential Calculus for Functions of Single Variable.

Optimization Lecture 1 Review of Differential Calculus for Functions of Single Variable. Optimiztion Lecture 1 Review of Differentil Clculus for Functions of Single Vrible http://users.encs.concordi.c/~luisrod, Jnury 14 Outline Optimiztion Problems Rel Numbers nd Rel Vectors Open, Closed nd

More information

Mathematics. Area under Curve.

Mathematics. Area under Curve. Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding

More information

Math Calculus with Analytic Geometry II

Math Calculus with Analytic Geometry II orem of definite Mth 5.0 with Anlytic Geometry II Jnury 4, 0 orem of definite If < b then b f (x) dx = ( under f bove x-xis) ( bove f under x-xis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem

More information

a 2 +x 2 x a 2 -x 2 Figure 1: Triangles for Trigonometric Substitution

a 2 +x 2 x a 2 -x 2 Figure 1: Triangles for Trigonometric Substitution I.B Trigonometric Substitution Uon comletion of the net two sections, we will be ble to integrte nlyticlly ll rtionl functions of (t lest in theory). We do so by converting binomils nd trinomils of the

More information

( ) 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.

( ) 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 information

Advanced Algebra & Trigonometry Midterm Review Packet

Advanced Algebra & Trigonometry Midterm Review Packet Nme Dte Advnced Alger & Trigonometry Midterm Review Pcket The Advnced Alger & Trigonometry midterm em will test your generl knowledge of the mteril we hve covered since the eginning of the school yer.

More information

T M S C A M I D D L E S C H O O L M A T H E M A T I C S R E G I O N A L T E S T M A R C H 9,

T M S C A M I D D L E S C H O O L M A T H E M A T I C S R E G I O N A L T E S T M A R C H 9, T M S C A M I D D L E S C H O O L M A T H E M A T I C S R E G I O N A L T E S T M A R C H 9, 0 GENERAL DIRECTIONS. Aout this test: A. You will e given 0 minutes to tke this test. B. There re 0 prolems

More information

Logarithms. Logarithm is another word for an index or power. POWER. 2 is the power to which the base 10 must be raised to give 100.

Logarithms. Logarithm is another word for an index or power. POWER. 2 is the power to which the base 10 must be raised to give 100. Logrithms. Logrithm is nother word for n inde or power. THIS IS A POWER STATEMENT BASE POWER FOR EXAMPLE : We lred know tht; = NUMBER 10² = 100 This is the POWER Sttement OR 2 is the power to which the

More information

The semester B examination for Algebra 2 will consist of two parts. Part 1 will be selected response. Part 2 will be short answer.

The semester B examination for Algebra 2 will consist of two parts. Part 1 will be selected response. Part 2 will be short answer. ALGEBRA B Semester Em Review The semester B emintion for Algebr will consist of two prts. Prt will be selected response. Prt will be short nswer. Students m use clcultor. If clcultor is used to find points

More information