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

Size: px
Start display at page:

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

Transcription

1 . 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. The grph of n eponentil function depends f ( ) on the vlue of. > 1 0 < < 1 y 5 (-1, 1/) (1,) y 5 (-1, 1/) 1 (1,) Points on the grph: (-1, 1/), (0,1), (1, ) Properties of eponentil functions 1. The domin is the set of ll rel numbers: Df = R. The rnge is the set of positive numbers: Rf = (0, +). (This mens tht is lwys positive, tht is > 0 for ll. The eqution = negtive number hs no solution). There re no -intercepts. The y-intercept is (0, 1) 5. The -is (line y = 0) is horizontl symptote 6. An eponentil function is incresing when > 1 nd decresing when 0 < < 1 7. An eponentil function is one to one, nd therefore hs the inverse. The inverse of the eponentil function f() = is rithmic function g() = () 8. Since n eponentil function is one to one we hve the following property: If u = v, then u = v. (This property is used when solving eponentil equtions tht could be rewritten in the form u = v.) Nturl eponentil function is the function f() = e, where e is n irrtionl number, e n The number e is defined s the number to which the epression ( 1 1 ) n pproches s n becomes lrger nd lrger. Since e > 1, the grph of the nturl eponentil function is s below

2 5 y (1,e) (-1, 1/e) Emple: Use trnsformtions to grph f() = - -. Strt with bsic function nd use one trnsformtion t time. Show ll intermedite grphs. This function is obtined from the grph of y = by first reflecting it bout y-is (obtining y = - ) nd then shifting the grph down by units. Mke sure to plot the three points on the grph of the bsic function! Remrk: Function y = hs horizontl symptote, so remember to shift it too when performing shift up/down y = y = y = - Emple: Use trnsformtions to grph f() = e -1. Strt with bsic function nd use one trnsformtion t time. Show ll intermedite grphs. Bsic function: y = e y = e -1 (shift to the right by1)

3 y= e -1 (horizontl compression times) y = e -1 ( verticl stretch times) Emple: Solve (i) (ii) (iii) Rewrite the eqution in the form u = v Since =, we cn rewrite the eqution s Using properties of eponents we get. Use property 8 of eponentil functions to conclude tht u = v Since we hve =. Solve the eqution u = v ( 1) / Solution set = {0, ½ }. Logrithmic functions A rithmic function f() = (), > 0, 1, > 0 (rithm to the bse of ) is the inverse of the eponentil function y =. Therefore, we hve the following properties for this function (s the inverse function) (I) y = () if nd only if y = Emple: This reltionship gives the definition of (): () is n eponent to which the bse must be rised to obtin ) (8) is n eponent to which must be rised to obtin 8 (we cn write this s = 8) Clerly this eponent is, thus (8) = b) 1/ (9) is n eponent to which 1/ must be rised to obtin 9: ( 1/ ) = 9. Solving this eqution for, we get =, nd = or = -. Thus 1/ (9) = -.

4 c) () is n eponent to which must be rised to obtin : =. We know tht such number eists, since is in the rnge of the eponentil function y = (there is point with y-coordinte on the grph of this function) but we re not ble to find it using trditionl methods. If we wnt to refer to this number, we use (). The reltionship in (I) llows us to move from eponent to rithm nd vice vers Emple: - Chnge the given rithmic epression into eponentil form: = The eponentil form is: =. Notice tht this process llowed us to find vlue of, or to solve the eqution () = - Chnge the given eponentil form to the rithmic one: =. Since is the eponent to which is rised to get, we hve = (). Note tht the bse of the eponent is lwys the sme s the bse of the rithm. Common rithm is the rithm with the bse 10. Customrily, the bse 10 is omitted when writing this rithm: 10 () = () Nturl rithm is the rithm with the bse e (the inverse of y = e ): ln() = e () (II) Domin of rithmic function = (0, ) (We cn tke rithm of positive number only.) Rnge of rithmic function = (-, + ) (III) ( ) =, for ll rel numbers ( ), for ll > 0 Emple 5 = 5, lne () =,, e ln7 = 7 (IV) Grph of f() = () is symmetric to the grph of y = bout the line y= > 1 0 < < 1 y 5 y = y= (1,) (-1, 1/) 1 (,1) y = () (1/, -1) y y = 5 y= (-1, 1/) (,1) 1 (1,) (1/, -1) - y = () Points on the grph of y = () : (1/, -1), (1,0), (, 1) (V) The -intercept is (1, 0). (VI) There is no y-intercept (VII) The y-is (the line = 0) is the verticl symptote (VIII) A rithmic function is incresing when > 1 nd decresing when 0 < < 1 (IX) A rithmic function is one to one. Its inverse is the eponentil function

5 (X) Becuse rithmic function is one to one we hve the following property: If (u) = (v), then u = v (This property is used to solve rithmic equtions tht cn be rewritten in the form (u) = (v).) Emple: Use trnsformtions to grph f() = - (-1) +. Strt with bsic function nd use one trnsformtion t time. Show ll intermedite grphs. Plot the three points on the grph of the bsic function ) y = () b) y = (-1) c) y = (-1) d) y =- (-1) e) y = - (-1) + Remrk: Since rithmic function hs verticl symptote, do not forget to shift it when shifting left/right Emple: Find the domin of the following functions (A rithm is defined only for positive (> 0) vlues) ) f() = 1/ ( ) Df: > 0 = 0 = =

6 Df = (, ) (, ) b) g()= ln 9 Dg: = 0 9 = 0 = - = 9 = -/ = use the test points to determine the sign in ech intervl Dg= (, / ) (, ) Emple: Solve the following equtions ) 5 ( + + ) = (i) Find the domin of the rithm(s) + + > = 0 = 1 1 (1)() 1 15 not rel number (ii) Since y = + + hs no -intercepts nd the grph is prbol tht opens up, the grph must lwys sty bove -is. Therefore, + + > 0 for ll Chnge the eqution to the eponentil form nd solve + + = = = 0 = 1 1 (1)( 1) 1 85 since there re no restrictions on, bove numbers re solutions of the eqution.

7 b) e -+1 = 1 This is n eponentil eqution tht cn be solved by chnging it to the rithmic form -+ 1 = e (1) -+1 = ln(1) -= -1 + ln1 = 1 ln1 1 ln1 Since this is n eponentil equtions, there re no restrictions on. Solution is = Properties of rithms: Suppose > 0, 1 nd M, N > 0. Properties of rithms 1 ln1 (i) (1) = 0 () = 1 Emple: (1) = 0 15 (15)= 1 ln(1) = 0 ln(e) = 1 (ii) M M Emple: (7) e ln() = (iii) ( r ) = r Emple: ( ) = ln(e ) = (iv) (MN) = (M) + (N) Emple : 5 (10)= 5 (5)+ 5 () (M) + (N) = (MN) ln(+1) + ln(-1)= ln[(+1)(-1)] (v) Emple: M ( M ) ( N) N 15 (15) () M 1 ( M ) ( N) (1) () N (vi) (M r )= r (M) Emple: ( ) = () r (M) = (M r ) 5 (+1)= [(+1) 5 ] (vii) If M = N, then (M) = (N) Emple: if =, then () = () If (M) = (N), then M = N if (-1) = (-5), then -1 = -5 (viii) Chnge of the bse formul b M ( M ) ( ), where b is ny positive number different thn 1 b In prticulr, M lnm ( M ) nd ( M ) ( ) ln( ) This formul is used to find vlues of rithms using clcultor.

8 Emple: Evlute () ln () ln() ( ) Emple : Write s sum/difference of rithms. Epress powers s product. 1 ( ) [ ( ) ] 1 1 ( ) [( ) ] ( ) ( ) 1 ( Emple: Write s single rithm 1 1/ 1) (+1) (-1)- () = = [(+1) ] [(-1) ] ()= ( 1) = ( 1) ( 1) ( 1) ( ) ( 1) ( 1). Eponentil nd rithmic equtions A rithmic eqution is n eqution tht contins vrible inside rithm. Since rithm is defined only for positive numbers, before solving rithmic eqution you must find its domin ( lterntively, you cn check the pprent solutions by plugging them into the originl eqution nd checking whether ll rithms re well defined). There re two types of rithmic equtions: (A) Equtions reducible to the form (u) = r, where u is n epression tht contins vrible nd r is rel number To solve such eqution chnge it to the eponentil form r = u nd solve. Emple: Solve (-1)+ () = 5 (i) (ii) (iii) Determine the domin of the eqution. (Wht is inside of ny rithm must be positive) -1 > 0 > 1 (Only numbers greter thn 1 cn be solutions of this eqution) Use properties of rithms to write the left hnd side s single rithm (-1) + () = 5 ((-1) ) = 5 Chnge to the eponentil form 5 = (-1)

9 (iv) Solve = (-1) / = (-1) -1 = / = 1 + / (v) Since = 1 + / is greter thn 1, it is the solution (B) Equtions reducible to the form (u) = (v). To solve such eqution use the (vii) property of rithms to get the eqution u = v. Solve the eqution. Emple: Solve 5 () + 5 (-)= 5 (+ ). (i) Determine the domin of the eqution. (Wht is inside of ny rithm must be positive) > 0 nd > 0 nd + > 0 >0 nd > nd > - If is to stisfy ll these inequlities, then > (Only numbers greter thn cn be solutions of this eqution) (ii) Use properties of rithms to write ech side of the eqution s single rithm 5 ((-)) = 5 ( + ) (iii) Since the rithms re equl ( (M) = (N), we must hve (M = N) (-) = + (iv) Solve (-) = + = + - = 0 (-)(+1) = 0 = or = -1 (v) Since ny solution must be greter thn, only = is the solution Eponentil equtions These re equtions in which vrible ppers in the eponent. Since eponentil functions re defined for ll rel numbers, there re no restrictions on vrible nd we do not hve to check the solutions. There re three types of eponentil equtions: (A) Equtions tht cn be reduced to the form u = r, where u is n epression tht contins vrible nd r is positive rel number. If r is negtive or 0, the eqution hs no solution.

10 To solve such eqution, chnge into rithmic form nd solve Emple: Solve -1 = 5 (i) Write the eqution in the desired form (eponent = number) -1 = 5/ (ii) Chnge to the rithmic form -1 = (5/) (iii) Solve = 1 + (5/) = 1 (5 / ) To find n pproimte vlue, use the chnge of the bse formul to rewrite (5/) s (5/)/ (B) Equtions tht cn be reduced to the form u = v. To solve such n eqution use the property of eponentil functions tht sys tht if u = v, then u = v nd solve it. 6 Emple Solve (i) 16 Use the properties of eponents to write the eqution in the desired form. Notice tht ll bses (16,, ) re powers of, 16 =, = 1, = (ii) Use the property (7) + = 1 6 (iii) Solve + 1 = 0 (+6)(-) = 0 = - 6 or = Solutions: -6, (C) Equtions tht cn be reduced to the form u = b v To solve such eqution pply the (or ln ) to both sides of the eqution (property (vii) of rithms), use the property of rithms to bring the u nd v outside of the rithms nd solve for the vrible. Keep in mind tht () nd (b) re just numbers ( like 1. or ) Emple: Solve +1 = 5 1- (i) Apply to both sides ( +1 ) = (5 1- ) (ii) Use properties of rithms. (Enclose the powers into the prentheses) (+1)() = (1-)(5) (iii) Solve

11 Eliminte prentheses ()+ () = (5) -(5) Bring the terms with to the left hnd side () + (5) = (5) () Fctor out (()+(5)) = (5) () ( 5) ( ) Divide, to find = ( ) (5) You could use properties of rithms to write the solution s = (5 / ) ( 5 ) (5 / ) ( 50) If n eponentil eqution cnnot be trnsformed to one of the types bove, try to substitute by u n eponentil epression within the eqution. This might reduce the eqution to n lgebric one, like qudrtic or rtionl. Emple: Solve = 0 (i) Rewrite the eqution so tht ppers eplicitly ( ) + 1 = 0 ( ) + ( ) 1 = 0 (ii) Substitute u = u + u 1 = 0 (iii) Solve the eqution for u (u+6)(u-) = 0 u = -6 or u = (iv) Bck- substitute nd solve for = -6 or = No solution = 1 Solution: = 1

### 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)

### 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

### 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

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

### 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

### 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

### than 1. It means in particular that the function is decreasing and approaching the x-

6 Preclculus Review Grph the functions ) (/) ) log y = b y = Solution () The function y = is n eponentil function with bse smller thn It mens in prticulr tht the function is decresing nd pproching the

### 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

### 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

### Mathematics Number: Logarithms

plce of mind F A C U L T Y O F E D U C A T I O N Deprtment of Curriculum nd Pedgogy Mthemtics Numer: Logrithms Science nd Mthemtics Eduction Reserch Group Supported y UBC Teching nd Lerning Enhncement

### 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

### 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

### 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

### UNIT 1 FUNCTIONS AND THEIR INVERSES Lesson 1.4: Logarithmic Functions as Inverses Instruction

Lesson : Logrithmic Functions s Inverses Prerequisite Skills This lesson requires the use of the following skills: determining the dependent nd independent vribles in n exponentil function bsed on dt from

### Pre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs

Pre-Session Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:

### 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

### 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

### 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

### 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.

### 4.1 One-to-One Functions; Inverse Functions. EX) Find the inverse of the following functions. State if the inverse also forms a function or not.

4.1 One-to-One Functions; Inverse Functions Finding Inverses of Functions To find the inverse of function simply switch nd y vlues. Input becomes Output nd Output becomes Input. EX) Find the inverse of

### 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:

### SESSION 2 Exponential and Logarithmic Functions. Math 30-1 R 3. (Revisit, Review and Revive)

Mth 0-1 R (Revisit, Review nd Revive) SESSION Eponentil nd Logrithmic Functions 1 Eponentil nd Logrithmic Functions Key Concepts The Eponent Lws m n 1 n n m m n m n m mn m m m m mn m m m b n b b b Simplify

### 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

### 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

### We know that if f is a continuous nonnegative function on the interval [a, b], then b

1 Ares Between Curves c 22 Donld Kreider nd Dwight Lhr We know tht if f is continuous nonnegtive function on the intervl [, b], then f(x) dx is the re under the grph of f nd bove the intervl. We re going

### 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

### 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

### 3.1 EXPONENTIAL FUNCTIONS & THEIR GRAPHS

. EXPONENTIAL FUNCTIONS & THEIR GRAPHS EXPONENTIAL FUNCTIONS EXPONENTIAL nd LOGARITHMIC FUNCTIONS re non-lgebric. These functions re clled TRANSCENDENTAL FUNCTIONS. DEFINITION OF EXPONENTIAL FUNCTION The

### 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!

### 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

### SUMMER KNOWHOW STUDY AND LEARNING CENTRE

SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18

### Duality # Second iteration for HW problem. Recall our LP example problem we have been working on, in equality form, is given below.

Dulity #. Second itertion for HW problem Recll our LP emple problem we hve been working on, in equlity form, is given below.,,,, 8 m F which, when written in slightly different form, is 8 F Recll tht we

### 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

### 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

### (i) b b. (ii) (iii) (vi) b. P a g e Exponential Functions 1. Properties of Exponents: Ex1. Solve the following equation

P g e 30 4.2 Eponentil Functions 1. Properties of Eponents: (i) (iii) (iv) (v) (vi) 1 If 1, 0 1, nd 1, then E1. Solve the following eqution 4 3. 1 2 89 8(2 ) 7 Definition: The eponentil function with se

### 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

### Operations with Polynomials

38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils

### 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

### 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.

### The First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).

The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples

### 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)

### 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

### Chapter 6 Notes, Larson/Hostetler 3e

Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn

### 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

### NAME: MR. WAIN FUNCTIONS

NAME: M. WAIN FUNCTIONS evision o Solving Polnomil Equtions i one term in Emples Solve: 7 7 7 0 0 7 b.9 c 7 7 7 7 ii more thn one term in Method: Get the right hnd side to equl zero = 0 Eliminte ll denomintors

### 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

### Math 153: Lecture Notes For Chapter 5

Mth 5: Lecture Notes For Chpter 5 Section 5.: Eponentil Function f()= Emple : grph f ) = ( if = f() 0 - - - - - - Emple : Grph ) f ( ) = b) g ( ) = c) h ( ) = ( ) f() g() h() 0 0 0 - - - - - - - - - -

### AQA Further Pure 1. Complex Numbers. Section 1: Introduction to Complex Numbers. The number system

Complex Numbers Section 1: Introduction to Complex Numbers Notes nd Exmples These notes contin subsections on The number system Adding nd subtrcting complex numbers Multiplying complex numbers Complex

### Advanced Functions Page 1 of 3 Investigating Exponential Functions y= b x

Advnced Functions Pge of Investigting Eponentil Functions = b Emple : Write n Eqution to Fit Dt Write n eqution to fit the dt in the tble of vlues. 0 4 4 Properties of the Eponentil Function =b () The

### The Regulated and Riemann Integrals

Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue

### 5.2 Exponent Properties Involving Quotients

5. Eponent Properties Involving Quotients Lerning Objectives Use the quotient of powers property. Use the power of quotient property. Simplify epressions involving quotient properties of eponents. Use

### 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

### 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

### 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

### 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

### 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

### 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)

### Equations and Inequalities

Equtions nd Inequlities Equtions nd Inequlities Curriculum Redy ACMNA: 4, 5, 6, 7, 40 www.mthletics.com Equtions EQUATIONS & Inequlities & INEQUALITIES Sometimes just writing vribles or pronumerls in

### MATHS NOTES. SUBJECT: Maths LEVEL: Higher TEACHER: Aidan Roantree. The Institute of Education Topics Covered: Powers and Logs

MATHS NOTES The Institute of Eduction 06 SUBJECT: Mths LEVEL: Higher TEACHER: Aidn Rontree Topics Covered: Powers nd Logs About Aidn: Aidn is our senior Mths techer t the Institute, where he hs been teching

### 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

### 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,

### 5.5 The Substitution Rule

5.5 The Substitution Rule Given the usefulness of the Fundmentl Theorem, we wnt some helpful methods for finding ntiderivtives. At the moment, if n nti-derivtive is not esily recognizble, then we re in

### 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

### 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

### and that at t = 0 the object is at position 5. Find the position of the object at t = 2.

7.2 The Fundmentl Theorem of Clculus 49 re mny, mny problems tht pper much different on the surfce but tht turn out to be the sme s these problems, in the sense tht when we try to pproimte solutions we

### MATHEMATICS AND STATISTICS 1.2

MATHEMATICS AND STATISTICS. Apply lgebric procedures in solving problems Eternlly ssessed 4 credits Electronic technology, such s clcultors or computers, re not permitted in the ssessment of this stndr

### PART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point.

PART MULTIPLE CHOICE Circle the pproprite response to ech of the questions below. Ech question hs vlue of point.. If in sequence the second level difference is constnt, thn the sequence is:. rithmetic

### 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

### 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

### 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

### Math 360: A primitive integral and elementary functions

Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:

### fractions Let s Learn to

5 simple lgebric frctions corne lens pupil retin Norml vision light focused on the retin concve lens Shortsightedness (myopi) light focused in front of the retin Corrected myopi light focused on the retin

### A. Limits - L Hopital s Rule. x c. x c. f x. g x. x c 0 6 = 1 6. D. -1 E. nonexistent. ln ( x 1 ) 1 x 2 1. ( x 2 1) 2. 2x x 1.

A. Limits - L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( ) lim where lim f or lim f limg. c g = c limg( ) = c = c = c How to find it: Try nd find limits by

### QUA DR ATIC EQUATION

J-Mthemtics. INTRODUCTION : QUA DR ATIC QUATION The lgebric epression of the form + b + c, 0 is clled qudrtic epression, becuse the highest order term in it is of second degree. Qudrtic eqution mens, +

### 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

### 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

### 2.4 Linear Inequalities and Interval Notation

.4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or

### 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

### 11.1 Exponential Functions

. Eponentil Functions In this chpter we wnt to look t specific type of function tht hs mny very useful pplictions, the eponentil function. Definition: Eponentil Function An eponentil function is function

### 0.1 THE REAL NUMBER LINE AND ORDER

6000_000.qd //0 :6 AM Pge 0-0- CHAPTER 0 A Preclculus Review 0. THE REAL NUMBER LINE AND ORDER Represent, clssify, nd order rel numers. Use inequlities to represent sets of rel numers. Solve inequlities.

### 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

### Calculus 2: Integration. Differentiation. Integration

Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is

### Exponentials & Logarithms Unit 8

U n i t 8 AdvF Dte: Nme: Eponentils & Logrithms Unit 8 Tenttive TEST dte Big ide/lerning Gols This unit begins with the review of eponent lws, solving eponentil equtions (by mtching bses method nd tril

### 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,

### Bridging the gap: GCSE AS Level

Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions

### 20 MATHEMATICS POLYNOMIALS

0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of

### Functions and transformations

Functions nd trnsformtions A Trnsformtions nd the prbol B The cubic function in power form C The power function (the hperbol) D The power function (the truncus) E The squre root function in power form

### SECTION 9-4 Translation of Axes

9-4 Trnsltion of Aes 639 Rdiotelescope For the receiving ntenn shown in the figure, the common focus F is locted 120 feet bove the verte of the prbol, nd focus F (for the hperbol) is 20 feet bove the verte.

### 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

### Basic 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

### 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

### 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

### Algebra Readiness PLACEMENT 1 Fraction Basics 2 Percent Basics 3. Algebra Basics 9. CRS Algebra 1

Algebr Rediness PLACEMENT Frction Bsics Percent Bsics Algebr Bsics CRS Algebr CRS - Algebr Comprehensive Pre-Post Assessment CRS - Algebr Comprehensive Midterm Assessment Algebr Bsics CRS - Algebr Quik-Piks

### SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mthemtics Bsic Algebr Opertions nd Epressions Common Mistkes Division of Algebric Epressions Eponentil Functions nd Logrithms Opertions nd their Inverses Mnipulting

### Calculus. Rigor, Concision, Clarity. Jishan Hu, Jian-Shu Li, Wei-Ping Li, Min Yan

Clculus Rigor, Concision, Clrity Jishn Hu, Jin-Shu Li, Wei-Ping Li, Min Yn Deprtment of Mthemtics The Hong Kong University of Science nd Technology ii Contents Rel Numbers nd Functions Rel Number System

### 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

### MAA 4212 Improper Integrals

Notes by Dvid Groisser, Copyright c 1995; revised 2002, 2009, 2014 MAA 4212 Improper Integrls The Riemnn integrl, while perfectly well-defined, is too restrictive for mny purposes; there re functions which