Std. XII Commerce Mathematics and Statistics I
|
|
- Philip Jenkins
- 7 years ago
- Views:
Transcription
1
2 ` Written according to the New Tet book (-4) published by the Maharashtra State Board of Secondary and Higher Secondary Education, Pune. Std. XII Commerce Mathematics and Statistics I Third Edition: April 6 Salient Features : Precise Theory for every Topic. Ehaustive coverage of entire syllabus. Topic-wise distribution of all tetual questions and practice problems at the beginning of every chapter. Relevant and important formulae wherever required. Covers answers to all Tetual Questions. Practice problems based on Tetual Eercises and Board Questions (March 8 March 6) included for better preparation and self evaluation. Multiple Choice Questions at the end of every chapter. Two Model Question papers based on the latest paper pattern. Includes Board Question Papers of March and October 4, 5 and March 6. Printed at: Repro India Ltd., Mumbai No part of this book may be reproduced or transmitted in any form or by any means, C.D. ROM/Audio Video Cassettes or electronic, mechanical including photocopying; recording or by any information storage and retrieval system without permission in writing from the Publisher. 4_57_JUP P.O. No. 659
3 Preface Mathematics is not just a subject that is restricted to the four walls of a classroom. Its philosophy and applications are to be looked for in the daily course of our life. The knowledge of mathematics is essential for us, to eplore and practice in a variety of fields like business administration, banking, stock echange and in science and engineering. With the same thought in mind, we present to you "Std. XII Commerce: Mathematics and Statistics-I" a complete and thorough book with a revolutionary fresh approach towards content and thus laying a platform for an in depth understanding of the subject. This book has been written according to the revised syllabus and includes two model question papers based on the latest paper pattern. At the beginning of every chapter, topic-wise distribution of all tetual questions including practice problems have been provided for simpler understanding of various types of questions. Every topic included in the book is divided into sub-topics, each of which are precisely eplained with the associated theories. We have provided answer keys for all the tetual questions and miscellaneous eercises. In addition to this, we have included practice problems based upon solved eercises which not only aid students in self evaluation but also provide them with plenty of practice. We've also ensured that each chapter ends with a set of Multiple Choice Questions so as to prepare students for competitive eaminations. We are sure this study material will turn out to be a powerful resource for students and facilitate them in understanding the concepts of Mathematics in the most simple way. The journey to create a complete book is strewn with triumphs, failures and near misses. If you think we ve nearly missed something or want to applaud us for our triumphs, we d love to hear from you. Please write to us on: mail@targetpublications.org Yours faithfully Publisher Best of luck to all the aspirants! BOARD PAPER PATTERN Time: Hours Total Marks: 8. One theory question paper of 8 marks and duration for this paper will be hours.. For Mathematics and Statistics, (Commerce) there will be only one question paper and two answer papers. Question paper will contain two sections viz. Section I and Section II. Students should solve each section on separate answer books.
4 Section I Q.. This Question will have 8 sub-questions, each carring two marks. Students will have to attempt any 6 out of the given 8 sub-questions. Q.. This Question carries 4 marks and consists of two sub parts (A) and (B) as follows: (A) It contains sub-questions of marks each. Students will have to attempt any out of the given sub-questions. (B) It contains sub-questions of 4 marks each. Students will have to attempt any out of the given sub-questions. Q.. This Question carries 4 marks and consists of two sub parts (A) and (B) as follows: (A) It contains sub-questions of marks each. Students will have to attempt any out of the given sub-questions (B) It contains sub-questions of 4 marks each. Students will have to attempt any out of the given sub-questions. Section II Q.4. This Question will have 8 sub-questions, each carring two marks. Students will have to attempt any 6 out of the given 8 sub-questions. Q.5. This Question carries 4 marks and consists of two sub parts (A) and (B) as follows: (A) It contains sub-questions of marks each. Students will have to attempt any out of the given sub-questions. (B) It contains sub-questions of 4 marks each. Students will have to attempt any out of the given sub-questions. Q.6. This Question carries 4 marks and consists of two sub parts (A) and (B) as follows: (A) It contains sub-questions of marks each. Students will have to attempt any out of the given sub-questions (B) It contains sub-questions of 4 marks each. Students will have to attempt any out of the given sub-questions. Evaluation Scheme for Practical i. Duration for practical eamination for each batch will be one hour. ii. Total marks : MARKWISE DISTRIBUTION [ Marks] [4 Marks] [4 Marks] [ Marks] [4 Marks] [4 Marks] Unitwise Distribution of Marks Section - I Sr.No. Units Marks with Option Mathematical Logic Matrices Continuity Differentiation Application of Derivative Integration Definite Integrals Total 58
5 Unitwise Distribution of Marks Section - II Sr. No. Units Marks with Option. Commercial Arithmetic: Ratio, Proportion, Partnership Commission, Brokerage, Discount Insurance, Annuity. Demography 8. Bivariate Data Correlation 8 4. Regression Analysis 7 5. Random Variable and Probability Distribution 8 6. Management Mathematics 4 Total 58 Weightage of Objectives Sr. No. Objectives Marks Marks with Option Percentage 4 Knowledge Understanding Application Skill Total 8 6. Weightage of Types of Questions Sr. No. Types of Questions Marks Marks with Option Percentage Objective Type Short Answer Long Answer Total 8 6. No. Topic Name Page No. Mathematical Logic Matrices 4 Continuity 4 Differentiation 5 5 Applications of Derivative 88 6 Integration 8 7 Definite Integrals 8 Model Question Paper - I Model Question Paper - II 5 Board Questions Paper March 4 7 Board Questions Paper October 4 9 Board Questions Paper March 5 Board Questions Paper October 5 Board Questions Paper March 6 5
6 Continuity Chapter : Continuity Type of Problems Eercise Q. Nos. Continuity of Standard Function Eamine the Continuity of a Function at a given point Types of Discontinuity (Removable Discontinuity/ Irremovable Discontinuity) Find the value of Function if it is Continuous at given point Find the value of k/a/b if the Function is Continuous at a given point/points. Find the points of Discontinuity for the given Functions. Q. Practice Problems (Based on Eercise.) Q.. Q.,, Practice Problems (Based on Eercise.) Miscellaneous Q. Practice Problems (Based on Miscellaneous). Q.4 Practice Problems (Based on Eercise.) Q.,, 9,,, 4, 5, 6, 7 Q.,, 8 Q.4 Miscellaneous Q., Practice Problems (Based on Miscellaneous) Q.. Q.5, 7 Practice Problems (Based on Eercise.) Miscellaneous Q., 4 Practice Problems (Based on Miscellaneous) Q.5, 7,, 4 Q.4. Q.6, 8, 9 Practice Problems (Based on Eercise.) Miscellaneous Q.5, 6, 7, 8 Practice Problems (Based on Miscellaneous) Miscellaneous Q.9 Q.6, 8,, 8, 9,,,,, 5, 6 Q.5, 6, 7
7 Std. XII : Commerce (Maths I) Syllabus:. Continuity of a function at a point. Algebra of continuous functions. Types of discontinuity.4 Continuity of some standard functions Introduction Continuity is the state of being continuous and continuous means without any interruption or disturbance. For eample, the price of a commodity and its demand are inversely proportional. The graph of demand curve of a commodity is a continuous curve without any breaks or gaps. Y X X O Demand Y Note: A graph consisting of jumps is not a graph of continuous function.. Continuity of a function at a point Definition: A function f is said to be continuous at a point a in the domain of f, if a i.e. if f() eists and a f() a a f() f (a). f() f (a) If any of the above conditions is not satisfied by the function, then it is discontinuous at that point. The point is known as point of discontinuity. eg., Consider the function, f() + 7, 4 5 5, 4 Since, f() has different epressions for the value of left hand and right hand its have to be found out. 4 Price f() 4 Also, f (4) 5 (4) 5 5 and f() 4 (5 5) ( + 7) f() 4 4 f() f (4) f() is continuous at 4. Continuity of a function on its domain Definition: A real valued function f : D R where D R is said to be a continuous function on D, if it is continuous at every point in the domain D. eg., Consider the functions, i. f() ii. f() sin These two functions are continuous on every domain D, where D R.. Algebra of continuous functions If f and g are two real valued functions defined on the same domain, which are continuous at a, then. the function kf is continuous at a, for any constant k R.. the function f g is continuous at a. the function f. g is continuous at a 4. the function f is continuous at a, when g g (a) 5. composite functions, f[g()] and g[f()], if well defined are continuous functions at a.. Types of discontinuity. Removable discontinuity: A real valued function f is said to have removable discontinuity at a in its domain, if f() eists but f() f (a) a a i.e. f() f() f(a) a a This type of discontinuity can be removed by redefining the function f at a as f (a) f(). a eg., Consider the function, 5 6 f(),, Here, f() [,, ] f() eists
8 Chapter : Continuity Also, f () f() f (). (given) function f is discontinuous at, This discontinuity can be removed by redefining f as follows: 56 f(),, is a point of removable discontinuity.. Irremovable discontinuity: A real valued function f is said to have irremovable discontinuity at a, if f() does not eist i.e. a f() a a f() or one of the its does not eist. Such a function can not be redefined as continuous function. eg., Consider the function, f() + +,, Here, and f() f() + + () + () + 8 f() () 8 f() it of the function does not eist. f has irremovable discontinuity at.4 Continuity of some standard functions. Constant function: The constant function f() k (where k R is a constant). The function is continuous for all belonging to its domain. eg., f(), f() log, f() e 7. Polynomial function: The function f() a + a + a +. + a n n, where n N, a, a. a n R is continuous for all belonging to domain of. eg., f() , f() 5 + 9, f() 4 6, R.. Rational function: If f and g are two polynomial functions having same domain then the rational function f is continuous in its g domain at points where g(). eg., 56 Consider the function, 9 Here, f() and g() 9 Given function is continuous on its domain, where 9 i.e., ( + ) ( ) i.e., +, i.e.,, The function is continuous on its domain ecept at,. 4. Trigonometric function: sin (a + b) and cos (a + b), where a, b R are continuous functions for all R. eg., sin (5 + ), cos (7 ) R. Note: Tangent, cotangent, secant and cosecant functions are continuous on their respective domains. 5. Eponential function: f() a, a >, a, R is an eponential function, which is continuous for all R. eg., f(), f(), f() e R, where a >, a. 6. Logarithmic function: f() log a where a >, a is a logarithmic function which is continuous for every positive real number i.e. for all R + eg., f() log a 7, f() log a 9 R, where a >, a. Some Important Formulae Algebra of its: If f() and g() are any two functions,. [f() + g()] f() + g() a [f() g()] a [f()g()] a f( ) a g( ) a a a f() a a f() g() a a f ( ) a g( ), where a g() g() a [k.f()] k f(), where k is a constant. a
9 Std. XII : Commerce (Maths I) Limits of Algebraic functions:. a.. 4 a a a k k, where k is a constant. n a a n na n Limits of Trigonometric functions: sin. tan.. cos Limits of Eponential functions:.. a log a, where (a >, a ) ( + ) e Limits of Logarithmic functions: log. Eercise.. Are the following functions continuous on the set of real numbers? Justify your answers. i. f() 7 Given, f() 7 It is a constant function. f() is continuous on the set of real numbers i.e., R ii. f() e Given, f () e It is a constant function. [ e.788] f() is continuous on the set of real numbers i.e., R. iii. f () log 9 Given, f() log 9 Here, log 9 is a constant f() is a constant function f() is continuous on the set of real numbers i.e., R. iv. f() Given, f() It is a polynomial function f() is continuous on the set of real numbers i.e., R v. g() sin (4 ) Given, g() sin (4 ) It is a sine function f() is continuous on the set of real numbers i.e., R vi. h() Given, h() It is a rational function and is discontinuous if But, R, h() is continuous on the set of real numbers, ecept when vii. g() 69 Given, g() It is a rational function and is discontinuous, if + But R, + g () is continuous on the set of real numbers i.e., R viii. f() 5 Given, f() 5 It is an eponential function It is continuous on the set of real numbers i.e., R i. f() 5 Given, f() 5 It is the difference of two eponential functions It is continuous on the set of real numbers i.e., R
10 Chapter : Continuity (5 + 7). f() e Given, f() e (5 + 7) It is an eponential function It is continuous on the set of real numbers i.e., R. Eamine the continuity of the following functions at the given point. (All functions are defined on R R) i. f() + 9, for 4 +, for > ; at. [Mar 5] f() ( + 9) () (i) and f() (4 + ) 4() + 5. (ii) Also, f () () (iii) f() f() f(). [From (i), (ii) and (iii)] f is continuous at. ii. f() 6, for 4 4 8, for 4; at 4. [Oct 5] 6 f() ( 4)( 4) 4 ( 4) 4 ( + 4).[ 4, 4, 4 ] 4 f() (i) Also, f (4) 8. (ii)(given) f() f (4). [From (i) and (ii)] 4 f is continuous at 4. iii. f() 4, for 79, for ; at Consider, By synthetic division, we get ( ) ( + + 9) f() [,, ] ( ) 9 9 f.(i) Also, f( ).(ii)(given) f() f ( ).[From (i) and (ii)] f is discontinuous at. sin 5 iv. f(), for, for ; at. sin5 sin5 f() 5 5 sin () sin. [, 5, ] f() 5 Also, f(). (given) f() f () f is discontinuous at. 5
11 Std. XII : Commerce (Maths I) 7 v. f() f(), for, for ; at. [Oct 4] () [,, ] 7 6 f(). (i) Also, f(). (ii)(given) f() f().[from (i) and (ii)] f is continuous at. vi. f() 5 5, for 4 8, for 4; at 4. 5 f() [ 4, 4, 4 ] f() Also, f(4) 8 4 f() f(4). (i). (ii)(given) 6. [From (i) and (ii)] f is discontinuous at 4. vii. f() cos, for, for ; at [Oct 5] f() cos cos [, ] for all R, also, when, cos eists. Let cos finite number k (say) cos where k [, ] f(). (i) k 6
12 Also, f(). (ii)(given) f() f(). [From (i) and (ii)] f is continuous at. tan sin viii. f() sin sin, for < Consider, f() sin sin, for ; at tan sin sin sin tan sin sin 4sin sin [sin sin 4sin ] tan sin 4sin sin sin cos 4sin sin sin cos 4sin sin cos 4sin sin sin 4sin [ cos sin ] Also, Chapter : Continuity sin sin 4 sin 4 8 [,, f() 8 f() sin sin sin sin sin sin sin ]. (i) sin sin () (). [ sin ] sin.sin sin sin sin 4 4 sin f() f() f() does not eist f is discontinuous at. (ii) f().[from (i) and (ii)] 7
13 Std. XII : Commerce (Maths I) 9 i. f(), for < 6 4, for > 4 for ; at Consider, 9 f() f() does not eist. But f() 4 f() f() f is discontinuous at. 6. f(), for < 4 4, for >, for, at 6 f() log 6 + log log a [ log a ] log 6 f() log 9. (i) 4 4 f() (log 4). 4 a [ log a ] f() (log 4)... (ii) f() f()... [From (i) and (ii)] f() does not eist f is discontinuous at. Discuss the continuity of the following functions. a i. f(),, a & a log a, ; at. a f() a a f() log a. (i).[,, a log a] Also, f() log a. (ii)(given) f() f().[from (i) and (ii)] f is continuous at 5 ii. f() 4, log 5, ; at. 4 5 f() 4 8
14 log5 log a...[ log a] log 4 log 5 log f() 4 log. (i) Also, f () log 5. (ii) (given) 4 f() f() f is discontinuous at iii. g() 5, e 5/, ; at. g() g() e 5. (i)...[, 5 5 e Also g() g() g() g is discontinuous at 5. (ii)(given) e].[from (i) and (ii)] Chapter : Continuity log + iv. h(),, ; at. log h() log log () log...[,, ] h(). (i) Also, h(). (ii)(given) h() h().[from (i) and (ii)] h is continuous at v. f() tan, <, > e e, at. f() tan tan log...[,, a f() log f() e e e e tan tan log a, ]. (i) 9
15 Std. XII : Commerce (Maths I) f() f() e e e e e e e loge e e...[... (ii) f() does not eist f is discontinuous at log e] f().[from (i) and (ii)] 4. Discuss the continuity of the following functions at the points given against them. If the function is discontinuous, determine whether the discontinuity is removable. In that case, redefine the function, so that it becomes continuous. i. f(), for <, for ; at. f() ( ) () () 4 4 and f() f() 5 f() f() ( ) () 6. (i). (ii) f().[from (i) and (ii)] it of the function does not eist. f has irremovable discontinuity at 7 ii. f(), for < 8, for ; at. 7 f() and 9...[a b (a b)(a + ab + b )] ( + + 9) [, ] () + () + 9 f() 7.(i) f() f() 4 8 8().(ii) f() f().[from (i) and (ii)] it of the function does not eist f has irremovable discontinuity at iii. f() sin9, for, for ; at. sin9 f() sin sin9 9 9 [, 9, f() 9. (i) sin ] Also, f(). (ii) (given) f() f().[from (i) and (ii)]
16 Chapter : Continuity f has removable discontinuity at This discontinuity can be removed by redefining the function as: f() sin9, for 9, for ; at e iv. f(), for 5, for ; at e f() 5 e 5 e 5 log e 5 e [,, log e ] f() 5 Also, f() f() f(). (i). (ii)(given).[from (i) and (ii)] f has removable discontinuity at This discontinuity can be removed by redefining the function as: e f() 5, for, for ; at 5 v. f(), for 5, for ; at f() 4...[, ] f(). (i) Also, f() 5. (ii)(given) f() f().[from (i) and (ii)] f has removable discontinuity at This discontinuity can be removed by redefining the function as: f(), for, for ; at 5. If f is continuous at, then find f() 5 5 i. f(), Given, f is continuous at f() f() log 5 f() (log 5) 5...[ a log a ]
17 Std. XII : Commerce (Maths I) sin ii. f() log Given, f is continuous at f() f() sin... [, sin,, [Mar ] log sin log( ) sin sin sin sin sin log + sin sin sin log( + ) log sin, a loga] iii. f() 5 5, [Mar 5] tan Given, f is continuous at f() f() 5 5 tan 5 5 tan 5 5 tan 5 5 tan 5 tan 5.[,, tan 5 tan 5 tan log5 log f() (log 5) (log ) a log a, tan ] cos cos iv. f(), [Mar 6] Given, f is continuous at f() f() f () 4 cos cos 4cos coscos.[cos 4cos cos] 4cos 4cos 4coscos 4cos cos 4cossin 4 cos. 4. cos(). () sin.[ sin ]
18 Chapter : Continuity 6. Find the value of k, if the function i. g(), for k, for is continuous at Given, g is continuous at and g() k g() g() k ()... [ a k n a a n na ii. h() +, for < 5 + k, for is continuous at Given, h is continuous at h() h() h ().(i) Now, h() + () + and h() h() 5 k 5 k h() 5 + k 5 + k...[from (i)] k 4 n iii. f() tan 7, for k, for is continuous at [Mar 5] Given, f is continuous at and f () k f() f() k 7 tan7 tan tan7 7 ] 7 () k 7.[,7, tan iv. h() + k, for 7, for 7 is continuous at 7 Given h is continuous at 7, h(7) h(7) h() k ( + k) 7 h(7) (7 + k) 7 + k or (7 + k) k 7 or 7 k k or k 7 k or k 7. ] sin 7. If f(), for, is continuous at, then find f. [Oct 5] Given, f is continuous at f f sin Put h h as,,h sin h f h h h cosh h... [sin cos ]
19 Std. XII : Commerce (Maths I) 4 h h cosh cosh h cosh cos h 4h cosh sin h 4h cosh h sin h 4 h h h cosh sin 4 ().[ cos ] 4 4 f 8 e 8. If f() for <, a a for log 7 for >, b b is continuous at, then find a and b. [ Mar 6] Given, f is continuous at and f() f() f() f() f() f().(i) Now, e f() a e a e a loge a a.[,, log a ] f()...(ii) a log Also, f() 7 b log 7 7 b 7 f() 7 b Now, a a and 7 b b 7 a, b 7 7 log 7 b 7 7 b log... [,7, ]...(iii) 9. If f is continuous at and f() + a, for < + a + b, for and f(), then find a, b. Given, f is continuous at f() f() Now, f() + a.[from (i) and (ii)].[from (i) and (iii)]. (i) a f() + a and f() + a + b + a + b f() a + b + a a + b.[from (i)] b Also, f() + a + b, for and f() f() () + a + b + a + b a + b. (ii) Substituting b in (ii) we get a + a a, b
20 . Is the function f() + 5 cos + continuous at? Justify. Given, f() + 5 cos + f() ( + + ) 5 cos Let + + p() and 5 cos q() f() p() q()...(i) Now, p() + + p() is a polynomial function It is continuous at each value of and q() 5 cos Here, 5 is constant function and cos is cosine function which is continuous. q() is continuous at each value of f() is a difference of two continuous function which is always continuous f() is continuous at Miscellaneous Eercise. Discuss the continuity of the function. f() 64, for 4 9 5, for 4; at 4 f() Chapter : Continuity [ 4, 4, 4 ] f() 6 4. (i) Also, f(4). (ii)(given) f() f(4) 4.[From (i) and (ii)] f is discontinuous at 4. Eamine the continuity of the function e f() log, for, for ; at. If discontinuous, then state whether the discontinuity is removable. If so, redefine and make it continuous. e e f() log 9 log 9 e 9 log 9 loge e log( ) e...,, loge, log () f(). (i) Also, f(). (ii)(given) 5
21 Std. XII : Commerce (Maths I) f() f().[from (i) and (ii)] f has removable discontinuity at This discontinuity can be removed by redefining the function as: f() e, for log( ), for ; at. The function f is defined as 7 8 f(), for 5 /5 /5, for > / / f() Eamine, if f is continuous at. 7 8 f() n n 7 a... 5 na a 5 a f() 8 5 and f() n 5 5 n a...[ a a f() 5 () f() f() f() does not eist f is discontinuous at n na 4. The function f defined as 8 8 f(), for 5 7 is continuous at. Find f() Given, function is continuous at 8 8 f() f() n ] 6
22 [, ] f () 7 5. Find k if the function given below is continuous at cos sin f(), for k, for Given, f is continuous at f f() cos sin k cos sin cos [sin sin cos] cos sin Put h h as,, h cos h sin h h h h sinhcosh h h [cos sin,sin cos] h sinhsin h 8h h sin sin h h h h 4 4 h sin sinh. 8 h h h h 8 Chapter : Continuity h sinh. h,, h h 6. If the function given below is continuous at as well as at 4, then find the values of a and b. f() + a + b, +, 4 a + 5b, 4 [Oct 4] Given, f is continuous at f() f().(i) Now, f() + a + b () + a() + b f() 4 + a + b and f() ( + ) () + f() a + b 8.[From (i)] a + b 4.(ii) Also, f is continuous at 4 f() f().(iii) 4 Now, 4 and 4 4 f() 4 f() 4 f() 4 f() 8a + 5b + (4) a + 5b a(4) + 5b 4 8a + 5b.[From(iii)] 8a + 5b 4.(iv) 7
23 Std. XII : Commerce (Maths I) By eq. (iv) 5 eq. (iii), we get 8a 5b 4 a 5b a 6 a Substituting a in eq. (ii), we get + b 4 b 4 6 b a or b 7. Find a and b if f is continuous at, where sin f() + a,, cos + b, Given, f is continuous at and f () f() f() f () 8 f() Now, f() f() Put + h h as,, h f() h.(i) sin a sin h a h sin h a h h sin h h a h.[sin(+) sin] sin h h a h sin h a h h h () + a sin h.[h, h, ] h h f() + a Also f().(ii) cos b Put + h h as,, h f() cosh b h h cosh b h h h h cosh b h.[cos(+) cos] h sin b h.[ cos sin ] h sin h b h 4 4 h sin h b 4 h 4 h sin b h h h b h sin... [ h,, ] f() b.(iii) + a.[from (i) and (ii)] a + a and b.[from (i) and (iii)]
24 Chapter : Continuity b b a, b 8. Find k, if the function f is continuous at, where e sin i. f(), for k, for Given, f is continuous at and f() k f() f() k k e sin e sin e. log e. ().[ ii. f() sin e log e, sin ] 7, for k, for Given, f is continuous at and f() f() f() 7 9 k k 9 9 k k 9 k 9. k log9 log k log k log9... [ a log a ] log k log 9... [n log a log a n ] log k log k log iii. f(), for 5 k, for Given, f is continuous at and f() k f() f() log k k 5 log 5 log log...[,, ] 9. Find the points of discontinuity, if any, for the following: i. f() cos cos Given, f() Let, + cos p() and + q() Consider, p() + cos Here, is always continuous for all real values of and cosine is a continuous function p() is a continuous function and q() + It is a polynonimal function It is continuous for all real values of f() is a continuous function. 9
25 Std. XII : Commerce (Maths I) 5 4 ii. f() Given, f() f() f() is a rational function f() will be discontinuous if ( + ) ( ) i.e., + or i.e., or f() is discontinuous at and 4 9 iii. f() 6 49 Given, f() 6 f() is a rational function f() will be discontinuous if i.e., i Value of is a comple number f() is continuous for all real values of 9 iv. f() sin 9 9 Given, f() sin 9 Let 9 p() and sin 9 q() Consider, p() 9 It is a polynomial function It is continuous function and q() sin 9 Here, sine is a continuous function and 9 is a constant function q() is continuous as sin f() is continuous function.. If possible, redefine the function to make it continuous. i. f(), for e, for ; at. f Put + h h as,, h f() h h h h h h f() e...[ e ]...(i) Also, f() e. (ii) (given) f() f().[from (i) and (ii)] f has removable discontinuity at This discontinuity can be removed by redefining the function as: f(), for e, for ; at 6 ii. f() tan, for sin log 5, for ; at. tan 6 f() sin tan 6 sin cos cos tan 6 tan cos tan 6. tan cos log 6 cos... [, tan, f() log 6 Also, f() log 5 f() f() a log a ].(given) f has removable discontinuity at 4
26 Chapter : Continuity This discontinuity can be removed by redefining the function as: tan 6 f() sin, for log 6, for ; at sin 5 iii. f(), for 5, for ; at. [Oct 4] sin 5 f() sin () f() 5 Also, f() 5 f() f() sin5 5 [, 5, sin ]. (given) f has removable discontinuity at This discontinuity can be removed by redefining the function as: sin 5 f(), for 5, for ; at cos iv. f() sin, for < sin, for > cos, for ; at. f() cos sin sin sin sin sin sin sin sin...[a b (a b) (a + ab + b )] sin sin sin.[ sinsin sin, sin] sin sin sin f ( ) and f() sin cos sin sin cos sin sin cos sin sin sin sin sin sin sin sin sin sin.[ sinsin sin, sin] sin sin f ( ) f ( ) 4 f() it of the function does not eist f has irremovable discontinuity at 4
27 Std. XII : Commerce (Maths I) v. f(), for, for >, for ; at. f() and +...[, ] () 4 f() 5 4 f() 4.[, ] 4 f() f() f() it of the function does not eist. f has irremovable discontinuity at 4 4 vi. f(), for. 5, for ; at. 4 4 f() () () + () () + () () + (4) () [ a n a a n na 4 4 n f(). (i) Also, f() 5. (ii)(given) f() f() f has removable discontinuity at ] 4
28 Chapter : Continuity This discontinuity can be removed by redefining the function as: 4 4 f(), for, for ; at Additional Problems for Practice Based on Eercise.. Are the following functions continuous on the set of real numbers? Justify your answer. (All functions are defined on R R) i. f() ii. f() e 5 iii. f() log 9 iv. f() v. h() cos(9 + 5) vi. 5 g() 4 9 vii. g() + 7. Eamine the continuity of the following funtions at the given point: i. f() sin + cos, for, for ; at ii. f() sin, for, for ; at iii. f() ( + ) /, for e, for ; at 6 iv. f(), for 7, for ; at v. f() + 6 +, for 4 + 8, for > 4; at 4 y e.sin y vi. f(y), for y y 4, for y ; at y 4 vii. f(), for e, for at viii. f() sin ( + ), for tan sin, for > ; at 5 i. f(), for < 4, for ; at. f(), for, for ; at. Discuss the continuity of the following functions: 5 a a i. f(), for log a, for ; at ii. f() a, for a e, for ; at 5 log iii. g(), for 5, for ; at iv. f() 5 e sin for (log 5 + ), for ; at v. sin a f(), for, for ; at vi. f() sin, for, for ; at vii. f() cos, for, for ; at 4 viii. f(), for, for ; at [Mar 6] 4. Discuss the continuity of the following functions at the points given against them. If the function is discontinuous, determine whether the discontinuity is removable. In that case, redefine the function, so that it becomes continuous. 4
29 Std. XII : Commerce (Maths I) i. f() cos tan, for 9, for ; at ii. 5 f(), for < 6, for ; at iii. f() sin 5 for 5, for ; at iv. f() 4, for sin 8, for ; at v. sin( ) f(), for, for ; at 5. If f is continuous at, then find f(). sin i. f() 4 log ( ), log( a ) log( b ) ii. f() log( ) log( ) iii. f() tan cos sin iv. f() 6. Find the value of k, if the function i. g(), for k, for is continuous at ii. f() 8, k for, for is continuous at iii. f() log( k ), for sin 5, for is continuous iv. f() + k, for k, for < is continuous at v. k f(), for ( ) 5 4, for is continuous at 7. If f() cos[7( )], for is 5( ) continuous at, find f() 8. If the function f() kcos, for, for be continuous at, then find k 9. Is the function f() + + cos + sin 5 + continuous at 4? Justify. If the function f is continuous at, then find f(). Where f() for. [Mar 4]. If the function f is continuous at, then 5 find k where f(), for < k +, for > [Mar 4]. Discuss the continuity of the function f defined as 8 f (),, ; at [Mar 8]. Discuss the continuity of the function f defined as: 5 f (), if 5 e, if ; at [Oct 8] 4. Discuss the continuity of f () at, where 4 f () for 4, for [Mar 9] 5. Discuss the continuity of the function f defined as, f() +, if 6, if < ; at [Oct ] 44
30 Chapter : Continuity 4 6. If f(), for 4, for Discuss the continuity of f() at [Mar, Oct ] 7. Discuss the continuity of the following function: f, for e, for ; at [Oct ] 8. If f is continuous at, where f () + a, for b, for < Find a, b given that f (). [Mar 8] 9. Find k, if the function f defined as: cosk f (),, is continuous at [Oct 8]. Find k, if the function 64 f (), for 4 4 k, for 4 is continuous at 4 [Oct 9]. If f() is continuous at and it is defined as a a f, k, find k. [Mar ]. The function f defined as sin p f(), if > q + 5 6, if is continuous at. Find the values of p and q, given that f (). [Oct ]. If f() tan + a, for <, for + 4 b, for > is continuous at, then find the values of a and b. [Mar ] 9 4. Find f() if f(), is contiuous at. [Oct ] 5. If f is continuous at where e f(), a, then find a. [Mar ] 6. If f is continuous at where f() + a, b, <, find a and b. Given that f() [Oct ] Based on Miscellaneous Eercise. Eamine the continuity of the following funtions at the given point: i. f(), for cos ii., 7 for ; at f() sin tan for <, for log( ), for > e 4 iii. f() ( 6), for 6, for 6; at 6 5 iv. f(), for < +, for ; at 6 v. f() ( ), for 7, for ; at. Discuss the continuity of the following functions: i. f() 5 4 for log, for ; at ii. ( ) f() tan.log( ) log 4, for 45
31 Std. XII : Commerce (Maths I). Discuss the continuity of the following functions at the points given against them. If the function is discontinuous, determine whether the discontinuity is removable. In that case, redefine the function, so that it becomes continuous: i. f() 4 e, 6 for log, for ; at ii. f(), for log, for ; at iii. f() iv. f() 6 64, for 8, for ; at (8 ), for sin log 4 8 log 8, for ; at 4. If f is continuous at, then find f(). i. f() ii. f() cos 4 e e sin 5, 5. Find the value of k, if the function sin f(), for k, for is continuous at 6. If f() sin 4 + a, for > b, for <, for is continuous at, find a and b. [Mar 9,, Oct 9] 7. cos4 If f(), for < a, for, for > 6 4 is continuous at, then find the value of a. 8. Discuss the continuity of the function f at 5 5, where f(), cos cos6 for log5, for 8 [Mar 4] Multiple Choice Questions., If f() c, is continuous at, then c (A) (B) 4 (C) (D)., if If f() a b, if 5 is continuous, 7, if 5 then the value of a and b is (A), 8 (B), 8 (C), 8 (D), 8. The sum of two discontinuous functions (A) is always discontinuous. (B) may be continuous. (C) is always continuous. (D) may be discontinuous. 4. For what value of k the function 5 44,if f() is k,if continuous at? (A) (B) (C) (D) 4 4 log ( a ) log ( b ) 5. The function f() is not defined at. The value which should be assigned to f at so that it is continuous at, is (A) a b (B) a + b (C) log a + log b (D) log a log b 46
32 Chapter : Continuity 6. In order that the function f() ( + ) cot is continuos at, f() must be defined as (A) f() e (B) f() (C) f() e (D) None of these 7. If f() sin, sin is k, a continuous function, then k (A) (B) (C) (D) 8. A function f is continuous at a point a in the domain of f if (A) f() eists (B) a a f() f(a) (C) f() f(a) a (D) both (A) and (B). 9. Which of the following function is discontinuous? (A) f() (B) g() tan (C) h() (D) none of these. If the function f() continuous at, then k kcos,when, when (A) (B) 6 (C) (D) None of these. The points at which the function f() is discontinuous, are (A),4 (B), 4 (C),,4 (D),,4. Which of the following statement is true for graph f() log (A) Graph shows that function is continuous (B) Graph shows that function is discontinuous (C) Graph finds for negative and positive values of (D) Graph is symmetric along -ais is,when. If f(), then, when (A) f() (B) f() (C) f() is continuous at (D) All the above are correct 4. a,when a If f() a, then, when a (A) f() is continuous at a (B) f() is discontinuous at a (C) f() (D) None of these 5. cos4, when < If f() a when,,when 6 4 is continuous at, then the value of a will be (A) 8 (B) 8 (C) 4 (D) None of these ,when If f(), then 6, when (A) f() is continuous at (B) f() is discountinuous at (C) f() 6 (D) None of these 7. The values of A and B such that the function f() sin, AsinB,, is continuous cos, everywhere are (A) A, B (B) A, B (C) A, B (D) A, B 47
33 Std. XII : Commerce (Maths I) 8. If f() k k,for <, is,for continuous at, then k (A) 4 (B) (C) (D) 9. The function f() sin is (A) Continuous for all (B) Continuous only at certain points (C) Differentiable at all points (D) None of these sin cos. The function f() sin cos is not defined at. The value of f(), so that f() is continuous at, is (A) (B) (C) (D). 7 The function f() is discontinuous for (A) only (B) and only (C),, only (D),, and other values of. The function ' f is defined by f(), if >, f() k if and, if < is continuous, then the value of k is equal to (A) (B) (C) 4 (D). cos4 Function f(), where and 8 f() k, where is a continous function at then the value of k will be? (A) k (B) k (C) k (D) None of these 4., when/ If f(), when /, then,when/ (A) /f() (B) /f() (C) f() is continuous at 5. If f() 5 for 5 and f is 7 continuous at 5, then f(5) (A) (B) 5 (C) (D) 5 Answers to Additional Practice Problems Based on Eercise.. i. Polynomial function continuous ii. Constant function continuous iii Constant function continuous iv. Polynomial function continuous v. Cosine function continuous vi. Rational function continuous for all R, ecept when vii. Addition of eponential functions continuous. i. Continuous ii. Continuous iii. Continuous iv. Discontinuous v. Continuous vi. Discontinuous vii. Continuous viii. Continuous i. Discontinuous. Continuous. i. Discontinuous ii. Continuous iii. Continuous iv. Discontinuous v. Discontinuous vi. Continuous vii. Discontinuous viii. Discontinuous 4. i. Discontinuous, removable ii. Discontinuous, irremovable iii. Discontinuous, removable iv. Discontinuous, removable v. Discontinuous, removable 5. i. (log 4) ii. a + b iii. iv i. ii. iii. 5 iv. v (D) f() is discontinuous at 7. 49
34 Chapter : Continuity Addition of continuous functions. f() is continuous.. f(). k 4. Discontinuous. Discontinuous 4. Continuous 5. Discontinuous 6. Continuous 7. Discontinuous 8. a, b 9. k ± 4. k 48. log a. p, q Answers to Multiple Choice Question. (B). (C). (B) 4. (C) 5. (B) 6. (C) 7. (B) 8. (D) 9. (B). (B). (B). (A). (D) 4. (B) 5. (A) 6. (B) 7. (C) 8. (C) 9. (A). (C). (C). (B). (B) 4. (D) 5. (A). a, b a 6. a, b 4 Based on Miscellaneous Eercise. i. Discontinuous ii. Continuous iii. Discontinuous iv. Continuous v. Continuous. i. Discontinuous ii. Discontinuous. i. Discontinuous, removable ii. Discontinuous, removable iii. Discontinuous, removable iv. Discontinuous, removable 4. i. (log ) ii a 5, b Discontinuous 49
35 Board Question Paper : March 6 BOARD QUESTION PAPER : MARCH 6 Notes: i. All questions are compulsory. ii. Figures to the right indicate full marks. iii. Answer to every question must be written on a new page. iv. L.P.P. problem should be solved on graph paper. v. Log table will be provided on request. vi. Write answers of Section I and Section II in one answer book. Section I Q.. Attempt any SIX of the following: [] i. If y (sin ), find d y d () ii. If A show that A A is a scalar matri. () iii. Write the negation of the following statements: (a) y N, y + 7 (b) If the lines are parallel then their slopes are equal. () iv. The total revenue R 7 where is number of items sold. Find for which total revenue R is increasing. () v. sec Evaluate: d tan 4 () vi. Find d y d (sin 5) () vii. Discuss the continuity of function f at Where f () 4, for, for () viii. State which of the following sentences are statements. In case of statement, write down the truth value: (a) Every quadratic equation has only real roots. (b) 4 is a rational number. () Q.. (A) Attempt any TWO of the following: [6][4] i. Solve the following equations by the inversion method: + y 5 and + y () ii. iii. Find and y, if y Evaluate: tan d. () () 5
36 Std. XII : Commerce (Maths I) (B) Attempt any TWO of the following: [8] i. (a) Epress the truth of each of the following statements using Venn diagram. () All teachers are scholars and scholars are teachers. () If a quadrilateral is a rhombus then it is a parallelogram. (b) Write converse and inverse of the following statement: If Ravi is good in logic then Ravi is good in Mathematics. (4) ii. Find the area of the region bounded by the lines y + 8, and 4. (4) iii. 9 Evaluate: d (4) Q.. (A) Attempt any TWO of the following: [6][4] i. e If f () a, for <, a, for log ( 7 ), for >, b b Is continuous at then find a and b. () ii. If the function f is continuous at, then find f() cos cos where f (), () iii. If f() k and f(), f() 4, find f(). () (B) Attempt any TWO of the following: [8] i. Find MPC (Marginal Propensity to Consume) and APC (Average Propensity to Consume) if the ependiture E c of a person with income I is given as E c (.) I + (.75) I when I. (4) ii. Cost of assembling wallclocks is 4 and labour charges are 5. Find the number of wallclocks to be manufactured for which marginal cost is minimum. (4) iii. If cos y k, y show that y d y d k. (4) 6
Book-Keeping & Accountancy
Written as per the revised syllabus prescribed by the Maharashtra State Board of Secondary and Higher Secondary Education, Pune. STD. XII Commerce Book-Keeping & Accountancy Fourth Edition: March 2016
More informationLIMITS AND CONTINUITY
LIMITS AND CONTINUITY 1 The concept of it Eample 11 Let f() = 2 4 Eamine the behavior of f() as approaches 2 2 Solution Let us compute some values of f() for close to 2, as in the tables below We see from
More informationCore Maths C3. Revision Notes
Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...
More informationAlgebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123
Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from
More informationGeorgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1
Accelerated Mathematics 3 This is a course in precalculus and statistics, designed to prepare students to take AB or BC Advanced Placement Calculus. It includes rational, circular trigonometric, and inverse
More informationCore Maths C1. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the
More informationUNIT 1: ANALYTICAL METHODS FOR ENGINEERS
UNIT : ANALYTICAL METHODS FOR ENGINEERS Unit code: A/60/40 QCF Level: 4 Credit value: 5 OUTCOME 3 - CALCULUS TUTORIAL DIFFERENTIATION 3 Be able to analyse and model engineering situations and solve problems
More informationSAT Subject Math Level 2 Facts & Formulas
Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Arithmetic Sequences: PEMDAS (Parentheses
More informationMEMORANDUM. All students taking the CLC Math Placement Exam PLACEMENT INTO CALCULUS AND ANALYTIC GEOMETRY I, MTH 145:
MEMORANDUM To: All students taking the CLC Math Placement Eam From: CLC Mathematics Department Subject: What to epect on the Placement Eam Date: April 0 Placement into MTH 45 Solutions This memo is an
More information376 CURRICULUM AND SYLLABUS for Classes XI & XII
376 CURRICULUM AND SYLLABUS for Classes XI & XII MATHEMATICS CLASS - XI One Paper Time : 3 Hours 100 Marks Units Unitwise Weightage Marks Periods I. Sets Relations and Functions [9 marks] 1. Sets Relations
More informationPrentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary)
Core Standards of the Course Standard 1 Students will acquire number sense and perform operations with real and complex numbers. Objective 1.1 Compute fluently and make reasonable estimates. 1. Simplify
More informationG. GRAPHING FUNCTIONS
G. GRAPHING FUNCTIONS To get a quick insight int o how the graph of a function looks, it is very helpful to know how certain simple operations on the graph are related to the way the function epression
More informationALGEBRA 2/TRIGONOMETRY
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Thursday, January 9, 015 9:15 a.m to 1:15 p.m., only Student Name: School Name: The possession
More informationContinuity. DEFINITION 1: A function f is continuous at a number a if. lim
Continuity DEFINITION : A function f is continuous at a number a if f(x) = f(a) REMARK: It follows from the definition that f is continuous at a if and only if. f(a) is defined. 2. f(x) and +f(x) exist.
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationSouth Carolina College- and Career-Ready (SCCCR) Pre-Calculus
South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationGeometry Notes RIGHT TRIANGLE TRIGONOMETRY
Right Triangle Trigonometry Page 1 of 15 RIGHT TRIANGLE TRIGONOMETRY Objectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right
More informationGraphing Trigonometric Skills
Name Period Date Show all work neatly on separate paper. (You may use both sides of your paper.) Problems should be labeled clearly. If I can t find a problem, I ll assume it s not there, so USE THE TEMPLATE
More informationPRE-CALCULUS CONCEPTS FUNDAMENTAL TO CALCULUS. A Thesis. Presented to. The Graduate Faculty of The University of Akron. In Partial Fulfillment
PRE-CALCULUS CONCEPTS FUNDAMENTAL TO CALCULUS A Thesis Presented to The Graduate Faculty of The University of Akron In Partial Fulfillment of the Requirements for the Degree Master of Science Michael Matthew
More informationMPE Review Section III: Logarithmic & Exponential Functions
MPE Review Section III: Logarithmic & Eponential Functions FUNCTIONS AND GRAPHS To specify a function y f (, one must give a collection of numbers D, called the domain of the function, and a procedure
More informationExponential and Logarithmic Functions
Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,
More informationFunctions: Piecewise, Even and Odd.
Functions: Piecewise, Even and Odd. MA161/MA1161: Semester 1 Calculus. Prof. Götz Pfeiffer School of Mathematics, Statistics and Applied Mathematics NUI Galway September 21-22, 2015 Tutorials, Online Homework.
More informationFunctions and their Graphs
Functions and their Graphs Functions All of the functions you will see in this course will be real-valued functions in a single variable. A function is real-valued if the input and output are real numbers
More informationFive 5. Rational Expressions and Equations C H A P T E R
Five C H A P T E R Rational Epressions and Equations. Rational Epressions and Functions. Multiplication and Division of Rational Epressions. Addition and Subtraction of Rational Epressions.4 Comple Fractions.
More informationD.3. Angles and Degree Measure. Review of Trigonometric Functions
APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric
More information36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous?
36 CHAPTER 1. LIMITS AND CONTINUITY 1.3 Continuity Before Calculus became clearly de ned, continuity meant that one could draw the graph of a function without having to lift the pen and pencil. While this
More informationThe numerical values that you find are called the solutions of the equation.
Appendi F: Solving Equations The goal of solving equations When you are trying to solve an equation like: = 4, you are trying to determine all of the numerical values of that you could plug into that equation.
More informationSection 6-3 Double-Angle and Half-Angle Identities
6-3 Double-Angle and Half-Angle Identities 47 Section 6-3 Double-Angle and Half-Angle Identities Double-Angle Identities Half-Angle Identities This section develops another important set of identities
More informationTrigonometry Review Workshop 1
Trigonometr Review Workshop Definitions: Let P(,) be an point (not the origin) on the terminal side of an angle with measure θ and let r be the distance from the origin to P. Then the si trig functions
More informationMathematics 31 Pre-calculus and Limits
Mathematics 31 Pre-calculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals
More informationALGEBRA 2/TRIGONOMETRY
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Tuesday, January 8, 014 1:15 to 4:15 p.m., only Student Name: School Name: The possession
More informationMATH ADVISEMENT GUIDE
MATH ADVISEMENT GUIDE Recommendations for math courses are based on your placement results, degree program and career interests. Placement score: MAT 001 or MAT 00 You must complete required mathematics
More informationMath Placement Test Practice Problems
Math Placement Test Practice Problems The following problems cover material that is used on the math placement test to place students into Math 1111 College Algebra, Math 1113 Precalculus, and Math 2211
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationTrigonometric Functions: The Unit Circle
Trigonometric Functions: The Unit Circle This chapter deals with the subject of trigonometry, which likely had its origins in the study of distances and angles by the ancient Greeks. The word trigonometry
More informationDIPLOMA IN ENGINEERING I YEAR
GOVERNMENT OF TAMILNADU DIRECTORATE OF TECHNICAL EDUCATION DIPLOMA IN ENGINEERING I YEAR SEMESTER SYSTEM L - SCHEME 0-0 I SEMESTER ENGINEERING MATHEMATICS - I CURRICULUM DEVELOPMENT CENTER STATE BOARD
More informationChapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs
Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More informationAngles and Quadrants. Angle Relationships and Degree Measurement. Chapter 7: Trigonometry
Chapter 7: Trigonometry Trigonometry is the study of angles and how they can be used as a means of indirect measurement, that is, the measurement of a distance where it is not practical or even possible
More informationFunction Name Algebra. Parent Function. Characteristics. Harold s Parent Functions Cheat Sheet 28 December 2015
Harold s s Cheat Sheet 8 December 05 Algebra Constant Linear Identity f(x) c f(x) x Range: [c, c] Undefined (asymptote) Restrictions: c is a real number Ay + B 0 g(x) x Restrictions: m 0 General Fms: Ax
More informationExtra Credit Assignment Lesson plan. The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam.
Extra Credit Assignment Lesson plan The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam. The extra credit assignment is to create a typed up lesson
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationopp (the cotangent function) cot θ = adj opp Using this definition, the six trigonometric functions are well-defined for all angles
Definition of Trigonometric Functions using Right Triangle: C hp A θ B Given an right triangle ABC, suppose angle θ is an angle inside ABC, label the leg osite θ the osite side, label the leg acent to
More informationRIGHT TRIANGLE TRIGONOMETRY
RIGHT TRIANGLE TRIGONOMETRY The word Trigonometry can be broken into the parts Tri, gon, and metry, which means Three angle measurement, or equivalently Triangle measurement. Throughout this unit, we will
More informationAppendix 3 IB Diploma Programme Course Outlines
Appendix 3 IB Diploma Programme Course Outlines The following points should be addressed when preparing course outlines for each IB Diploma Programme subject to be taught. Please be sure to use IBO nomenclature
More informationBiggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More informationAlgebra I Credit Recovery
Algebra I Credit Recovery COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics,
More informationBy Clicking on the Worksheet you are in an active Math Region. In order to insert a text region either go to INSERT -TEXT REGION or simply
Introduction and Basics Tet Regions By Clicking on the Worksheet you are in an active Math Region In order to insert a tet region either go to INSERT -TEXT REGION or simply start typing --the first time
More informationNEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document
More information096 Professional Readiness Examination (Mathematics)
096 Professional Readiness Examination (Mathematics) Effective after October 1, 2013 MI-SG-FLD096M-02 TABLE OF CONTENTS PART 1: General Information About the MTTC Program and Test Preparation OVERVIEW
More informationHomework 2 Solutions
Homework Solutions 1. (a) Find the area of a regular heagon inscribed in a circle of radius 1. Then, find the area of a regular heagon circumscribed about a circle of radius 1. Use these calculations to
More informationSection 5-9 Inverse Trigonometric Functions
46 5 TRIGONOMETRIC FUNCTIONS Section 5-9 Inverse Trigonometric Functions Inverse Sine Function Inverse Cosine Function Inverse Tangent Function Summar Inverse Cotangent, Secant, and Cosecant Functions
More informationExercises in Mathematical Analysis I
Università di Tor Vergata Dipartimento di Ingegneria Civile ed Ingegneria Informatica Eercises in Mathematical Analysis I Alberto Berretti, Fabio Ciolli Fundamentals Polynomial inequalities Solve the
More informationTSI College Level Math Practice Test
TSI College Level Math Practice Test Tutorial Services Mission del Paso Campus. Factor the Following Polynomials 4 a. 6 8 b. c. 7 d. ab + a + b + 6 e. 9 f. 6 9. Perform the indicated operation a. ( +7y)
More informationSection 5.0A Factoring Part 1
Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) (
More informationTOPIC 4: DERIVATIVES
TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the
More informationAMSCO S Ann Xavier Gantert
AMSCO S Integrated ALGEBRA 1 Ann Xavier Gantert AMSCO SCHOOL PUBLICATIONS, INC. 315 HUDSON STREET, NEW YORK, N.Y. 10013 Dedication This book is dedicated to Edward Keenan who left a profound influence
More informationC3: Functions. Learning objectives
CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms one-one and man-one mappings understand the terms domain and range for a mapping understand the
More informationStudents Currently in Algebra 2 Maine East Math Placement Exam Review Problems
Students Currently in Algebra Maine East Math Placement Eam Review Problems The actual placement eam has 100 questions 3 hours. The placement eam is free response students must solve questions and write
More informationChapter 7 Outline Math 236 Spring 2001
Chapter 7 Outline Math 236 Spring 2001 Note 1: Be sure to read the Disclaimer on Chapter Outlines! I cannot be responsible for misfortunes that may happen to you if you do not. Note 2: Section 7.9 will
More informationIntegrating algebraic fractions
Integrating algebraic fractions Sometimes the integral of an algebraic fraction can be found by first epressing the algebraic fraction as the sum of its partial fractions. In this unit we will illustrate
More informationSelf-Paced Study Guide in Trigonometry. March 31, 2011
Self-Paced Study Guide in Trigonometry March 1, 011 1 CONTENTS TRIGONOMETRY Contents 1 How to Use the Self-Paced Review Module Trigonometry Self-Paced Review Module 4.1 Right Triangles..........................
More informationALGEBRA 2/TRIGONOMETRY
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Tuesday, June 1, 011 1:15 to 4:15 p.m., only Student Name: School Name: Print your name
More informationThe Method of Partial Fractions Math 121 Calculus II Spring 2015
Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method
More informationVersion. General Certificate of Education (A-level) January 2013. Mathematics MPC3. (Specification 6360) Pure Core 3. Final.
Version General Certificate of Education (A-level) January Mathematics MPC (Specification 66) Pure Core Final Mark Scheme Mark schemes are prepared by the Principal Eaminer and considered, together with
More information*X100/12/02* X100/12/02. MATHEMATICS HIGHER Paper 1 (Non-calculator) NATIONAL QUALIFICATIONS 2014 TUESDAY, 6 MAY 1.00 PM 2.30 PM
X00//0 NTIONL QULIFITIONS 0 TUESY, 6 MY.00 PM.0 PM MTHEMTIS HIGHER Paper (Non-calculator) Read carefully alculators may NOT be used in this paper. Section Questions 0 (0 marks) Instructions for completion
More informationX On record with the USOE.
Textbook Alignment to the Utah Core Algebra 2 Name of Company and Individual Conducting Alignment: Chris McHugh, McHugh Inc. A Credential Sheet has been completed on the above company/evaluator and is
More informationNegative Integral Exponents. If x is nonzero, the reciprocal of x is written as 1 x. For example, the reciprocal of 23 is written as 2
4 (4-) Chapter 4 Polynomials and Eponents P( r) 0 ( r) dollars. Which law of eponents can be used to simplify the last epression? Simplify it. P( r) 7. CD rollover. Ronnie invested P dollars in a -year
More informationPrecalculus REVERSE CORRELATION. Content Expectations for. Precalculus. Michigan CONTENT EXPECTATIONS FOR PRECALCULUS CHAPTER/LESSON TITLES
Content Expectations for Precalculus Michigan Precalculus 2011 REVERSE CORRELATION CHAPTER/LESSON TITLES Chapter 0 Preparing for Precalculus 0-1 Sets There are no state-mandated Precalculus 0-2 Operations
More informationPRE-CALCULUS GRADE 12
PRE-CALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.
More informationExact Values of the Sine and Cosine Functions in Increments of 3 degrees
Exact Values of the Sine and Cosine Functions in Increments of 3 degrees The sine and cosine values for all angle measurements in multiples of 3 degrees can be determined exactly, represented in terms
More information15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors
SECTION 5. Eact First-Order Equations 09 SECTION 5. Eact First-Order Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential
More informationCurriculum Map by Block Geometry Mapping for Math Block Testing 2007-2008. August 20 to August 24 Review concepts from previous grades.
Curriculum Map by Geometry Mapping for Math Testing 2007-2008 Pre- s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)
More informationSTD. XI Commerce Secretarial Practice
Written as per the revised syllabus prescribed by the Maharashtra State Board of Secondary and Higher Secondary Education, Pune. STD. XI Commerce Secretarial Practice Salient Features: Exhaustive coverage
More informationPolynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4.
_.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic
More informationVersion 1.0. General Certificate of Education (A-level) January 2012. Mathematics MPC4. (Specification 6360) Pure Core 4. Final.
Version.0 General Certificate of Education (A-level) January 0 Mathematics MPC (Specification 660) Pure Core Final Mark Scheme Mark schemes are prepared by the Principal Eaminer and considered, together
More informationMATH BOOK OF PROBLEMS SERIES. New from Pearson Custom Publishing!
MATH BOOK OF PROBLEMS SERIES New from Pearson Custom Publishing! The Math Book of Problems Series is a database of math problems for the following courses: Pre-algebra Algebra Pre-calculus Calculus Statistics
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More information9.3 OPERATIONS WITH RADICALS
9. Operations with Radicals (9 1) 87 9. OPERATIONS WITH RADICALS In this section Adding and Subtracting Radicals Multiplying Radicals Conjugates In this section we will use the ideas of Section 9.1 in
More informationMathematics Online Instructional Materials Correlation to the 2009 Algebra I Standards of Learning and Curriculum Framework
Provider York County School Division Course Syllabus URL http://yorkcountyschools.org/virtuallearning/coursecatalog.aspx Course Title Algebra I AB Last Updated 2010 - A.1 The student will represent verbal
More informationAlgebra 1 2008. Academic Content Standards Grade Eight and Grade Nine Ohio. Grade Eight. Number, Number Sense and Operations Standard
Academic Content Standards Grade Eight and Grade Nine Ohio Algebra 1 2008 Grade Eight STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express
More informationSOLVING TRIGONOMETRIC EQUATIONS
Mathematics Revision Guides Solving Trigonometric Equations Page 1 of 17 M.K. HOME TUITION Mathematics Revision Guides Level: AS / A Level AQA : C2 Edexcel: C2 OCR: C2 OCR MEI: C2 SOLVING TRIGONOMETRIC
More informationFINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether or not the relationship shown in the table is a function. 1) -
More informationHIGH SCHOOL: GEOMETRY (Page 1 of 4)
HIGH SCHOOL: GEOMETRY (Page 1 of 4) Geometry is a complete college preparatory course of plane and solid geometry. It is recommended that there be a strand of algebra review woven throughout the course
More informationBig Ideas in Mathematics
Big Ideas in Mathematics which are important to all mathematics learning. (Adapted from the NCTM Curriculum Focal Points, 2006) The Mathematics Big Ideas are organized using the PA Mathematics Standards
More informationPre Calculus Math 40S: Explained!
Pre Calculus Math 0S: Eplained! www.math0s.com 0 Logarithms Lesson PART I: Eponential Functions Eponential functions: These are functions where the variable is an eponent. The first tpe of eponential graph
More informationTaylor Polynomials. for each dollar that you invest, giving you an 11% profit.
Taylor Polynomials Question A broker offers you bonds at 90% of their face value. When you cash them in later at their full face value, what percentage profit will you make? Answer The answer is not 0%.
More informationPolynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter
Mathematics Learning Centre Polnomials Jackie Nicholas Jacquie Hargreaves Janet Hunter c 26 Universit of Sdne Mathematics Learning Centre, Universit of Sdne 1 1 Polnomials Man of the functions we will
More informationVersion 1.0. General Certificate of Education (A-level) June 2013. Mathematics MPC3. (Specification 6360) Pure Core 3. Final.
Version.0 General Certificate of Education (A-level) June 0 Mathematics MPC (Specification 660) Pure Core Final Mark Scheme Mark schemes are prepared by the Principal Eaminer and considered, together with
More informationPHILOSOPHY OF THE MATHEMATICS DEPARTMENT
PHILOSOPHY OF THE MATHEMATICS DEPARTMENT The Lemont High School Mathematics Department believes that students should develop the following characteristics: Understanding of concepts and procedures Building
More informationVersion 1.0. klm. General Certificate of Education June 2010. Mathematics. Pure Core 3. Mark Scheme
Version.0 klm General Certificate of Education June 00 Mathematics MPC Pure Core Mark Scheme Mark schemes are prepared by the Principal Eaminer and considered, together with the relevant questions, by
More informationIndiana State Core Curriculum Standards updated 2009 Algebra I
Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and
More informationParallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.
CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes
More informationReview of Fundamental Mathematics
Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools
More informationMathematics Pre-Test Sample Questions A. { 11, 7} B. { 7,0,7} C. { 7, 7} D. { 11, 11}
Mathematics Pre-Test Sample Questions 1. Which of the following sets is closed under division? I. {½, 1,, 4} II. {-1, 1} III. {-1, 0, 1} A. I only B. II only C. III only D. I and II. Which of the following
More informationMidterm 2 Review Problems (the first 7 pages) Math 123-5116 Intermediate Algebra Online Spring 2013
Midterm Review Problems (the first 7 pages) Math 1-5116 Intermediate Algebra Online Spring 01 Please note that these review problems are due on the day of the midterm, Friday, April 1, 01 at 6 p.m. in
More informationStart Accuplacer. Elementary Algebra. Score 76 or higher in elementary algebra? YES
COLLEGE LEVEL MATHEMATICS PRETEST This pretest is designed to give ou the opportunit to practice the tpes of problems that appear on the college-level mathematics placement test An answer ke is provided
More information