MINISTRY OF EDUCATION OF AZERBAIJAN REPUBLIC AZERBAIJAN UNIVERSITY OF ARCHITECTURE AND CONSTRUCTION Nigar M. Aslanova H I G H E R M A T H E M A T I C S - 1 TEXTBOOK (Dərs vəsaiti) Dərs vəsaiti Azərbaycan Memarlıq və İnşaat Universitetinin Tədris-Metodiki Şurasının 28 fevral 2017-ci il tarixli iclasının qərarı ilə nəşr olunur(protocol No3) BAKU – 2017 2 MSC 26-XX, 54 C30, 40-XX, 51-XX In textbook matrix theory, theory of system of linear equations, analytic geometry and differential and integral calculus of function of one variable are dcescribed. Throught the text given a lot of examples and exercises in order to make material suitable for the calculus audience. For first –year mathematics course of Azerbaijan University of Architecture and Conctruction . Textbook is also usefull for students and math teachers of technical and economical universities . (Bu dərs vəsaiti Azərbaycan Memarlıq və İnşaat Universitetinin tələbələri üçün nəzərdə tutulmuşdur. Lakin vəsaitdən, həmçinin, digər texniki və iqtisad universitetlərinin müəllim və tələbələri də istifadə edə bilərlər.) Reviewers 1.The principle researcher of department Differtential Equations of IMM of NASA prof. M.Bayramoglu 2.Chief of department Higher Mathematics prof. F.S.Latifov 3 Contents Preface .................................................... 6 CHAPTER 1 The space and the matrices ..... 8 §1. Space R n ............................................. 8 §2. Matrices .............................................. 13 §3.Inverse matrices ..................................... 22 §4. Determinants ........................................ 27 §5. Calculation of inverse matrix .................... 45 §6. Linear systems of equations ....................... 46 CHAPTER 2 Rectangular coordinate systems .................................................... 55 §1. The rectangular coordinate system in plane and space ................................................. 55 §2. Distance formula ................................... 59 §3. Polar and spherical coordinates ................. 63 CHAPTER 3 Vectors in plane and space ......... 67 §1. Vectors in plane .................................... 67 §2. Vectors in space .................................... 78 §3. Dot product of vectors ............................. 86 §4. The cross product of vectors ..................... 95 CHAPTER 4 Analytic geometry .................... 107 §1. Straight lines in plane ........................... 107 §2. Straight lines in space ........................... 115 §3. Planes in 3 space ................................... 123 §4. Quadric lines ....................................... 131 §5. Quadric surfaces ................................... 167 n R 4 CHAPTER 5 Number sets ............................ 180 §1. Sets, real numbers and operations on them ... 180 CHAPTER 6 Functions of one variable ........... 200 §1. Functions and operations on them ............... 200 §2. Limit of the function ................................. 215 §3. Continuity of the function .......................... 235 §4. Inverse function ...................................... 247 CHAPTER 7 Differentiation ......................... 251 §1. Derivative ............................................ 251 §2. Techniques of differentiation ...................... 260 §3. Higher order derivatives ............................ 265 §4. Derivatives of trigonometric and logarithmic functions .................................................. 267 §5. Differentials ......................................... 271 §6. Derivative of inverse function ..................... 276 §7. Derivatives involving exponential and inverse trigonometric functions ................................... 277 §8. The chain rule ....................................... 281 §9. Parametric representation of function and its differentiation ............................................ 285 CHAPTER 8 Applications of differentiation to function of one variable ............................... 286 §1. Extreme values of function ......................... 286 §2. Rolle’s theorem, Mean –value theorem ........ 292 §3.Intervals of increase and decrease. Concavity ... 299 §4. First and second derivative tests .................. 305 §5.Applied maximum and minimum problems ....... 315 §6. L’Hopital’s Rule .................................... 320 §7. Other indeterminate forms ......................... 331 5 CHAPTER 9 Integration ............................. 341 §1. Antiderivatives ...................................... 341 §2. Integration by substitution ....................... 350 §3. Integration by parts .............................. 357 §4. Reduction formulas ................................ 361 §5. Integrating of Rational functions ............... 363 §6. Miscellaneous substitutions ...................... 365 Answers to some Even-Numbered Problems ....... 368 Literature ................................................ 383 6 P R E F A C E The purpose of this book is to give an introduction to the theory of linear algebra, cover theory of differential and integral calculus of function of one variable and analytic geometry. I hope that students that study higher mathematics at their first semester will be able to read this book without major difficulties. There are numerous examples scattered through the text, which are intended to make understanding of material easier. In chapter 1 the space R n is introduced, given theory of matrices, determinats, and their applications to system of linear equations. Chapter 2 includes elements of vector algebra. In chapter 3 given description of vectors in plane and space. Chapter 4 covers main topics in analytic geometry. In chapter 5 given notions of sets, real numbers and operations on them. Chapters 6 and 7 are devoted to differential calculus of function of one variable. Fundamental theorems of calculus are given with detailed proofs. Chapter 8 covers appreciations of differential calculus of function of one variable. In chapter 9 theory of integral calculus of function of one variable is developed. There are a lot of problems throught the book which mainly serve to give opportunity for practice. 7 The level of difficulty of the problems varies widely. Because I give difficult ones with detailed hints, they can be solved in general without much difficulty. Now I want to very heartily thank all those who helped me with production of this book. Professor M.Bayramoglu and professor F.S.Latifov spent much time reading the whole manuscript and discussing with me their suggestions for improvement. Ms. A.Rahmanova turned my notes into an excellent typed manuscript. I thank all of them for their pleasant cooperation. 8 CHAPTER 1 The space n R and the matrices §1. Space n R 1 . Vectors from n R and their dot roduct. Definition1. An n -component row vector is an ordered set of n numbers written as ( , , ... , ) Definition 2. An n -component column vector is an ordered set of n numbers written as n b b b ... 2 1 In Definition 1 1 a is said to be the first coordinate of the vector, 2 a is the second coordinate and so on. Generally, i a is called the i -th component of the row vector. In Definition 2 1 b is the first component, 2 b is the second and so on. Sometimes we shall refer to n component row vector as a row vector or an n vector. Similar to that, the term column vector (or- n vector) will be used to denote an n component column vector. Vector with all zero entries is called a zero vector. Example 1. The following are examples of vectors: 1) (2,4) is a 2 component row vector 9 2) 4 6 3 is a 3- component column vector 3) (5,6,-8,0) is a 4-component row vector 4) 0 0 7 8 4 is a 5-component column vector 5) (0,0,0,0) is a 4- component zero vector We shall denote vectors with small letters like a,b,c,d. A zero vector is denoted to distinguish with number zero. Now we shall state main properties of vectors. Since it is the same we shall first derive all properties and definitions in terms of row vector. The coordinates of all considered vectors in this book are real or complex numbers. We use the notation n R to denote the set of n -vectors ) ,..., , ( 2 1 n x x x , here n i x i , 1 , is a real number. Definition 2. Two row vectors u and v are equal if and only if the number of coordinates of that vectors are the same and their coordinates with the same indices are equal. Thus, the vectors ) ,..., , ( 2 1 n u u u u and ) ,..., , ( 2 1 n v v v v are equal if and only if n n v u v u v u ,..., , 2 2 1 1 Definition 3. Take two ) ,..., , ( 2 1 n u u u u and ) ,..., , ( 2 1 n v v v v row vectors. Then the addition of u and v is by definition 10 ) ,..., , ( 2 2 1 1 n n v u v u v u v u Example 2. (2,3,4,5)+(-1,0,2,6)=(1,3,6,11) Example 3 . (-3,4,1)+(6,8,7)=(3,12,8 ) Definition 4 Let ) ,..., , ( 2 1 n u u u u be a vector and scalar (number). Then the scalar multiplication u is defined by ) ,..., , ( 2 1 n u u u u So to multiply a vector by a number, we only multiply each coordinate of the vector by that number. Example 4 . 4(-5,6,2)=(-20,24,8) Note. Combining Definition 3 and Definition 4, one can give the difference of two vectors by , ) 1 ( v u v u which means if ) ,..., , ( 2 1 n u u u u and ) ,..., , ( 2 1 n v v v v then ) ,.., , ( 2 2 1 1 n n u v u v u v u v Example 5. Let 0) (3,8,7,-1, u , v = (2,0,1,0,1). Evaluate 4 u +2 v = 4(3,8,7,-1,0) + 2(2,0,1,0,1) = (12,32,28,-4,0) + +(4,0,2,0,2) = (16,32,30,-4,2) Once knowing how to find the sum of two vectors and find their products by scalars, one can prove the next facts regarding that operations. We shall prove some parts and leave the remaining parts as exercises. Theorem 1. Let u, v and w be n-component vectors, and s , be numbers. Then 1) 0 u ; 3) u v v u (commutative low); 2) u 0 ; 4) ) ( ) ( w v u w v u (associative low); 11 5) v u v u ) ( (distributive low for scalar multiplication); 6) u u u ) ( ; 7) ) ( ) ( u u Proof. 2) If ) ,..., , ( 2 1 n u u u u then ) 0 ,..., 0 , 0 ( ) ,..., , ( 0 0 2 1 n u u u u 3) Let ) ,..., , ( 2 1 n v v v v . Then ) ,..., , ( ) ,..., , ( 2 2 1 1 2 2 1 1 u v u v u v u v v u v u v u v u n n n n Here it was used that for any two numbers x y y x have we y x , , and 0 = . Definition 5 Let ) ,..., , ( 2 1 n u u u u and ) ,..., , ( 2 1 n v v v v be two n-vectors. Then the dot product (it is also called the inner product or the scalar product) of u and v is denoted by v u and by definition is n n v u v u v u v u ... 2 2 1 1 Sometimes we shall take the dot product of a row and column vectors: n n n n v u v u v u v v v u u u ... ) ,... , ( 2 2 1 1 2 1 2 1 Example 6. Let u = (-3,0,4), v =(1,1,8) calculate v u Solution ∙ = − 3 ∙1 + 0 ∙1 + 4 ∙8 = 9 Example 7 . Let u =(1,2,-4,5) and 0 2 1 3 . Evaluate v u 12 Solution. We have 15 3 5 1 (-4) 2 2 0 1 v u Theorem 2. Let u,v and w be n- component vectors and and be numbers. Then 1. 0 u ; 2. u v v u (commutative low for scalar product) ; 3. w v u v w v u ) ( (distributive low for sclar product) ; 4. ) ( ) ( v u v u Now we introduce the term norm, which is also called the length. Definition 6. The norm of a vector u in n R , denoted u , is defined as u u u If ) ,..., , ( 2 1 n u u u u then 2 2 2 2 1 ... n u u u u u , so that 2 2 2 2 1 ... n u u u u Note that, the norm of vector in 2 R and 3 R is said to be the magnitude of vector. Example 8. Let ) 5 , 6 , 2 , 3 ( u compute u Solution. 74 25 49 5 ) 6 ( 2 3 2 2 2 2 u Problems In problems 1-10 perform the indicated computations with ) 7 , 4 , 5 ( ), 4 , 1 , 3 ( v u and ) 2 , 0 , 2 ( w 1. v u ; 2. v 3 ; 3. w 2 ; 4. w v 3 ; 5. v u 5 2 ; 6. w v 2 3 ; 13 7. c 0 ; 8. w v u ; 9. w v u 4 2 3 ; 10. v w v 2 7 3 11. Let ) ,..., , ( 2 1 n u u u u , ) ,..., , ( 2 1 n v v v v and ) ,..., , ( 2 1 n w w w w . Compute ) ( ) ( w v u w v u 12. Find numbers , and such that ) 0 , 0 , 0 ( ) ( ) 4 , 1 , 2 ( §2. Matrices 1. Matrices and operations on them. An × matrix A is a rectangular array of lm numbers placed in rows and m columns: = ... ... ... ... ... ... ... The component ij a of matrix A is the number located in the intersection of i- th row and j -th columns of matrix A Sometimes we will write ) ( ij a A . It is usual to denote matrices by uppercase letters A,B,C. If for matrix A l=m , then A is called a square matrix . A matrix whose elements equal to zero is called zero matrix. An × matrix is said to have order (or size, dimension) × . Two matrices are equal if they have the same orders and their elements with the same indices are equal. Example 1. 14 1) 6 5 4 3 , 2 2 (square matrix); 2) 5 3 0 2 1 4 , 2 3 matrix ; 3) 0 0 0 0 0 0 , 3 2 zero matrix; Each vector is a special kind of matrix. So, ) ,..., , ( 2 1 n x x x is n 1 matrix and n x x .. 1 is 1 n matrix. Linear operations on matrices (addition of two matrices and multiplication of matrix by number) is extension of that operations for vector to matrices. Definition 1. Let ) ( ij a A and ) ( ij b B be two m l matrices. Then their addition B A is defined by lm lm l l l l m m m m ij ij b a b a b a b a b a b a b a b a b a b a B A ... ... ... ... ... ... ... ) ( 2 2 1 1 2 2 22 22 21 21 1 1 12 12 11 11 Thus, A+B is the m l matrix found by adding the corresponding elements of matrices A and B. Example 2. 6 0 6 2 1 9 1 1 2 0 2 6 5 1 4 2 3 3 Definition 2. If A is an m l matrix, and if is a 15 number, then m l matrix A is given by lm l l m m a a a a a a a a a A ... ... ... ... ... ... ... 2 1 2 22 21 1 12 11 Thus, A is the matrix obtained by multiplying each element of A by number Example 3. If 2 3 1 4 6 2 5 2 4 1 2 3 A then 6 9 3 12 18 6 15 6 12 3 6 9 3 A Example 4. Let 1 0 2 5 4 3 B and 3 6 8 2 2 1 C Compute 2 B -4 C. Solution. 2 4 0 10 8 6 3 8 6 2 2 1 4 1 2 0 5 4 3 2 4 2 C B 10 28 24 2 0 10 12 32 24 8 8 4 The proof of the next theorem is left as an exercise. Theorem 1. Let A, B, and C be m l matrices and let be a scalar. Then 1) A O A ; 16 2) O A 0 ; 3) A B B A ; 4) ) ( C B A C B A ; 5) B A B A Problems In problems 1-7 perform the indicated computations. If , 2 1 5 2 3 1 A 5 7 4 1 0 2 B and , 3 7 6 4 1 1 C 1. 3 A ; 2. A+B ; 3. A-C ; 4. 2 C -5 A ; 5. A+B+C ; 6. C-A-B ; 7. Find a matrix D such that 2A+B-D is the 2 3 zero matrix. In problems 8-16 perform the indicated computation with 1 1 0 5 4 3 2 1 1 A , 0 6 7 5 0 3 1 2 0 B and 4 2 0 0 1 3 2 0 0 C 8. A -2 B; 9. 3 A-C ; 10. A+B+C; 11. 2 A-B+ 2 C ; 12. C-A-B; 13. 4 C -2 B +3 A 14. Let B=(b ij ) be n m matrix and let O is n m zero matrix. Use Definitions 1 and 2 to show that O A 0 and A A O 15. Let ) ( ij a A and ) ( ij b B be n m matrices compute A+B and B+A show that they are equal. 16. If ) ( ij a A , ) ( ij b B and ) ( ij c C are n m matrices, compute ( A+B )+ C and A +( B+C ) and show that 17 they are equal. 2.Multiplication of matrices. Definition 1. Let ) ( ij a A be an n m matrix whose i-th row is denoted i a Let ) ( ij b B be an p n matrix whose j-th column is denoted by ij b Then the product of A and B is an p m matrix ) ( ij c C where j i ij b a c That is the ij-th element of AB is the scalar product of the i-th row of ) ( i a A and the j-th column of ) ( j b B . In open form we have nj in j i j i ij b a b a b a c ... 2 2 1 1 Example 1. If 1 3 1 2 A and 3 2 4 2 B calculate AB and BA Solution. AB c C ij ) ( . Then, , 6 2 4 2 1 2 2 2 2 ) 1 2 ( 1 1 11 b a c , 11 3 8 3 1 4 2 3 4 ) 1 2 ( 2 1 12 b a c 4 2 6 2 1 2 3 2 2 ) 1 3 ( 1 2 21 b a c , 9 3 12 3 1 4 3 3 4 ) 1 3 ( 2 2 22 b a c Here shows -th row of matrix , and shows -th column of matrix 18 Thus 9 4 11 6 C Similarly, omitting the intermediate procedures, we get 4 5 6 8 2 2 9 4 4 2 12 4 1 3 1 2 3 2 4 2 ' BA C Remark. Example 1 illustrates an important property: Matrix products do not, in general, commute. So, BA AB in general. It sometimes happens that BA AB but this will be the exception, not the rule. The next example shows that it may happen that AB is defined but BA is not. Therefore, order of multiplying matrices is important. Example 2. Let 5 4 3 2 A and 2 1 6 2 1 0 2 1 B Calculate AB. Solution. We first note that A is 2 2 matrix and B is a 4 2 matrix. So the number of columns of A equals the number of rows of B . The product AB is therefore defined and is 4 2 matrix. Let ) ( ij c C AB .Then , 4 6 2 2 1 ) 3 2 ( 11 c , 22 18 4 6 2 ) 3 2 ( 12 c 3 1 0 ) 3 2 ( 13 c , , 8 6 2 2 1 ) 3 2 ( 14 c , 6 10 4 2 1 ) 5 4 ( 21 c 19 , 38 30 8 6 2 ) 5 4 ( 22 c , 5 5 0 1 0 ) 5 4 ( 23 c 14 10 4 2 1 ) 5 4 ( 24 c Hence, 14 5 38 6 8 3 22 4 A Note that the product BA is not defined since the number of columns of B which is not the same as the number of rows of A (which is two). Theorem 1. Let ) ( ij a A be an m n matrix, ) ( ij b B an p m matrix, and ) ( ij c C a q p matrix. Then the associative low C AB BC A ) ( ) ( (1) holds and ABC , defined by either side of (1), is q n matrix. The proof of that statement is not difficult, but it is tedious. We leave it to reader. Example 3 . Verify the associative law for , 1 1 2 0 A 2 2 2 1 B and 1 3 4 6 2 1 C Solution. First of all note that A is 2 2 B is 2 2 , and C is 3 2 matrices. We then calculate 0 1 4 4 2 2 2 1 1 1 2 0 AB 20 6 2 1 28 20 20 1 3 4 6 2 1 0 1 4 4 ) ( C AB In similar way, 14 10 10 8 8 9 1 3 4 6 2 1 0 2 2 1 BC 6 2 1 28 20 20 14 10 10 8 8 9 1 1 2 0 ) ( BC A Thus ) ( ) ( BC A C AB The associative law can be extended to longer products. For example, suppose AB , BC and CD are defined. Then ABCD=A(B(CD))=C((AB)C)D=(AB)(CD) The next is distributive law for matrix multiplication. Theorem 2. If all the following sums and products are defined, then A(B+C)=AB+AC (2) and (A+B)C=AC+BC. (3) Proof. Let A be m n and bet B and C be p m . Then kj -th component of B + C is kj kj c b and the ij -th component of A ( B + C ) is m k m k kj ik kj ik m k kj kj ik th ij c a b a c b a 1 1 1 ) ( component of AB plus ij -th component of AC and this proves (2). The proof of (3) is identical and is therefore omitted. Problems In problems 1-15 perform the indicated computations 1. 6 0 1 4 2 1 3 2 ; 2. 3 1 6 5 4 1 2 3 ;