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E-grāmata: Calculus Problems

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  • Formāts: PDF+DRM
  • Sērija : La Matematica per il 3+2 101
  • Izdošanas datums: 01-Nov-2016
  • Izdevniecība: Springer International Publishing AG
  • Valoda: eng
  • ISBN-13: 9783319154282
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Calculus ProblemsPalielināt
  • Formāts: PDF+DRM
  • Sērija : La Matematica per il 3+2 101
  • Izdošanas datums: 01-Nov-2016
  • Izdevniecība: Springer International Publishing AG
  • Valoda: eng
  • ISBN-13: 9783319154282
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This book is intended as a practical working guide for students in Engineering, Mathematics, Physics, or any other field where rigorous calculus is needed. Each chapter starts with a concise summary of the main results that should be kept in mind and used for the exercises in that chapter, which is then followed by a selection of solved problems. A selection of exercises without solutions rounds out each chapter. The primary focus is on functions of one real variable, but some differential calculus of functions of two or more variables is also included. As for basic ordinary differential equations, separation of variables, linear first order and constant coefficients ODEs are discussed.

1Manipulation of Graphs.- 2 Invertible Mappings.- 3 Sequences.- 4 Maxima,minima, least upper bound, greatest lower bound.- 5 Limits of functions.- 6 ContinuousFunctions.- 7 Differentiable Functions.- 8 Taylor Expansions.- 9 Geometry offunctions: extrema and convexity.- 10 Indefinite and Definite Integrals.- 11 ImproperIntegrals and Integral Functions.- 12 Numerical Series.- 13 Separation ofVariables.- 14 First Order Linear Differential Equations.- 15 ConstantCoefficient Differential Equations.
1 Manipulation of Graphs
1(28)
1.1 Operations on Graphs
1(4)
1.2 Guided Exercises on Graphs
5(20)
1.3 Problems on Graphs
25(4)
2 Invertible Mappings
29(12)
2.1 Injective, Surjective and Bijective Mappings
29(1)
2.2 Inversion of a Map
30(1)
2.3 Monotone Functions
31(1)
2.4 Guided Exercises on Invertible Mappings
32(6)
2.5 Problems on Invertible Mapings
38(3)
3 Maximum, Minimum, Supremum, Infimum
41(12)
3.1 Upper and Lower Bounds, Maximum and Minimum
41(1)
3.2 Supremum, Infimum
42(1)
3.3 Guided Exercises on Maximum, Minimum, Supremum, Infimum
43(7)
3.4 Problems on Maximum, Minimum, Supremum, Infimum
50(3)
4 Sequences
53(12)
4.1 Lists of Real Numbers
53(1)
4.2 Convergence Notions
54(1)
4.3 Results on Limits
55(3)
4.4 Guided Exercises on Sequences
58(5)
4.5 Problems on Sequences
63(2)
5 Limits of Functions
65(24)
5.1 Convergence Notions
65(3)
5.2 Results on Limits
68(3)
5.3 Local Comparison of Functions
71(2)
5.4 Orders
73(3)
5.5 Guided Exercises on Limits of Functions
76(10)
5.6 Problems on Limits
86(3)
6 Continuous Functions
89(16)
6.1 Basic Properties of Continuous Functions
89(1)
6.2 Discontinuities, Continuous Extensions
90(1)
6.3 Global Properties of Continuous Functions
91(2)
6.4 Continuous Monotonic Functions
93(1)
6.5 Guided Exercises on Continuous Functions
94(7)
6.6 Problems on Continuity
101(4)
7 Differentiable Functions
105(26)
7.1 The Derivative of a Function
105(4)
7.2 Derivatives of Elementary Functions
109(1)
7.3 The Classical Theorems of Differential Calculus
110(3)
7.4 Guided Exercises on Differentiable Functions
113(14)
7.5 Problems on Differentiability
127(4)
8 Taylor Expansions
131(16)
8.1 Taylor Expansions
131(3)
8.2 Guided Exercises on Taylor Expansions
134(8)
8.3 Problems on Taylor Expansions
142(5)
9 The Geometry of Functions
147(20)
9.1 Asymptotes
147(2)
9.2 Convexity and Concavity, Inflection Points
149(3)
9.3 The Nature of Critical Points
152(1)
9.4 Guided Exercises on the Geometry of Functions
153(12)
9.5 Problems on the Geometry of Functions
165(2)
10 Indefinite and Definite Integrals
167(24)
10.1 Primitive Functions
167(2)
10.2 Computing Indefinite Integrals
169(3)
10.2.1 General Techniques
169(1)
10.2.2 Rational Functions
170(2)
10.3 Riemann Integrals
172(3)
10.4 The Fundamental Theorem of Calculus
175(1)
10.5 Guided Exercises on Integration
176(11)
10.6 Problems on Integration
187(4)
11 Improper Integrals and Integral Functions
191(26)
11.1 Improper Integrals
191(3)
11.2 Convergence Criteria
194(1)
11.3 Integral Functions
195(1)
11.4 Guided Exercises on Improper Integrals and Integral Functions
196(17)
11.4.1 Improper Integrals
196(5)
11.4.2 Integral Functions
201(12)
11.5 Problems on Improper Integrals and Integral Functions
213(4)
12 Numerical Series
217(20)
12.1 Convergence
217(3)
12.2 Positive Series: Criteria
220(2)
12.3 Order, Series and Integrals
222(1)
12.4 Alternating Series
223(1)
12.5 Guided Exercises on Numerical Series
224(9)
12.6 Problems on Series
233(4)
13 Separation of Variables
237(22)
13.1 Differential Equations
237(1)
13.2 The Method of Separation of Variables
238(2)
13.3 Guided Exercises on Separation of Variables
240(14)
13.4 Problems on Separation of Variables
254(5)
14 First Order Linear Differential Equations
259(16)
14.1 First Order Linear Equations with Continuous Coefficients
259(1)
14.2 Guided Exercises on Linear First Order Equations
260(9)
14.3 Problems on Linear First Order Equations
269(6)
15 Constant Coefficient Linear Differential Equations
275(22)
15.1 Linear Equations with Constant Coefficients
275(1)
15.2 The Homogeneous Equation
276(2)
15.3 The Nonhomogeneous Equation
278(3)
15.4 Guided Exercises on Constant Coefficient Differential Equations
281(14)
15.5 Problems on Constant Coefficient Differential Equations
295(2)
16 Miscellaneous
297(10)
16.1 Problems
297(10)
Appendix A Basic Facts and Notation 307(12)
Appendix B Calculus 319(6)
Solutions 325(36)
Further Reading 361(2)
Index 363
Marco Baronti was born in Genova in 1956. Since 1990 he is Associate Professor in Mathematical Analysis at the University of Genova. His scientific interests are mainly in Functional Analysis and in particular in Geometry of Banach Spaces.  Filippo De Mari was born in Genova in 1959. In 1987 he received his Ph.D. from Washington University in St. Louis, USA. Since 1998 he is Associate Professor in Mathematical Analysis at the University of Genova. His scientific interests are mainly in Harmonic Analysis, Representation Theory and Lie Groups. Robertus van der Putten was born in Sanremo in 1959. In 1989 he received his Ph.D. from the University of Milan. Since 1990 he is Researcher in Mathematical Analysis at the University of Genova. His scientific interests are mainly in Calculus of Variations.  Irene Venturi was born in Viareggio in 1978. In 2009 she received her Ph.D. from the University of Genova and in 2011 a Masters in Security Safety and Sustainibility in Transportation Systems. She is a teacher in Mathematics and has several editorial collaborations.
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