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Baronti Marco - Calculus Problems

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Baronti Marco Calculus Problems

Calculus Problems: summary, description and annotation

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This book, intended as a practical working guide for students in Engineering, Mathematics, Physics, or any other field where rigorous calculus is needed, includes 450 exercises. Each chapter starts with a summary of the main definitions and results, which is followed by a selection of solved exercises accompanied by brief, illustrative comments. A selection of problems with indicated solutions rounds out each chapter. A final chapter explores problems that are not designed with a single issue in mind but instead call for the combination of a variety of techniques, rounding out the books coverage. Though the books primary focus is on functions of one real variable, basic ordinary differential equations (separation of variables, linear first order and constant coefficients ODEs) are also discussed. The material is taken from actual written tests that have been delivered at the Engineering School of the University of Genoa. Literally thousands of students have worked on these problems, ensuring their real-world applicability.

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Springer International Publishing Switzerland 2016
Marco Baronti , Filippo De Mari , Robertus van der Putten and Irene Venturi Calculus Problems UNITEXT 10.1007/978-3-319-15428-2_1
1. Manipulation of Graphs
Marco Baronti 1
(1)
Dipartimento di Matematica, Universit di Genova, Genova, Italy
(2)
DIME, Universit di Genova, Genova, Italy
Marco Baronti (Corresponding author)
Email:
Filippo De Mari
Email:
Robertus van der Putten
Email:
Irene Venturi
Email:
1.1 Operations on Graphs
A coordinate system determines a bijective correspondence between Picture 1 and the plane. Assume that an orthogonal system of coordinates has been fixed and henceforth systematically identify Picture 2 with the plane. Given a subset I of for example an interval and a function the graph of f is the subset of - photo 3 , for example an interval, and a function the graph of f is the subset of If the graph of f is known it is - photo 4 , the graph of f is the subset
of If the graph of f is known it is possible to understand the graphs of - photo 5
of Picture 6 . If the graph of f is known, it is possible to understand the graphs of the functions that can be obtained from f using the sum and the product, that is, the functions Picture 7 , Picture 8 , f ( ax ) and af ( x ), where a is a fixed real number. More precisely, if Calculus Problems - image 9 and Calculus Problems - image 10 are given, then the function Calculus Problems - image 11 defined on the set
Calculus Problems - image 12
by the relation
Calculus Problems - image 13
is the translation of f by a . The graph of Picture 14 is obtained from the graph of f by translating it to the right by a if and to the left by a if see Fig Fig 11 The graph of f left - photo 15 and to the left by | a | if see Fig Fig 11 The graph of f left and the graph of right - photo 16 , see Fig..
Calculus Problems - image 17
Fig. 1.1
The graph of f (left) and the graph of Calculus Problems - image 18 (right)
The graph of the function Calculus Problems - image 19 , defined by
Calculus Problems - image 20
is obtained from Picture 21 by translating it upwards by a if and downwards by a if see Fig Fig 12 The graph of f below and - photo 22 and downwards by | a | if see Fig Fig 12 The graph of f below and the graph of above - photo 23 , see Fig..
Fig 12 The graph of f below and the graph of above Similarly if - photo 24
Fig. 1.2
The graph of f (below) and the graph of Calculus Problems - image 25 (above)
Similarly, if Calculus Problems - image 26 , the dilate of f by a is the function defined on the set
Calculus Problems - image 27
by
Calculus Problems - image 28
The graph of Picture 29 is obtained from the graph of f by a horizontal dilation or contraction, according as dilation or contraction see Fig Fig 13 The original domain is - photo 30 (dilation) or Calculus Problems - image 31 (contraction), see Fig..
Calculus Problems - image 32
Fig. 1.3
The original domain is Calculus Problems - image 33 , that of Calculus Problems - image 34 is Calculus Problems - image 35
The graph of the function Calculus Problems - image 36 defined by
Calculus Problems - image 37
is obtained from Picture 38 by a vertical dilation or contraction, according as dilation or contraction see Fig Fig 14 The graph of 3 f is three - photo 39 (dilation) or contraction see Fig Fig 14 The graph of 3 f is three times wider - photo 40 (contraction), see Fig..
Fig 14 The graph of 3 f is three times wider than that of f In many - photo 41
Fig. 1.4
The graph of 3 f is three times wider than that of f
In many applications the notions of even function and that of odd function are important. In order to avoid formal complications, it is advisable to consider functions defined on symmetric domains.
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