Ritow - Capsule Calculus
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Abstract: This brief introductory text presents the basic principles of calculus from the engineering viewpoint. Excellent either as a refresher or as an introductory course, it focuses on developing familiarity with the basic principles rather than presenting detailed proofs. Topics include differential calculus, in terms of differentiation and elementary differential equations; integral calculus, in simple and multiple integration forms; time calculus; equations of motion and their solution; complex variables; complex algebra; complex functions; complex and operational calculus; and simple and inverse transformations. Advanced subjects comprise integrations and differentiation techniques, in addition to a more sophisticated variety of differential equations than those previously discussed. It is assumed that the reader possesses an acquaintance with algebra and trigonometry as well as some familiarity with graphs. Additional background material is presented as needed
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TABLE I
SOME BASIC DERIVATIVES
 | ( e is any constant) |
 | |
 | |
 | |
 |
TABLE II
DERIVATIVES OF INVOLVED EXPRESSIONS
| Rule 1. |  |
| Rule 2. |  |
| Rule 3. |  |
| Rule 4. |  |
COMMONLY USED DERIVATIVES
 |
 |
 |
| ex dx = ex + c |
| sin x dx = cos x + e |
| cos x dx = sin x + e |
 |
 |
 |
| x is a variable; n is an integers; c and a are numbers. |
TABLE IV
LAWS OF INTEGRALS
| Rule 1. | az dx = a z dx |
| Rule 2. | (z + y) dx = z dx + y dx |
| Rule 3. |  |
| Rule 4. |  |
| Symbols x, y , and z are variables. a is a number. |
TABLE V
COMMONLY USED SYMBOLS FOR DIFFERENTIATION AND INTEGRATION WITH RESPECT TO TIME
ELEMENTARY LAPLACE TRANSFORMS
| Real Quantity | Transformed Quantity |
|---|---|
| V 2 | V 2 p , |
| V 2 | pV 2 p V 2(0) |
| 1 | 1/ p |
| t | 1/ p 2 |
| e at |  |
TABLE VII
LAPLACE TRANSFORMS
| Time Function ( x ) | Laplace transform ( X ) |
| 1 |  |
| unit step at t=0+ |  |
| e kt |  |
| t |  |
| t 2 |  |
| tn |  |
| te kt |  |
| sin kt |  |
| cos kt |  |
 |  |
 |  |
TABLE VIII
LAPLACE TRANSFORM THEOREMS
DEFINITIONS: Assume that x and y are related numbers with some physical meaning. For example, x might be the distance driven by a car and y the amount of gas in the gas tank.
As shown in , for every value of x there exists a value of y . This can also be stated as: y is a function of x [sometimes abbreviated as y = f ( x )]. With but a single value of y for every value of x in this case, it can be said that y is a single valued function of x . If for every value of x there were two or more possible values of y , the statement of their relationship would be: y is a multiple-valued function of x . If y is a single-valued function of x , then x may or may not also be a single-valued function of y.
In plotting the relationship between distance and gasoline for a long trip, such as shown in , x is a multiple-valued function of y.
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