By Dr. Adi Ben-Israel, Dr. Robert Gilbert (auth.)
This is a brand new form of calculus e-book: scholars who grasp this article is going to be good versed in calculus and, additionally, own an invaluable operating wisdom of 1 of crucial mathematical software program platforms, specifically, MACSYMA. this may equip them with the mathematical competence they want for technological know-how and engi neering and the aggressive office. the alternative of MACSYMA isn't crucial for the didactic objective of the ebook. in reality, any of the opposite significant mathematical software program platforms, e. g. , AXIOM, MATHEMATICA, MAPLE, DERIVE, or lessen, might have been taken for the examples and for buying the ability in utilizing those structures for doing arithmetic on pcs. The symbolic and numerical calcu lations defined during this booklet may be simply played in any of those platforms via mild amendment of the syntax once the scholar knows and masters the MACSYMA examples during this publication. what's very important, although, is that the scholar will get the entire info essential to layout and execute the calculations in at the very least one concrete implementation language as this is often performed during this e-book and likewise that using the mathematical software program process is totally built-in with the textual content. In those occasions of globalization, agencies that are not able to rent safely informed know-how specialists won't prosper. for firms which rely seriously on sci ence and engineering, ultimate aggressive within the worldwide economic climate would require hiring staff having had a historically rigorous mathematical education.
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Assume(a>O)$ c8. (a-x)-y d8. a XY The corresponding logarithmic transformation is performed by default, even when we do not explicitly tell MACSYMA that a > O. c9. log (a-x) d9. log(a) x We can suppress this automatic logarithmic transformation by setting the MACSYMA option variable logexpand: false. clO. log(a-x), logexpand:false dlO. log aX MACSYMA works internally with natural logarithms, written as log(x). To specify a logarithm to the base a, enter 10gb (a, x). This logarithm simplifies as follows.
3 . 10. 22) is computed (recursively) by the MACSYMA function cl. polynomial(a,x) if length(a) = 1 then part(a,1) else part(a,1)+x*polynomial(rest(a),x)$ where: a is the list of coefficients lao, aI, ... , anJ, length(a) is the number of elements in the list a, part (a, n) is the nth term in the list a, rest (a,n) gives what is left of the list a after the first n elements are removed. Here n = I is the default value. Note that checking the length on each pass is not very efficient. We can overcome that by testing for an empty list [ ] instead: c2.
However, MACSYMA interprets sin - (~1) x as 1/ sin x, in the same way that it takes sin -2 x to mean sin 2 x. - -....... --_ ............... . '. - I x ----------_: ..... x ______ . _wa ___ - I tr 2 b a Fig. 3. (Restricted) sine function (a) and its inverse (b) To iIIu Irate how the inverse function a in x combine with other trigonometric function , con ider the expre ion co (a in x) over the domain [-I I] of the function asin. U ing the Pythagorean theorem co 2 y + in 2 y = I, for y = a inx, and the fact that sin(asio x) = x, we get co (asinx) = ±jl- in2(asinx) = ±~.