Students’ conceptual understanding of a function and its derivative in an experimental calculus course

Calculus has been witnessing fundamental changes in its curriculum, with an increased emphasis on visualization. This mode for representing mathematical concepts is gaining more strength due to the advances in computer technology and the development of dynamical mathematical software. This paper focuses on the understanding of the function and its derivative as viewed by students of a reformed Calculus 1 course offered in two experimental sections at the Lebanese American University in Beirut, Lebanon. Results have shown that the general approach adopted in the course proved to be unpopular for a great majority of the students, but rewarding for others. Interviews conducted with some students and a study of their performance on very specific exam questions reveal that for most students, the algebraic representation of a function still dominated their thinking; however, these students showed an almost complete understanding of the derivative, particularly the idea of the instantaneous rate of change and/or the slope of a curve at a given point. Furthermore, very few of these students referred to the mechanical methods for finding derivatives.     

The site of protonation of bifunctional bases with competing basic centers. I. Aromatic nitriles

The experimental values of the gas-phase proton affinities for a variety of 4-substituted benzonitriles, 4-substitutedN, N-dimethylanilines, and 4-substituted benzaldehydes have been examined by means of correlation analysis techniques and by ab initio quantum mechanical methods (MP2/ 6-31G(d) level). From this study it is concluded that in the gas phase, 4-(dimethylamino)-benzonitrile essentially protonates on the dimethylamino group, while protonated 4-cyanobenzaldehyde is very nearly a 21 mixture of the carbonyl- and cyano-protonated forms.This work is dedicatedin memoriam to Professor Robert W. Taft.     

Additions of enantiopure alpha-sulfinyl carbanions to (S)-N-sulfinimines: asymmetric synthesis of beta-amino sulfoxides and beta-alphamino alcohols

The addition of the lithium anions derived from (R)- and (S)-methyl and -ethyl p-tolyl sulfoxides to (S)-N-benzylidene-p-toluenesulfinamide provides an easy access route to enantiomerically pure beta-(N-sulfinyl)amino sulfoxides. Stereoselectivity can be achieved when the configurations at the sulfur atoms of the two reagents are opposite (matched pair), thus resulting in only one diastereoisomer, even for the case in which two new chiral centers are created. The N-sulfinyl group primarily controls the configuration of the carbon bonded to the nitrogen, whereas the configuration of the alpha-sulfinyl carbanion seems to be responsible for the level of asymmetric induction, as well as for the configuration of the new stereogenic C-SO carbon in the reactions with ethyl p-tolyl sulfoxides. An efficient method for transforming the obtained beta-(N-sulfinyl)amino sulfoxides into optically pure beta-amino alcohols, based on the stereoselective non-oxidative Pummerer reaction, is also reported.