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.     

Coupling of an aldehyde or ketone to pyridine mediated by a tungsten imido complex

The reactivity of W(NPh)(o-(Me3SiN)2C6H4)(py)2 and W(NPh)(o-(Me3SiN)2C6H4)(pic)2 (py=pyridine; pic=4-picoline) with unsaturated substrates has been investigated. Treatment of W(NPh)(o-(Me3SiN)2C6H4)(py)2 with diphenylacetylene or 2,3-dimethyl-1,3-butadiene generates W(NPh)(o-(Me3SiN)2C6H4)(eta2-PhCCPh) and W(NPh)(o-(Me3SiN)2C6H4)(eta4-CH2=C(Me)C(Me)=CH2), respectively, while the addition of ethylene to W(NPh)(o-(Me3SiN)2C6H4)(py)2 generates the known metallacycle W(NPh)(o-(Me3SiN)2C6H4)(CH2CH2CH2CH2). The addition of 2 equiv of acetone to W(NPh)(o-(Me3SiN)2C6H4)(pic)2 provides the azaoxymetallacycle W(NPh)(o-(Me3SiN)2C6H4)(OCH(Me)2)(OC(Me)2-o-C5H3N-p-Me), the result of acetone insertion into the ortho C-H bond of picoline. Similarily, the addition of 2 equiv of RC(O)H [R=Ph, tBu] to W(NPh)(o-(Me3SiN)2C6H4)(py)2 generates W(NPh)(o-(Me3SiN)2C6H4)(OCH2R)(OCHR-o-C5H4N) [R=Ph, tBu,]. In contrast, reaction between W(NPh)(o-(Me3SiN)2C6H4)(py)2 and 2-pyridine carboxaldehyde yields the diolate W(NPh)(o-(Me3SiN)2C6H4)(OCH(C5H4N)CH(C5H4N)O). The synthesis of W(NPh)(o-(Me3SiN)2C6H4)(PMe3)(py)(eta2-OC(H)C6H4-p-Me), formed by the addition of p-tolualdehyde to a mixture of W(NPh)(o-(Me3SiN)2C6H4)(py)2 and PMe3, suggests that an eta2-aldehyde intermediate is involved in the formation of the azaoxymetallacycle, while the isolation of W(NPh)(o-(Me3SiN)2C6H4)(Cl)(OC(Me)(CMe3)-o-C5H4N), formed by the reaction of pinacolone with W(NPh)(o-(Me3SiN)2C6H4)(py)2, in the presence of adventitious CH2Cl2, suggests that the reaction proceeds via the hydride W(NPh)(o-(Me3SiN)2C6H4)(H)(OC(Me)(CMe3)-o-C5H4N).     

A dimeric constrained‐geometry titanium complex linked by double Ti–CH2–Al–CH2 heterocycles

In the structure of bis({N‐[di­methyl(1η⁵‐2,3,4,6‐tetra­methyl­in­den­yl)­silyl]­cyclo­hexyl­amido‐1κN}(methyl‐3κC)‐di‐μ₃‐methyl­ene‐1:2:3κ³C;1:3:3′κ³C‐tris(pentafluorophenyl‐2κC)titanium) benzene disolvate, [Me₂Si(η⁵‐2,3,4,6‐Me₄C₉H₂)(C₆H₁₁N)]Ti[(μ₃‐CH₂)Al(C₆F₅)₃][AlMe(μ₃‐CH₂)]₂ or [Ti₂(C₂₁H₇AlF₁₅)₂(C₂₁H₃₁NSi)₂]·2C₆D₆, the dimer is located on an inversion center, and the two Ti centers are linked by double Ti(μ₃‐CH₂)Al(C₆F₅)₃AlMe(μ₃‐CH₂) heterocycles. The electron‐deficient Ti centers are further stabilized by two α‐agostic interactions between Ti and one H atom of each bridging methyl­ene group.