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.     

K-Dependence and temperature dependence of N2-, H2-, and He-broadening coefficients for the J = 12-11 transition of acetonitrile CH3CN located near 220.7 GHz

Extensive experiments on the K = 3 component of the J = 12-11 rotational transition of acetonitrile CH₃C¹⁴N, located near 220.7 GHz, were performed at different temperatures in the range 235-350 K. They allow determining the N₂-, H₂-, and He-broadening coefficients, as well as their temperature dependences. More specific measurements on all the K-components of the involved transition perturbed by N₂ at 303 K allow to point out a clear decreasing of the broadening coefficient with increasing K. Narrowing effects are clearly observed, and experimental lines were analysed both with Voigt and speed dependent Voigt profiles; but no exhaustive lineshape study was carried out. All the experimental parameters are compared with results derived from a semiclassical calculation of collisional interactions, including electrostatic, induction, and dispersion energy contributions.     

四种硝基苯的电化学行为及其反应活性的理论计算

在滴汞电极上研究了硝基苯、对硝基甲苯、间二硝基苯及2，4-二硝基氯苯的电化学行为，并用量子理论对四种硝基苯化合物的反应活性进行了预测，理论与试验结果基本一致，据此提出了预测分子的电化学活性的方法。     

Synthesis,Vibrational Spectrum and Structure of the Tetracyanoborate K[B(CN)4]·CH3CN

The new tetracyanoborate K[B(CN)₄]·CH₃CN was synthesized by dissolution of the solvent‐free K[B(CN)₄] in acetonitrile and subsequent careful crystallization. The crystal structure has been determined by single‐crystal X‐ray diffraction. It crystallizes in the orthorhombic space group P2₁2₁2₁ with Z = 4. Some comparisons with related structures are made, and the vibrational spectrum is discussed.     

Complexing properties of methyl and phenyl glycine derivatives in their compounds with H+, Ni(II), Cu(II), and Zn(II)

The influence of the solvent and the substituents on the complexing properties of methyl and phenyl glycine derivatives is discussed. On the basis of a computer analysis of potentiometric titration results, the composition and the stability constants of the complexes ofN-methylglycine,N,N-dimethylglycine,N-phenylglycine and phenylglycine with H⁺ and with Ni(II), Cu(II), Zn(II) were determined. The ligand-metal coordination mode as well as the zwitterion level in percent in ligand/proton systems were determined by spectral analyses and equilibria studies.

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Komplexbildung der Methyl- und Phenylglycin-Derivate in ihren Verbindungen mit dem Proton und Ni(II), Cu(II) und Zn(II)

Zusammenfassung Der Einfluß des Lösungsmittels und der Substituenten auf die komplexbildenden Eigenschaften der Methyl- und Phenylglycin-Derivate wird diskutiert. Anhand einer Computer-Analyse von potentiometrischen Daten wurden die Zusammensetzung und die Beständigkeitskonstanten der Komplexe vonN-Methylglycin,N,N-Dimethylglycin,N-Phenylglycin und Phenylglycin mit H⁺, Ni(II), Cu(II) und Zn(II) festgestellt. Mittels spektroskopischer Methoden und Gleichgewichtstudien wurde der Koordinationstyp des Liganden mit dem Metall festgestellt sowie der Prozentanteil des Zwitterions im Ligand/Proton-System.

     

A new method for the direct calculation of resonance parameters with application to the quasibound states of the H2X1Σ g + system

A new method for the direct calculation of resonance parameters is presented. It is based upon searching for poles of the scattering matrix at complex energies. This search is expedited by the use of analytic derivatives of the scattering matrix with respect to the total energy. This procedure is applied initially to a single channel problem, but is generalizable to more complicated systems. Using the most accurate available potential energy data, we calculate resonance parameters for all of the physically important quasibound states of the ground electronic state of the hydrogen molecule. Corrections to the Born-Oppenheimer potential are included and assessed. The new method has no difficulty locating resonances with widths greater than about 1×10^–7 cm^–1. It is easier to find narrow resonances by monitoring the dependence of the imaginary part of the reactance matrix on the real part of a complex energy than to monitor the dependence of the eigenphase sum on energy at real energies.     

Analysis of carboxylic acids by gas chromatography of acetonyl esters. Application to aromatic,dicarboxylic and higher fatty acids

Summary The gas chromatographic analysis of the acetonyl esters of aromatic, dicarboxylic and higher fatty acids is reported and discussed together with their electron impact and chemical ionisation mass spectra. An improved method for the preparation of acetonyl esters is discussed.     

Partial molar enthalpy properties and correlation of excess molar enthalpy data of acetonitrile + diethylamine or S-butylamine mixtures at various temperatures and atmospheric pressure

Experimental data of excess molar enthalpy of binary mixtures of acetonitrile + diethylamine or S-butylamine mixtures as a function of composition at 288.15, 293.15, 298.15 and 303.15 K at atmospheric pressure have been used to calculate excess partial molar enthalpy and partial molar enthalpy of each component as a function of composition as well as partial molar enthalpy properties at infinite dilution. The Flory and Prigogine–Flory–Patterson (PFP) theories were applied to correlate the data. The results of the calculations as well as the influence of temperature and isomers chain on the partial molar enthalpy properties are discussed.     

What crystals of small analogs are trying to tell us about cellulose structure

The molecular geometries from crystal structures of 23 small molecules such as cellobiose were reviewed and extrapolated to give model cellulose chains. Within a given model, all monosaccharide units and their linkages are identical so the models are regular helices. Despite fairly large ranges for the glycosidic linkage torsion angles and , 29° and 57°, respectively, there is little variation in the n and h parameters of the model helices. They are extended, with h values (the advance per residue along the helix axis) of 5.04–5.27 Å. Some models were slightly right-handed, with n values up to 2.12 residues per helix turn. Left-handed models were in the majority, and their n values were as large as –2.91. These results are consistent with known structures of cellulose and its derivatives. An exception comes from a heavily derivatized cellobiose molecule. It yields right-handed helices with n 4.5 and h 3 Å. Because one half turn of this helix reverses the direction of the chain in a compact region, the linkage geometry is a model for chain-folding. Other derivatives that are unable to form the O3O5 hydrogen bond gave left-handed helices. The puckering of the glucose rings was also surveyed. A number of rings in small molecule structures are puckered to a degree that is similar to the puckering determined for methyl cellotrioside, cellotetraose, cellulose I and cellulose II.     

Electrochemical Behavior of Diphenyl Disulfide and Thiophenol on Glassy Carbon and Gold Electrodes in Aprotic Media

The investigation of the electrochemical reduction processes of C₆H₅SSC₆H₅ and C₆H₅SH in CH₃CN using cyclic voltammetry indicates a different behavior on GC and Au electrodes. On GC surface adsorption phenomena are absent, the electrochemical reduction process is irreversible and diffusion controlled. For both the starting molecules the same species, C₆H₅S^?, is formed upon reduction. The E° values of the reduction processes were determined by convolution method and the standard free energy of the S? S bond of C₆H₅SSC₆H₅ estimated. On Au surface instead, a self‐assembled monolayer of C₆H₅SAu_ads originated after the S? S or S? H bond breaking can be observed by simply dipping the electrode in solution of C₆H₅SSC₆H₅ and C₆H₅SH, respectively. The properties of the SAM were investigated by electrochemical reduction of the adsorbed thiolates. On Au electrode the reduction processes involve C₆H₅SAu_ads and give rise to desorbed C₆H₅S^?. A neutral radical is obtained by electrochemical oxidation of thiolate anion. It reacts rapidly with the electrode surface to give the S‐Au bond again.