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Conceptual understanding of calculus is crucial in the fields of applied sciences, business, and engineering and technology subjects. However, the current status indicates that students possess only procedural knowledge developed from rote learning of procedures in calculus without insight of core ideas. Hence, this paper aims to assess students' challenges and to get insight on common barriers towards attaining conceptual knowledge of calculus. A purposive sample of 238 grade 12 natural sciences students from four different schools in one administrative Zone of Ethiopia were selected to participate in this study. An open ended test about limit of functions at a point and at infinity was administered and analyzed quantitatively and qualitatively. The findings reveal a number of factors about students' knowledge such as: lack of conceptual knowledge in limit of functions, knowledge characterized by a static view of dynamic processes, over generalization, inconsistent cognitive structure, over dependence on procedural learning, lack of coherent and flexibility of reasoning, lack of procedural fluency and wrong interpretation of symbolic notations. Students' thinking strategies influencing these challenges originate from their arithmetic thinking rather than algebraic, linguistic ambiguities, compartmentalized learning, dependence on concept image than concept definition, focus in obtaining correct answers for wrong reasons, and attention given to lower level cognitive demanding exercises.

Background: CD4 cell counts is widely used as a biomarker for treatment progression when studying the efficacy of drugs to treat HIV-infected patients. In the past, it had been also used in determining eligibility to initiate antiretroviral therapy. The main aim of this was to model the evolution of CD4 counts over time and use this model for an early prediction of subject-specific time to cross a pre-specified CD4 threshold. Methods: Hospital based retrospective cohort study of HIV-infected patients was conducted from January 2009 to December 2014 at University of Gondar hospital, Northwest Ethiopia. Fractional polynomial random effect model is used to model the evolution of CD4 counts over time in response to treatment and to estimate the individual probability to be above a pre-selected CD4 threshold. Human subject research approval for this study was received from University of Gondar Research Ethics Committee and the medical director of the hospital. Results: A total of 1347 patients were included in the analysis presented in this paper. The cohort contributed a total of 236.58 per 100 person-years of follow-up. Later the data were divided into two periods: the first is the estimation period in which the parameters of the model are estimated and the second is the prediction period. Based on the parameters from the estimation period, model based prediction for the time to cross a threshold was estimated. The correlations between observed and predicted values of CD4 levels in the estimation period were 0.977 and 0.982 for Neverapine and Efavirenz containing regimens, respectively; while the correlation between the observed and predicted CD4 counts in the prediction period are 0.742 and 0.805 for NVP and EFV, respectively. Conclusions: The model enables us to estimate a subject-specific expected time to cross a CD4 threshold and to estimate a subject-specific probability to have CD4 count above a pre-specified threshold at each time point. By predicting long-term outcomes of CD4 count of the patients one can advise patient about the potential ART benefits that accrue in the long-term. [ABSTRACT FROM AUTHOR]