Raja-arvo ja derivaatta pisteessä

Tiivistelmä. Tämä pro gradu -tutkielma on osa Oulun yliopiston projektia, jonka tarkoituksena on luoda avoin oppikirja lukion matematiikan kurssista MAA6. Oppimateriaali on suunnattu opetussuunnitelman mukaan vain pitkän matematiikan opiskelijoille. Materiaalin sisältö perustuu vuoden 2019 opetussuunnitelmaan sekä tieteellisiin artikkeleihin ja didaktisiin tutkimuksiin. Tutkielmani koostuu viidestä osasta. Tutkielmani ensimmäisessä osassa esitellään projektiryhmässä ja opetussuunnitelmassa asetetut yleiset tavoitteet oppimateriaalille. Toisessa osassa perustellaan tarkasti oppimateriaalissa tehdyt päätökset tieteellisten artikkeleiden ja didaktisten tutkimusten kautta. Itse oppimateriaali on kolmannessa osassa. Neljännessä osassa on opettajan opas, jossa on opettajille lisätietoa oppimateriaalista. Viimeisenä on vastausosio oppimateriaalin tehtäville. Oppimateriaalin aiheena on raja-arvo ja derivaatta pisteessä. Tämä on jaettu kahteen kappaleeseen. Ensimmäisessä kappaleessa tutkitaan tarkemmin mikä raja-arvo on sekä määritellään siihen liittyviä keskeisiä laskusääntöjä sekä käsitteitä. Raja-arvon osaamista pohjustetaan erilaisten pohdintatehtävien avulla. Asioiden osaamista vahvistetaan ja varmistetaan erilaisten harjoitustehtävien avulla. Suurin osa pohdintatehtävistä ja harjoitustehtävistä tehdään Geogebralla piirrettyjen kuvaajien avulla. Täsmällistä määritelmää raja-arvolle ei anneta, vaan tarkoituksena on pyrkiä saamaan havainnollinen käsitys raja-arvosta ja oppia muutamia laskutekniikoita. Raja-arvon tarkka määritelmä ei kuulu lukiomatematiikkaan. Toisessa kappaleessa käsitellään funktion derivaattaa yksittäisessä pisteessä. Raja-arvon avulla johdatellaan derivaatan käsitteeseen ja johdetaan siihen liittyviä laskusääntöjä. Kappaleesta löytyy sekä teoriaosuus että esimerkkiosuus. Derivaatan teoriaa pohjustetaan erilaisten pohdintatehtävien avulla. Asioiden osaamista vahvistetaan ja varmistetaan erilaisten harjoitustehtävien avulla. Suurin osa pohdintatehtävistä ja harjoitustehtävistä tehdään Geogebralla piirrettyjen kuvaajien avulla. Aiheeseen ei sisälly derivaattafunktioita. Itse laskemista tässä oppimateriaalissa on melko vähän, sillä tavoitteena on aktivoida opiskelijat pohtimaan, keskustelemaan ja käsittelemään matematiikkaa syvällisesti pelkän rutiininomaisen laskemisen sijaan. Laskuja tulee kuitenkin laskettua väistämättä, mutta ensisijaisina tavoitteina on antaa opiskelijoille parempi kuva matematiikasta tieteenä sekä motivoida opiskelijoita laajempaan ymmärrykseen matematiikan periaatteista

Impact of vaccination on the COVID-19 pandemic in U.S. states

Governments worldwide are implementing mass vaccination programs in an effort to end the novel coronavirus (COVID-19) pandemic. Here, we evaluated the effectiveness of the COVID-19 vaccination program in its early stage and predicted the path to herd immunity in the U.S. By early March 2021, we estimated that vaccination reduced the total number of new cases by 4.4 million (from 33.0 to 28.6 million), prevented approximately 0.12 million hospitalizations (from 0.89 to 0.78 million), and decreased the population infection rate by 1.34 percentage points (from 10.10 to 8.76%). We built a Susceptible-Infected-Recovered (SIR) model with vaccination to predict herd immunity, following the trends from the early-stage vaccination program. Herd immunity could be achieved earlier with a faster vaccination pace, lower vaccine hesitancy, and higher vaccine effectiveness. The Delta variant has substantially postponed the predicted herd immunity date, through a combination of reduced vaccine effectiveness, lowered recovery rate, and increased infection and death rates. These findings improve our understanding of the COVID-19 vaccination and can inform future public health policies

Endogenous cross-region human mobility and pandemics

We study infectious diseases using a Susceptible-Infected-Recovered-Deceased model with endogenous cross-region human mobility. Individuals weigh the risk of infection against economic opportunities when moving across regions. The model predicts that the mobility rate of susceptible individuals declines with a higher infection rate at the destination. With cross-region mobility, a decrease in the transmission rate or an increase in the removal rate of the virus in any region reduces the global basic reproduction number (R0). Global R0 falls between the minimum and maximum of local R0s. A new method of Normalized Hat Algebra is developed to solve the model dynamics. Simulations indicate that a decrease in global R0 does not always imply a lower cumulative infection rate. Local and central governments may prefer different mobility control policies

Effects of mTOR inhibitor, everolimus, on proliferation, autophagy and temozolomide sensitivity of glioma cells

Purpose: To study the effect of the mTOR inhibitor, everolimus, on glioma cell proliferation, autophagy, and drug sensitivity to temozolomide (TMZ).Methods: Human glioma cell lines were cultured in vitro, and the effects of different concentrations of everolimus on the proliferation of brain glial cells were determined using CCK-8 method. The effect of different concentrations of everolimus on brain glial cell levels of autophagy protein were assayed by western blot method.Results: The results of CCK-8 analysis showed that everolimus inhibited the proliferation of glial cells in a time- and concentration-dependent manner. Western blot results showed that the expression levels of autophagy proteins, LC3-II and LC3-II/I, were gradually and concentration-dependently up-regulated, while p62 protein level was gradually decreased concentration-dependently, when compared with blank control (p < 0.05). Treatment with different concentrations of TMZ alone, and in combination with everolimus for 48 h inhibited the proliferation of brain glial cells in a concentration-dependent manner, but the inhibition due to TMZ-everolimus combination was significantly higher than that of TMZ singletreatment (p < 0.05). After 48 h, the expression level of Beclin-1 increased with the ratio of LC3-II/LC-I in TMZ-everolimus group, while the expression level of p62 decreased, when compared with TMZ alone, or control (p < 0.05).Conclusion: Everolimus significantly inhibits the proliferation of glioma cells and promotes the occurrence of autophagy. Combined use of TMZ and everolimus significantly enhances the sensitivity of TMZ to glioma cells, inhibits cell proliferation, and promotes autophagy better than TMZ alone. Keywords: mTOR inhibitor, Everolimus, Glioma cells, Proliferation, Autophag

Key Aquatic Environmental Factors Affecting Ecosystem Health of Streams in the Dianchi Lake Watershed, China

AbstractStreams in a lake watershed are important landscape corridors which link the lake and terrestrial ecosystems. Therefore, the ecosystem health of streams is usually used to indicate aquatic biodiversity of the lake ecosystem, as well as being affected by aquatic environmental factors in response to changes in land use cover of the terrestrial ecosystem due to natural geographic characteristics of the watershed with the closure of ridge lines. This study was carried out at a shallow freshwater lake watershed in the Yunnan-Guizhou Plateau of China, the Dianchi Lake watershed (DLW). Field survey of periphytic algal and macrozoobenthic biodiversity during July and August of 2009, as well as monthly monitoring of water temperature, pH, TSS, DO, TN, TP, NH3N, NO3N, CODMn, BOD, TOC, and the heavy metals Zn (II), Cd (II), Pb (II), Cu (II), and Cr (VI) from January to December 2009 was carried out in 29 streams flowing into Dianchi lake. Multivariate statistical techniques such as factor analysis and canonical correspondence analysis were applied to analyze the structure of the aquatic community in relation to aquatic environmental factors in order to provide controlling objectives for integrated watershed management and improvement of stream rehabilitation in the DLW. The results showed that the structure of the periphytic algal and macrozoobenthic communities were dominated by pollution-tolerant genera, namely the bacillariophytes Navicula and the annelids Tubificidae respectively, and TN, NH3N and TP were key aquatic environmental factors affecting the ecosystem health of streams in the DLW

Recent insight into the role of macrophage in alcohol-associated liver disease: a mini-review

Alcohol-associated liver disease (ALD) is a condition that develops due to prolonged and excessive alcohol consumption. It encompasses various stages of liver damage, including fatty liver, alcoholic hepatitis, and cirrhosis. Immune cells, particularly macrophages, of various types play a significant role in the onset and progression of the disease. Macrophages observed in the liver exhibit diverse differentiation forms, and perform a range of functions. Beyond M1 and M2 macrophages, human macrophages can polarize into distinct phenotypes in response to various stimuli. Recent advancements have improved our understanding of macrophage diversity and their role in the progression of ALD. This mini-review provides a concise overview of the latest findings on the role and differentiation of macrophages in ALD. Additionally, it discusses potential therapeutic targets associated with macrophages and explores potential therapeutic strategies