<prompt> Hlalosa bohlokoa ba theorem e fanoeng ea motheo.Theorem ea motheo ea Calculus  <response>: Instruction in English: Explain the importance of the given fundamental theorem. . Response in English: The Fundamental Theorem of Calculus is one of the most important results in all of mathematics, as it forms a bridge between the two major branches of calculus: differentiation and integration. It provides a connection between the antiderivative and the definite integral, showing that the two are essentially inverse operations. In practical terms, it allows us to evaluate definite integrals without having to use the often-tedious technique of finding the limit of Riemann sums. Instead, we can use the far simpler method of finding an antiderivative and substituting the limits of the integral. The theorem has two main parts. The first part states that if a function is continuous on a closed interval, then it is possible to construct an antiderivative of the function by integrating it over that interval. This is a powerful result that allows us to find the exact value of definite integrals very easily, and it is the basis for many of the techniques used to evaluate integrals. The second part of the theorem states that if a function is defined and continuous on an interval and has an antiderivative, then the definite integral of the function over the interval is equal to the difference between the values of the antiderivative at the endpoints of the interval. This allows us to solve problems involving rates of change and accumulation, such as problems involving motion or the accumulation of a quantity over time. Overall, the Fundamental Theorem of Calculus provides a crucial connection between differentiation and integration that allows us to solve a wide range of important problems in mathematics, the sciences, and engineering. . Response in Sesotho: Theory Theorem of Calculus ke e ngoe ea liphetho tsa bohlokoahali ho lipalo tsohle, kaha e theha borokho lipakeng tsa makala a mabeli a maholo a lipalo: phapang le kopanyo. E fana ka khokahano lipakeng tsa antiderivative le karolo e hlakileng, e bontsa hore bobeli ke tsebetso e fapaneng. Ka mantsoe a sebetsang, e re lumella ho lekola likarolo tse hlakileng ntle le ho sebelisa mokhoa o khathatsang hangata oa ho fumana moeli oa lipalo tsa Riemann. Ho e-na le hoo, re ka sebelisa mokhoa o bonolo haholo oa ho fumana antiderivative le ho fetola meeli ea motsoako. Theorem e na le likarolo tse peli tse kholo. Karolo ea pele e bolela hore haeba tsebetso e ntse e tsoela pele ka nako e koetsoeng, joale ho ka khoneha ho haha antiderivative ea mosebetsi ka ho e kopanya ka nako eo. Sena ke sephetho se matla se re lumellang ho fumana boleng bo nepahetseng ba lintlha tse hlakileng habonolo, me ke motheo oa mekhoa e mengata e sebelisoang ho lekola likarolo. Karolo ea bobeli ea theorem e bolela hore haeba tsebetso e hlalosoa me e tsoela pele ka nako e itseng me e na le antiderivative, joale karolo e tobileng ea tsebetso nakong ea nako e lekana le phapang pakeng tsa litekanyetso tsa antiderivative qetellong ea nako. . Sena se re fa monyetla oa ho rarolla mathata a amanang le litefiso tsa phetoho le ho bokella, joalo ka mathata a amanang le ho sisinyeha kapa ho bokella bongata ha nako e ntse e ea. Ka kakaretso, Theorem ea Motheo ea Calculus e fana ka kamano ea bohlokoa lipakeng tsa phapang le kopanyo e re lumellang ho rarolla mathata a mangata a bohlokoa a lipalo, mahlale le boenjiniere. 