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Fundamental Theorem of Calculus
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In summary, the Fundamental Theorem of Calculus is a mathematical theorem that connects differentiation and integration. It states that if a function is continuous on a closed interval and has an antiderivative, then the definite integral of that function over that interval is equal to the difference of the antiderivative evaluated at the upper and lower bounds of the interval. This theorem is important because it allows for easy calculation of definite integrals, which are essential in many applications of mathematics and science. It also provides a deeper understanding of the relationship between differentiation and integration. In real life, it has numerous applications in fields such as physics, engineering, economics, and statistics. The theorem has two parts, with the first stating that the definite integral can be calculated by
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SammyS
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Since u = b(x) your final answer is equivalent to [itex]f(b(x))\cdot b'(x)\ .[/itex]planauts said:Homework Statement
Homework Equations
The Attempt at a Solution
I know this is not right.
Could someone help me out here?
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FAQ: Fundamental Theorem of Calculus
What is the Fundamental Theorem of Calculus?
The Fundamental Theorem of Calculus is a mathematical theorem that connects the concepts of differentiation and integration. It states that if a function is continuous on a closed interval and has an antiderivative, then the definite integral of that function over that interval is equal to the difference of the antiderivative evaluated at the upper and lower bounds of the interval.
Why is the Fundamental Theorem of Calculus important?
The Fundamental Theorem of Calculus is important because it provides a way to easily calculate definite integrals, which are essential in many applications of mathematics and science. It also gives a deeper understanding of the relationship between differentiation and integration.
How is the Fundamental Theorem of Calculus used in real life?
The Fundamental Theorem of Calculus has numerous real-life applications, including in physics, engineering, economics, and statistics. For example, it can be used to calculate work done by a variable force, find the area under a curve, or determine the average value of a function over an interval.
What is the difference between the first and second parts of the Fundamental Theorem of Calculus?
The first part, also known as the Fundamental Theorem of Calculus (FTC) Part I, states that the definite integral of a function can be calculated by finding its antiderivative and evaluating it at the upper and lower bounds of the interval. The second part, also known as FTC Part II, states that the derivative of the definite integral of a function is equal to the original function.
Can the Fundamental Theorem of Calculus be proved?
Yes, the Fundamental Theorem of Calculus can be proved using other mathematical concepts and theorems, such as the Mean Value Theorem and the Intermediate Value Theorem. However, the proof is quite complex and requires a strong understanding of calculus and mathematical logic.