The first fundamental theorem of calculus

The fundamental theorem of calculus states that if a function F is an antiderivative of another function f, then the integral of f from a to x is equal to F(x) - F(a). In summary, the fundamental theorem of calculus makes the statement ##\int \frac{\mathrm{d} y}{\mathrm{d} x}\mathrm{d}x = y## true because it is a direct result of the Leibniz rule. This rule shows that the integral of a function's derivative is equal to the original function, which applies to both the indefinite and definite forms of the integral.
  • #1
Mr Davis 97
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Say I have the statement ##\int \frac{\mathrm{d} y}{\mathrm{d} x}\mathrm{d}x = y##. How does the fundamental theorem of calculus make this necessarily true? When I see the formal statement of the theorem, it is usually in terms of a definite integral such as ##F(x) = \int_{a}^{x}f(t)dt##. How does the later apply to the former if the former is an antiderivative?
 
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  • #2
BUT Y= F(X)
SO, dy/dx=f(x)
la segunda es integral definida
 
  • #3
Mr Davis 97 said:
Say I have the statement ##\int \frac{\mathrm{d} y}{\mathrm{d} x}\mathrm{d}x = y##. How does the fundamental theorem of calculus make this necessarily true? When I see the formal statement of the theorem, it is usually in terms of a definite integral such as ##F(x) = \int_{a}^{x}f(t)dt##. How does the later apply to the former if the former is an antiderivative?
Do you know the Leibniz rule?
 

What is the first fundamental theorem of calculus?

The first fundamental theorem of calculus is a fundamental concept in calculus that states that the derivative of a function can be calculated by evaluating the function at a certain point and subtracting the value of the function at another point.

What is the difference between the first and second fundamental theorem of calculus?

The first fundamental theorem of calculus deals with the relationship between a function and its derivative, while the second fundamental theorem of calculus deals with the relationship between a definite integral and its corresponding antiderivative.

How is the first fundamental theorem of calculus applied in real-world situations?

The first fundamental theorem of calculus is used in various fields such as physics, engineering, and economics, to calculate the rate of change of a quantity over time or to find the area under a curve.

What is the importance of the first fundamental theorem of calculus in mathematics?

The first fundamental theorem of calculus is considered one of the most important theorems in calculus as it allows for the calculation of derivatives without using limits, making it a powerful tool in solving mathematical problems.

What are some common mistakes students make when using the first fundamental theorem of calculus?

Some common mistakes students make when using the first fundamental theorem of calculus include incorrect interpretation of the theorem, using the wrong function for the derivative, and not considering the limits of integration when evaluating a definite integral.

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