Is There a Fundamental Theorem of Calculus for Squared Derivatives?

In summary, the FTC for quadratic function is a special case of the regular FTC that allows us to easily find the area under quadratic curves by taking their integrals. It simplifies the process of finding integrals for quadratic functions and has many practical applications in various fields. To use it, we need to identify the function and limits of integration, take the integral using the power rule, and then subtract the resulting values. However, the FTC can only be applied to continuous quadratic functions, as non-continuous functions require different methods to find the area under the curve.
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Bruno Tolentino
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Fundamental theorem of calculus:[tex] \int_a^b \frac{df}{dx} dx = f(b) - f(a) [/tex]
All right? Everybody knows...

BUT, BUT, exist some analog of the FTC for this case:

[tex] \int_a^b \left( \frac{df}{dx} \right)^2 dx = ? [/tex]

Hum?
 
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  • #2
Bruno Tolentino said:
Fundamental theorem of calculus:[tex] \int_a^b \frac{df}{dx} dx = f(b) - f(a) [/tex]
All right? Everybody knows...

BUT, BUT, exist some analog of the FTC for this case:

[tex] \int_a^b \left( \frac{df}{dx} \right)^2 dx = ? [/tex]
Not that I'm aware of.
 
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FAQ: Is There a Fundamental Theorem of Calculus for Squared Derivatives?

What is the FTC for quadratic function?

The Fundamental Theorem of Calculus (FTC) for quadratic functions states that the integral of a quadratic function from a to b is equal to the difference between the antiderivative of the function evaluated at b and a. In other words, it allows us to find the area under a quadratic curve by taking the integral of the function.

How is the FTC for quadratic function different from the regular FTC?

The FTC for quadratic function is a special case of the regular FTC, which applies to all continuous functions. While the regular FTC allows us to find the area under any curve, the FTC for quadratic function specifically deals with quadratic functions and simplifies the process of finding their integrals.

Why is the FTC for quadratic function important?

The FTC for quadratic function is important because it allows us to easily find the area under a quadratic curve, which has many practical applications in fields such as physics, engineering, and economics. It also helps us solve problems involving motion, optimization, and finding volumes of shapes.

What are the steps for using the FTC for quadratic function?

To use the FTC for quadratic function, we first need to identify the quadratic function and the limits of integration. Then, we take the integral of the function using the power rule and substitute the limits of integration. Finally, we subtract the resulting values to find the area under the curve.

Can the FTC for quadratic function be applied to all quadratic functions?

Yes, the FTC for quadratic function can be applied to all quadratic functions, as long as they are continuous. This means that the function does not have any breaks or gaps in its graph and can be drawn without lifting the pencil. If a quadratic function is not continuous, the FTC cannot be applied and other methods must be used to find the area under the curve.

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