Why doesn't the derivative of an integral give the value at the lower limit?

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Homework Help Overview

The discussion revolves around the concept of derivatives of integrals, specifically focusing on the behavior of the derivative of an integral with a constant lower limit and a variable upper limit. The subject area is calculus, particularly the Fundamental Theorem of Calculus.

Discussion Character

  • Conceptual clarification

Approaches and Questions Raised

  • The original poster attempts to understand why the derivative of an integral does not yield the value of the function at the lower limit, questioning the relationship between limits and derivatives in this context.
  • Some participants question the original poster's reasoning and suggest a different perspective on evaluating the limit of the integral as the upper limit approaches the lower limit.
  • Others explore the implications of having a fixed lower limit and how it affects the rate of change of the area under the curve.

Discussion Status

The discussion is ongoing, with participants providing clarifications and alternative viewpoints. Some guidance has been offered regarding the interpretation of the derivative in relation to the integral, and there appears to be a productive exchange of ideas.

Contextual Notes

There is a focus on the definitions and properties of derivatives and integrals, with an emphasis on the implications of fixed versus variable limits. The original poster's assumptions about the behavior of the integral as the upper limit approaches the lower limit are being examined.

barksdalemc
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Can someone explain a concept to me? The derivative of an integral ( whose lower limit is a real constant and whose upper limit is the variable x), is the intergrand evaluated at x as per the FTofC. I always thought about this as the limit of the integral as x approached the lower limit because by definition of the derivative we take limit as change in x approaches 0. So my question is why the derivate of an integral doesn't give the value of the function at the lower limit.
 
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Why did you think this? It is not correct. The limit of
\int_c^x f(t)dt

as x tends to c is zero.

What you shuld be thinkig about is

\frac{1}{h} (\int_c^{x+h} f(t)dt - \int_c^x f(t)dt)

as h tends to zero.
 
Last edited:
Yes that makes sense. So because the lower limit is fixed the rate of change of the area under the curve to the lower limit is zero, but the rate of change of the area up to the upper limit is changing by a value equal the integrand value evaluated at the upper limit?
 
I just saw the second equation you posted. That makes it 100% clear to me. Thanks.
 

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