True or False Integral Calculus Question #2

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Discussion Overview

The discussion revolves around a true or false question regarding integral calculus, specifically whether $F(2x)$ is an antiderivative of $f(2x)$ given that $F(x)$ is an antiderivative of $f(x)$. The scope includes mathematical reasoning and exploration of integration techniques.

Discussion Character

  • Mathematical reasoning
  • Exploratory

Main Points Raised

  • Post 1 presents the initial true or false statement regarding the relationship between $F(2x)$ and $f(2x)$.
  • Post 2 asks for the derivative of $F(2x)$ and suggests using substitution to integrate $f(2x)$.
  • Post 3 expresses confusion about how to apply substitution in this context.
  • Post 4 encourages the participant to attempt the substitution and share their findings.
  • Post 5 details a substitution method where $u = 2x$, leading to a transformation of the integral $\int f(2x)dx$ into $(1/2)F(2x)$.
  • Post 6 concludes that the original statement is false because the integral does not equal $F(2x)$ but rather $(1/2)F(2x)$.
  • Post 7 reiterates the substitution process and confirms the conclusion reached in Post 6, emphasizing the application of the chain rule.
  • Post 8 expresses gratitude for the clarification received.

Areas of Agreement / Disagreement

Participants appear to agree on the conclusion that $F(2x)$ is not an antiderivative of $f(2x)$, as indicated by the calculations presented. However, the initial true or false question remains a point of exploration rather than settled agreement.

Contextual Notes

The discussion includes assumptions about the properties of antiderivatives and the application of substitution in integration, which may not be universally agreed upon without further clarification.

MermaidWonders
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True or False: Let $F(x)$ be an antiderivative of a function $f(x)$. Then, $F(2x)$ is an antiderivative of the function $f(2x)$.
 
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What is the derivative of $F(2x)$? Or, perhaps more to the point, what do you get if you integrate $f(2x)$ using the substitution $u=2x,du=2\,dx$?
 
Kinda confused... how would you do that using substitution?
 
Try it and post your effort. :)
 
OK... Here's what I came up with... took me a while but I'm still kinda confused with myself. :(

Let $u = 2x$. Then $du/dx = 2$ --> $dx = du/2$.

$\int f(2x)dx$ then becomes $\int f(u)(du/2) = (1/2)f(u)du = (1/2)F(u) = (1/2)F(2x)$.
 
So I'm guessing that's false because $\int f(2x)dx \ne F(2x)$ but $(\frac{1}{2})F(2x)$, right?
 
MermaidWonders said:
OK... Here's what I came up with... took me a while but I'm still kinda confused with myself. :(

Let $u = 2x$. Then $du/dx = 2$ --> $dx = du/2$.

$\int f(2x)dx$ then becomes $\int f(u)(du/2) = (1/2)f(u)du = (1/2)F(u) = (1/2)F(2x)$.

That's right. Chain rule. :)

Excellent work!
 
Alright, thanks so much! :)
 

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