True or False Integral Calculus Question #2

In summary, the question is asking whether $F(2x)$ is an antiderivative of $f(2x)$, and the conversation explores using the substitution method to find the derivative of $F(2x)$ or the integral of $f(2x)$. The conclusion is that $F(2x)$ is not an antiderivative of $f(2x)$, but rather $(1/2)F(2x)$ is the antiderivative. The conversation also confirms the use of the chain rule in this process.
  • #1
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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  • #2
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 $u=2x,du=2\,dx$?
 
  • #3
Kinda confused... how would you do that using substitution?
 
  • #4
Try it and post your effort. :)
 
  • #5
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 $\int f(u)(du/2) = (1/2)f(u)du = (1/2)F(u) = (1/2)F(2x)$.
 
  • #6
So I'm guessing that's false because $\int f(2x)dx \ne F(2x)$ but $(\frac{1}{2})F(2x)$, right?
 
  • #7
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!
 
  • #8
Alright, thanks so much! :)
 

FAQ: True or False Integral Calculus Question #2

1. What is integral calculus?

Integral calculus is a branch of mathematics that deals with calculating the area under a curve. It involves finding the antiderivative of a function and using that to determine the area between the curve and the x-axis.

2. What is the purpose of true or false questions in integral calculus?

True or false questions in integral calculus are used to test a student's understanding of the concepts and principles of the subject. They help to reinforce knowledge and identify areas that need further study.

3. How do you determine if an integral calculus question is true or false?

To determine if an integral calculus question is true or false, you must first understand the concept being tested. Then, you can either solve the problem using the appropriate techniques or use your knowledge of the concept to determine if the statement is true or false.

4. Are there any tips for answering true or false integral calculus questions?

Yes, some tips for answering true or false integral calculus questions include carefully reading the question and identifying key terms, using your knowledge of the concepts and principles, and checking your work for any errors.

5. Can true or false integral calculus questions have more than one correct answer?

No, true or false integral calculus questions can only have one correct answer. The purpose of these questions is to test your understanding of a specific concept or statement, so there can only be one true or false answer.

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