How do I evaluate the derivative of G(x) for my Calc 2 final tomorrow?

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    Calc 2 Final
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Homework Help Overview

The discussion revolves around evaluating the derivative of the function G(x), defined as the integral from x to x^2 of sin(-t^2) dt. This topic falls under calculus, specifically focusing on differentiation and the application of the fundamental theorem of calculus.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss various methods for evaluating the derivative, including substitution and Taylor series. Some express uncertainty about their approaches and the correctness of their results, while others question the assumptions made during the evaluation process.

Discussion Status

The conversation includes attempts to clarify the derivative's evaluation, with some participants offering hints related to the fundamental theorem of calculus. There is ongoing exploration of different interpretations and methods, but no explicit consensus has been reached regarding the final answer.

Contextual Notes

Participants mention the urgency of the situation due to an upcoming final exam, which may influence their approaches and the pressure they feel to arrive at a solution quickly.

calchelp
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calc 2 final tomorrow need urgent help!

Evaluate the derivative of the function

G(x) = ({integral} from x to x^2) of sin(-t^2)dt

this was on my review for the final and i just cannot get an answer. any help will be greatly appreciated.
 
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need help evaluating derivative of sin(x^2)

sorry I'm new to this site.
i tried substitution which did not get me anywhere.

next, i did taylor series but could not get an answer.

now i am stuck. i am almost positive i have to use a taylor series.

with the series i got

x^2 - (x^6)/3! + (x^10)/5! ...

i plugged in x^2 for x
and i plugged in x for x and subtracted the two and got another series...
now what? antidifferentiate the series and say that that is my answer or did i mess up somewhere?
 
Welcome to PF!

Hi calchelp! Welcome to PF! :smile:

(have an integral: ∫ and try using the X2 tag just above the Reply box :wink:)
calchelp said:
Evaluate the derivative of the function

G(x) = ({integral} from x to x^2) of sin(-t^2)dt

this was on my review for the final and i just cannot get an answer. any help will be greatly appreciated.

Hint: fundamental theorem of calculus. :wink:
 


can i change the title of this thread??
 


is the answer:

sin(-x^4)2x - sin(-x^3)
 
calchelp said:
is the answer:

sin(-x^4)2x - sin(-x^3)

Almost … where did your x3 come from? :confused:
 


ok... so after about 20 minutes of looking at it i realized that i completely made up the x^3 i don't know why or where it came from...

so is the answer:
sin(-x^4)2x - sin(-x^2)

please tell me that is right. i am about to go insane!
 


can anyone confirm this please?
 


calchelp said:
ok... so after about 20 minutes of looking at it i realized that i completely made up the x^3 i don't know why or where it came from...

so is the answer:
sin(-x^4)2x - sin(-x^2)

please tell me that is right. i am about to go insane!

That's right.
 
  • #10


thank you! dick and tiny-tim!
 
  • #11


Essentially this is using the chain rule whereby when:

G(x) = (integral of f1(x) to f2(x) ) g(t) dt

G'(x) = g(f2(x))*f2'(x) - g(f1(x))*f1'(x)

so in your case, G'(x) = sin(-x^4)*2x - sin(-x^2)
 
  • #12
Hi calchelp! :smile:

(just got up :zzz: …)
calchelp said:
so is the answer:
sin(-x^4)2x - sin(-x^2)

You can tidy it up a bit by taking the minus out of the sin (because sin(-θ) = -sinθ) …

sin(x2) - 2x*sin(x4) :wink:
 

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