The discussion revolves around finding local and absolute extrema of a function derived from an integral, specifically g(x) = ∫₀ˣ t sin(t) dt, related to the function f(x) = x sin(x). Participants are exploring how to apply the Fundamental Theorem of Calculus to analyze the behavior of g(x) through its derivative g'(x).
Some participants have provided hints about using the first derivative to find critical points, while others express uncertainty about how to proceed with the analysis. There is an ongoing exploration of how to determine the nature of the extrema based on the sign of the derivative.
Participants mention a lack of clarity regarding the relationship between the integral and the original function, as well as uncertainty about how to evaluate the antiderivative directly. There is also a reference to previous knowledge from high school calculus that may not fully apply to the current problem.
Mark44 said:The graph you show is f(x) = xsinx. The function you're investigating is g(x)~=~\int_0^x t sin(t)dt
How would you ordinarily go about find the local max or min of a function?
Have you learned about the Fundamental Theorem of Calculus recently? (That's a hint.)
This is the integral you're interested inSlimsta said:<br /> g(x)~=~\int_0^{pi/4} x sin(x)dx<br />
Slimsta said:i would look at the graph and see which point looks like local min/max then plug it into the antiderivative of the integral. (which i don't really know how to get because its xsinx...)
so whatever i get for G(x) i will plug that number i got for the local min/max and that will be the value for local min/max.. am i making any sense? lol
and yes i have learned about the Fundamental Theorem of Calculus but how's that helping me?
If you're apologizing to me for jumping in, I don't mind. The more the merrier.Char. Limit said:Sorry, Mark, but it didn't seem to me as if he fully caught what you were saying...
Slimsta said:Char. Limit-
"If g(x) is at a min or max, what is the slope? "
its 0.
Mark44-
the FTC just tells me that g'(x)=f(x) (for my case)
which means that g'(x)=xsinx
now g(x) = antiderivative of g'(x) which is <i don't know how to get it, tried everything already>
but how does this help me with any of the question a,b,c or d?
Mark44 said:What I've been trying to steer you to is that you don't need to evaluate g(x) directly; all you need is g'(x), which you have already said is equal to x*sinx. If you want to find the max and min values of a function, say g, look at where its derivative g' is zero.
g'(x) = ?
g'(x) = 0 for what x?