Question

Let F(x)=integral of sin(t^2) dt from 0 to x for 0<=x<=3. On what interval or intervals is F increasing? Justify your answer.

Answer 1

I know that F'(x)=sin(x^2).

Answer 2

But what to do next?

Answer 3

set it equal to 0 and find solve for x. Then use number line test

Answer 4

sin(x^2)=0 but how do I solve for x though?

Answer 5

when is sine 0?

Answer 6

0, pi, 2pi.

Answer 7

techically it's 0, pi, 2pi, 3pi , .... k pi, and k is an integer. But it turns out that only 0, pi and 2pi work in this case.

so x^2 = 0, pi, 2pi
then x = 0, sqrt(pi), sqrt(2pi)

Answer 8

if your domain was, say, [0,10], then you might have to include sqrt(3pi), sqrt(4pi), ... and whichever one that is in that range.

Answer 9

So what's the answer for this problem?

Answer 10

so you have to check
is F'(x) positive in (0,sqrt(pi)) ?
is F'(x) positive in (sqrt(pi), sqrt(2pi)?
is F'(x) positive in (sqrt(2pi), 3)?

pick a number that is in each of those interval and plug it in F'(x) and see if you get a positive number. If you do, F(x) is increasing during that interval

Answer 11

Thanks.