Please answer this calculus question?

Question

Let F(x)=∫ sin(t^2) dt from 0 to x for 0≤x≤3. 

 

a) Use the Trapezoidal Rule with four equal subdivisions of the closed interval [0, 2] to approximate F(2).

b) On what interval or intervals is F increasing? Justify your answer.

c) If the average rate of change of F on the closed interval [0, 3] is k, find ∫ sin(t^2) dt from 0 to 3 in terms of k. 

 

Please show all your work.

Answer 1

1) F(2)=∫₀² sin(t²)dt

 

F(2)=0.5*[sin(0)+sin(0.5²)]/2+0.5*[sin(0.5²)+sin(1)]/2+0.5*[sin(1)+sin(1.5²)]/2+0.5*[sin(1.5²)+sin(2²)]/2=0.5*[sin(0.5²)+sin(1)+sin(1.5²)+sin(4)/2]≈0.744

 

2) F(x) increases on the intervals where F'(x)>0. F'(x)=sin(x²). Since 0≤x≤3, F'(x)≥0 on the intervals [0,√π)]∪[√(2π);3] Therefore, those are the intervals where F(x) increases.

 

3) Average rate of change of F(x) is simply this: (1/3)∫₀³ sin(t²)dt. It is k, by the problem statement. So, the answer here is simply 3k.