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(40 points)
Let
F(x)=int_(0)^(x)sin(t^(3))dt for
0 <= x <= 2.
On what intervals if
F(x) increasing?

( $40$ points) $\newline$ Let $F(x)=\int_{0}^{x} \sin \left(t^{3}\right) d t$ for $0 \leq x \leq 2$ . $\newline$ On what intervals if $F(x)$ increasing?

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Intermediate Value Theorem

Full solution

Q. (

40

points)

\newline

Let

F(x)=\int_{0}^{x} \sin \left(t^{3}\right) d t

for

0 \leq x \leq 2

.

\newline

On what intervals if

F(x)

increasing?

Define $F'(x)$ : To find where $F(x)$ is increasing, we need to look at the derivative $F'(x)$ because if $F'(x) > 0$ , then $F(x)$ is increasing.
Calculate $F'(x)$ : By the Fundamental Theorem of Calculus, $F'(x) = \sin(x^3)$ .
Find intervals for $\sin(x^3)$ : We need to find the intervals where $\sin(x^3) > 0$ .
Determine positive intervals: Since $\sin(x)$ is positive in the first and second quadrants, $\sin(x^3)$ will be positive when $x^3$ is in the first and second quadrants.
Identify first and second quadrants: The first quadrant for $x^3$ is from $0$ to $(\sqrt{\pi}/2)^{1/3}$ and the second quadrant is from $(\sqrt{\pi}/2)^{1/3}$ to $(\sqrt{3\pi}/2)^{1/3}$ .

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