Question

(a) Use a CAS to graph the solution curve of the initial-value

problem$S\left(x\right)=\underset{0}{\overset{x}{\int }}sin\left(\frac{\pi }{2}{t}^2\right)dt$ inon the interval$\left(-\infty ,\infty \right)$

(b) It is known that Fresnel sine integral$S\left(x\right)\to \frac{1}{2}$as$x\to \infty$andas$S\left(x\right)\to -\frac{1}{2}$.What $x\to -\infty$does the solution$y\left(x\right)$approach as$x\to \infty$As$x\to -\infty$?

(c) Use a CAS to find the values of the absolute maximum and the absolute minimum of the solution$y\left(x\right)$on the interval.

Short answer

b) $\underset{x\to \infty }{Limy}\left(x\right)\approx 9.357$

(c) The highest possible value in the coordinates $\left(1.772,12.235\right)$, The lowest possible value in the coordinates $\left(-1.772,2.004\right)$.

Step-by-step solution

(a) Given Information.

The given function is

$S\left(x\right)=\underset{0}{\overset{x}{\int }}sin\left(\frac{\pi }{2}{t}^2\right)dt$

Indicating value in graph

$y=5e^{\sqrt{\frac{\pi }{2}}}S\left(\sqrt{\frac{2}{\pi }}x\right)$

(b) Appling CAS

Finding $y\left(x\right)$

Set $\underset{x\to \infty }{Limy}$

$$\begin{gathered} \underset{x\to \infty }{Limy}\left(x\right)S\left(x\right)=\frac{1}{2} \\ \underset{x\to \infty }{Limy}\left(x\right)y\left(x\right)=5{e^{\sqrt{\frac{\pi }{2}}}}^{\left(\frac{1}{2}\right)} \\ \underset{x\to \infty }{Limy}\left(x\right)y\left(x\right)=5e^{\sqrt{\frac{\pi }{8}}}\approx 9.357 \end{gathered}$$

y (x) approaches an approximate value of 9.357, as can be seen in the graph. The curve is oscillating, with a steadily decreasing amplitude.

Set, role="math" localid="1663844263305" $\underset{x\to \infty }{Limy}\left(x\right)$

Then,

$$\begin{gathered} \underset{x\to \infty }{Limy}S\left(x\right)=-\frac{1}{2} \\ \underset{x\to \infty }{Limy}y\left(x\right)=5{e^{\sqrt{\frac{\pi }{2}.}}}^{\left(-\frac{1}{2}\right)} \\ \underset{x\to \infty }{Limy}y\left(x\right)=5e^{-\sqrt{\frac{\pi }{8}}} \\ \approx 2.672 \end{gathered}$$

approaches an approximate value of 2.672, as can be seen in the graph. The curve is oscillating, with a steadily decreasing amplitude.

(c) Determining the highest and lowest coordinates

The absolute maximum occurs at$x=1.7$ , and the absolute minimum occurs around $x=-1.8$, as shown in part(a).

The root is obtained using a CAS, and then the constraint y'(x)=0 is solved. The greatest value is found in the coordinates $\left(1.772,12.235\right)$, while the minimum value is found in the coordinates $\left(-1.772,2.004\right)$.

Determining the Result

(b)

$$\begin{gathered} \underset{x\to \infty }{Limy}y\left(x\right)\approx 9.357 \\ \underset{x\to \infty }{Limy}y\left(x\right)\approx 9-2.672 \end{gathered}$$

(c).

The highest possible value in the coordinates $\left(1.772,12.235\right)$. The lowest possible value in the coordinates$\left(-1.772,2.004\right)$