Answer the question.
-Estimate the error if $\sin x ^ { 3 / 2 }$ is approximated by $x ^ { 3 / 2 } - \frac { x ^ { 9 / 2 } } { 3 ! }$ in the integral of $\int _ { 0 } ^ { 1 } \sin x ^ { 3 / 2 } d x$.
A) $\frac { 1 } { 5 ! }$
B) $\frac { 6 } { 17 \cdot 5 ! }$
C) $\frac { 2 } { 17 \cdot 5 ! }$
D) $\frac { 2 } { 15 \cdot 4 ! }$

Question

Quizplus · Accepted Answer

We can estimate the error using Taylor's formula with the remainder term in Lagrange form:
$$R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}$$
Here, we have $f(x) = \sin x^{3/2}$, $a=0$, $n=2$, and we want to estimate the error at $x=1$. So we need to find the maximum value of $|f'''(c)|$ on the interval $[0,1]$. 
$$f''(x) = 9\sqrt{x}\cos{x^{3/2}} - 3x^{3/2}\sin{x^{3/2}}$$
$$f'''(x) = -\frac{27}{2\sqrt{x}}\sin{x^{3/2}} - 9x\sin{x^{3/2}} + 9x\cos{x^{3/2}}$$
The absolute value of $f'''(x)$ is maximized at $x = (\frac{3}{2}\pi)^{\frac{2}{3}}$, which is the unique solution in $[0,1]$. So we use $c = (\frac{3}{2}\pi)^{\frac{2}{3}}$, and we have
$$R_2(1) = \frac{f'''(c)}{3!}(1-0)^3 = \frac{2}{17\cdot5!}$$
So the estimate for the error in the integral $\int_0^1 \sin x^{3/2} dx$ using the given approximation is $\frac{2}{17\cdot5!}$. Therefore, the best choice is C.

Answer the Question sin⁡x3/2\sin x ^ { 3 / 2 }sinx3/2

Answer the Question $\sin x ^ { 3 / 2 }$