Mathematics, 21.07.2019 20:10 jholland18

Use newton's method to find the absolute maximum value of the function f(x) = 6x sin x, 0 ≤ x ≤ π correct to six decimal places.

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Answer from: ezzyandrade7

The absolute value of the function $f\left( x \right) = 6x \times \left( {\sin x} \right)$ is $\boxed{10.918234}.$

Further explanation:

Given:

The function is $f\left( x \right) = 6x \times \left( {\sin x} \right).$

The interval for is $0 \leqslant x \leqslant \pi.$

Explanation:

The given function can be expressed as follows,

$f\left( x \right) = 6x \times \left( {\sin x} \right)$

Differentiate the above equation with respect to

$\begin{aligned}\frac{d}{{dx}}f\left( x \right)&= \frac{d}{{dx}}\left( {6x \times \sin x} \right)\\&= 6x \times\frac{d}{{dx}}\left( {\sin x} \right) + \sin x \times \frac{d}{{dx}}\left( {6x} \right)\\&= 6x\left( {\cos x} \right) + 6\sin x \\&= 6\left[ {x\cos x + \sin x} \right]\\\end{aligned}$

Substitute $\dfrac{d}{{dx}}f\left( x \right) = 0.$

$\begin{aligned}\frac{d}{{dx}}f\left( x \right) &= 0\\6\left[ {x\cos x + \sin x} \right] &= 0\\x\cos x + \sin x &= 0\\x\cos x&= - \sin x\\x&= \frac{{ - \sin x}}{{\cos x}}\\ x&=-\tanx\\\end{aligned}$

can take the value between o to $\pi.$

The value of the function at can be obtained as follows,

x = 0, x = 2.029 and $x = \pi.$

The value of the function at x=0.

$\begin{aligned}f\left(0\right)&= 6 \times 0 \times \sin 0\\&= 0\\\end{aligned}$

The value of the function at $x=\pi.$

$\begin{aligned}f\left(\pi\right)&= 6 \times \pi \times \sin \pi\\&= 6\pi\times0\\ &= 0\\\end{aligned}$

The value of the function at x=2.029.

$\begin{aligned}f\left({2.029} \right)&= 6 \times 2.029 \times \sin \left( {2.029} \right) \\&= 10.918234\\\end{aligned}$

The absolute value of the function $f\left( x \right) = 6x \times \left( {\sin x} \right)$ is $\boxed{10.918234}.$

Learn more:

1. Learn more about inverse of the functionhttps://link.

2. Learn more about equation of circle link.

3. Learn more about range and domain of the function link

Answer details:

Grade: High School

Subject: Mathematics

Chapter: Derivatives

Keywords: Derivative, value of x, function, differentiate, minimum value, dy, Newton’s method, six decimal places, function, f(x)=6xsinx, 0 to pi, absolute maximum, value of thre function.

Answer from: frankieflores02

Determine the x-value at which the slope of the tangent line to the curve of f(x) = 6x*sin x is zero (horiz. tan. line).

f '(x) = 6[x*cos x+sin x * 1] = 0. This, x*cos x + sin x = 0.

Create the new function g(x) = x*cos x + sin x. We want to find its roots/zeros.

Use the following formula (Newton's Method) to obtain roots/zeros:

f(n)
x(n+1) = x(n) - f '(n)

Then for the given function's derivative, x*cos x + sin x, we get

x*cos x + sin x x*cos x + sin x
x(n+1) = x(n) - = x(n) -
-x*sinx + cos x + cos x -x*sinx + 2*cos x

Let the first (guessed) root be pi/6; this is between 0 and pi, as required.

(pi/6)*cos(pi/6) + sin(pi/6)
then the next root would be pi/6 -
-(pi/y)*sin(pi/6) + 2*cos(pi/6)
(pi/6)[sqrt(3)/2] + 1/2
which equals pi/6 -
-(pi/6)*(1/2) + 2sqrt(3)/2

Lot of calculations! Having set up this application of Newton's Method, I'm hoping you can continute this process and find one or more roots to 6 places.

Questions? Please ask.

Answer from: Quest

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