Of the infinitely many lines that are tangent to the curve y = −4 sin x and pass through the origin, there is one that has the largest slope. Use Newton's method to find the slope of that line correct to six decimal places.
see related questions below.
let the equation of the tangent be y = mx, where m is the slope
Here is a picture with m = 1/2
http://www.wolframalpha.com/input/?i=plot+y+%3D+-4sinx+%2C+y+%3D+.5x+%2C++from+0+to+20
Here is a picture with m = .9
http://www.wolframalpha.com/input/?i=plot+y+%3D+-4sinx+%2C+y+%3D+.9x+%2C++from+0+to+20
Wow, that is actually very close, but .....
let's intersect them
mx = -4sinx
m = -4sinx/x
we want m to be a max, so dm/dx = 0
dm/dx = (x(-4cosx) - (-4sinx)(1) )/x^2
= 0
4sinx - 4xcosx = 0
sinx = xcosx
sinx/cosx = x
tanx = x or tanx - x = 0
let f(x) = tanx - x
f ' (x) = sec^2 x - 1
Newton said that at the solution to f(x)
betterx = x - f(x)/f ' (x)
= x - (tanx - x)/(sec^2 x - 1)
using my 2nd graph to guess at x = 4.5
betterx = 4.5 - (tan(4.5) - (4.5) )/(sec^2 (4.5) - 1)
= 4.4936..
betterx = 4.4936.. - (tan(4.4936 ......
= 4.4934096..
betterx - next round
= 4.4934056..
next round
= 4.4934095
ok, say x = 4.4934095
then y = -4sin(4.4934095) =3.904479
the first point of intersection is (4.4934095,3.904479)
using (0,0) as the other point
the maximum slope = 3.9044../4.4934..
= .868935
To find the slope of the tangent line to the curve y = -4 sin x that passes through the origin using Newton's method, we need to find the derivative of the function y = -4 sin x.
Step 1: Find the derivative of the function y = -4 sin x.
The derivative of y = -4 sin x can be found using the chain rule and the derivative of sin x, which is cos x.
dy/dx = d/dx (-4sin x) = -4(cos x)
Step 2: Set up the equation for Newton's method.
Let m be the slope of the tangent line. We want to find the value of m that makes the equation y = mx pass through the origin and be tangent to the curve y = -4 sin x. This means that the value of m should satisfy the equation -4 sin x = mx at the point of tangency.
Step 3: Apply Newton's method.
Newton's method involves applying successive approximations to find the root of a function. In this case, we need to find the value of x at which -4 sin x - mx = 0 for the tangent line to intersect the curve.
Let's start with an initial guess, x_0 = 1, and use the iterative formula:
x_{n+1} = x_n - f(x_n)/f'(x_n)
where f(x) = -4 sin x - mx and f'(x) = -4 cos x.
Step 4: Iterate the formula to find the value of x at which -4 sin x - mx = 0.
Using the initial guess x_0 = 1:
x_1 = x_0 - (-4 sin x_0 - mx_0) / (-4 cos x_0)
= 1 - (-4 sin 1 - m * 1) / (-4 cos 1)
Repeat the iteration until the value of x converges to a solution:
x_2 = x_1 - (-4 sin x_1 - mx_1) / (-4 cos x_1)
x_3 = x_2 - (-4 sin x_2 - mx_2) / (-4 cos x_2)
...
Continue this process until the value of x converges to a solution.
Let's continue the iteration until we reach a desired level of accuracy, such as six decimal places.
Please let me know if you would like to continue the step-by-step process.
To find the slope of the line that is tangent to the curve y = -4 sin x and passes through the origin, we will use Newton's method.
Step 1: Find the derivative of the curve y = -4 sin x.
The derivative of -4 sin x is -4 cos x.
Step 2: Set up the equation of the line that passes through the origin.
The equation of a line passing through the origin (0,0) can be written as y = mx, where m is the slope.
Step 3: Use Newton's method to find the value of x where the line is tangent to the curve.
Let's assume that the line is tangent to the curve at x = a. That means the derivative of the curve at x = a should be equal to the slope of the line (m). So, we can set up the equation -4 cos a = m.
Step 4: Convert the equation into a form suitable for applying Newton's method.
Rearrange the equation to solve for a: cos a = -m/4.
To apply Newton's method, we need to find a function f(a) such that f(a) = 0. Since cos a = -m/4, we can define f(a) = cos a + m/4.
Step 5: Apply Newton's method iteratively to find the value of a.
Newton's method is an iterative process that uses the formula: a_next = a_current - f(a_current)/f'(a_current).
In our case, f(a) = cos a + m/4 and f'(a) = -sin a.
Starting with an initial guess for a, we can refine the value by repeatedly applying the formula until we reach the desired level of precision.
Let's assume an initial guess for a, say a_0 = 0.
Iteration 1:
a_1 = a_0 - (cos a_0 + m/4) / (-sin a_0)
= 0 - (cos 0 + m/4) / (-sin 0)
= -m/4
Iteration 2:
a_2 = a_1 - (cos a_1 + m/4) / (-sin a_1)
Continue this iterative process until we reach the desired level of precision.
Step 6: Calculate the slope of the line correct to six decimal places.
Once we have found the value of a using Newton's method, we can substitute it back into the equation of the line y = mx to get the slope (m).
Please note that the iterative process of Newton's method needs to be programmed or performed using numerical software, as it requires repeated calculations.