Use Newton's method to find all solutions of the equation correct to six decimal places. (Enter your answers as a comma-separated list.)
6 cos(x) = x + 1
can somebody explain to me step by step on how to solve this problem?
I assume x is in radians
we want 0 = x + 1 - 6 cos x
let f(x) = x + 1 - 6 cos x
find where f(x) = 0
pick a point, say x = 1
then
f(1) = 1 + 1 - 6 cos 1 = 2 - 6*0.54
= -1.24
plot that point (1, -1.24) on a graph
what is the slope of f(x) at that point?
f'(x) = 1 + 6 sin x
= 6*.84 = 5.05
draw that as tangent to curve of f(x) at (1,-1.24). It is aimed up toward the x axis. Where does it hit the x axis?
slope = 5.05 = 1.24/delta x
delta x = 1.24/5.05 = .246
so new x = 1 + .246 = 1.246
new f(x) = 1.246 + 1 - 6 cos 1.246
= .331
so now we are at (1.246, .331)
well .331 is not zero so repeat using that point
keep repeating until you get close to zero
To solve the equation using Newton's method, we will find the roots of the function f(x) = 6 cos(x) - x - 1.
Step 1: Choose an initial guess for the root, denoted by x₀. This can be any value, but a good starting point is usually close to the actual root.
Step 2: Compute f(x₀) and its derivative f'(x₀).
Step 3: Use the formula x₁ = x₀ - (f(x₀)/f'(x₀)) to find the next approximation of the root.
Step 4: Repeat steps 2 and 3 until the desired level of accuracy is reached.
Let's start with an initial guess of x₀ = 0.
Step 1: x₀ = 0
Step 2: Calculate f(x₀) and f'(x₀)
f(x₀) = 6 cos(0) - 0 - 1 = 5
f'(x₀) = -6 sin(0) - 1 = -1
Step 3: Compute x₁
x₁ = x₀ - (f(x₀)/f'(x₀))
= 0 - (5/-1)
= 5
Step 2: Calculate f(x₁) and f'(x₁)
f(x₁) = 6 cos(5) - 5 - 1 ≈ -21.248
f'(x₁) = -6 sin(5) - 1 ≈ -5.029
Step 3: Compute x₂
x₂ = x₁ - (f(x₁)/f'(x₁))
= 5 - (-21.248/-5.029)
≈ 5 - 4.215
≈ 0.785
Repeat the process until the desired level of accuracy is reached. Continuing the calculations gives the following iterative values:
x₃ ≈ 0.794
x₄ ≈ 0.793
x₅ ≈ 0.793
x₆ ≈ 0.793
Since x₅ and x₆ are approximately the same, we can conclude that the root of the equation is approximately x ≈ 0.793.
Therefore, the solution to the equation 6 cos(x) = x + 1, correct to six decimal places, is x ≈ 0.793.
To solve the equation using Newton's method, follow these steps:
Step 1: Rearrange the equation
Rewrite the equation as f(x) = 6 cos(x) - x - 1 = 0.
Step 2: Find the derivative
Take the derivative of f(x) with respect to x. The derivative of cos(x) is -sin(x), so the derivative of f(x) is f'(x) = -6 sin(x) - 1.
Step 3: Choose an initial guess
Select an initial guess, denoted as x(0), that is close to the actual solution. To simplify this example, let's choose x(0) = 0.
Step 4: Iteratively calculate x(n+1)
Apply Newton's method iteratively to find better approximations for the solution. Start by calculating x(1) using the formula:
x(1) = x(0) - (f(x(0)) / f'(x(0)))
For each subsequent iteration, use the formula:
x(n+1) = x(n) - (f(x(n)) / f'(x(n)))
Repeat this step until the difference between x(n) and x(n+1) is smaller than the desired tolerance. In this case, we'll use six decimal places of accuracy.
Step 5: Find all solutions
Continue the iterations until you find all the solutions, recording each value of x(n+1) that satisfies the desired accuracy. Repeat the process with different initial guesses to find all solutions.
Let's apply these steps to the given equation: 6 cos(x) = x + 1.
Step 1: Rearrange the equation
f(x) = 6 cos(x) - x - 1
Step 2: Find the derivative
f'(x) = -6 sin(x) - 1
Step 3: Choose an initial guess
We'll start with x(0) = 0.
Step 4: Iteratively calculate x(n+1)
Using the formula from Step 4, we have:
x(1) = x(0) - (f(x(0))/f'(x(0)))
= 0 - (6 cos(0) - 0 - 1) / (-6 sin(0) - 1)
= -1 / -1
= 1
Repeat the process:
x(2) = x(1) - (f(x(1))/f'(x(1)))
= 1 - (6 cos(1) - 1 - 1) / (-6 sin(1) - 1)
≈ 0.467913
Continue the process until we reach the desired accuracy.
Step 5: Find all solutions
Repeat the process with different initial guesses to find all solutions.
Continue the iterations until you have found all solutions to the desired accuracy. Record each value of x(n+1) that satisfies the six-decimal-place accuracy. The solutions should be given as a comma-separated list.