solve each equation over the given interval 2x=cosx for all values of x
There is no easy method to solve equations of this type.
One way is to graph the left side and the right side as individual functions and see where they intersect
e.g.
graph : y = 2x and y = cosx
you will see that they intersect appr. at x = 1/2
As a matter of fact that is how "Wolfram" shows the solution
http://www.wolframalpha.com/input/?i=2x%3Dcos%28x%29+
With a good calculator at hand, you can then pinpoint the answer more accurately
e.g.
x = .5, LS = 1, RS = .877 , RS < LS so lower the x
x = .4, LS = .8, RS = .921 RS > LS so raise the x
x = .45 , LS = .9, RS = .90044 , not bad , RS > LS , so raise the x a bit
x = .452, LS = .904 , RS = .8995 , RS < LS, lower the x
x = .4505, LS = .901 , RS = .9002
do you get the idea.
Wolfram had it at x = .450184
then LS = .900368
RS = .900367 , how is that for close
To solve the equation 2x = cos(x) over the given interval, we need to find the values of x that satisfy the equation.
One approach to solving this equation is by using trial and error or by graphing the functions 2x and cos(x) to find their points of intersection. However, this might not always yield an accurate solution.
A more systematic approach is to use algebraic methods. To do this, let's rewrite the equation as:
2x - cos(x) = 0
Now, we want to find the values of x for which this equation holds true. Unfortunately, there is no straightforward algebraic method to solve this equation. However, we can use numerical methods or approximation techniques to estimate the values of x.
One common numerical method to solve equations like this is called the "Newton-Raphson method." This method requires an initial guess value for x, which we will call x₀.
1. Make an initial guess for x, let's say x₀ = 0.
2. Find the derivative of the equation: f'(x) = 2 + sin(x)
3. Use the Newton-Raphson formula to refine the guess:
x₁ = x₀ - f(x₀)/f'(x₀)
4. Repeat step 3 until you reach a satisfactory level of precision for x.
However, keep in mind that the Newton-Raphson method, along with other numerical methods, might not always provide exact solutions but rather approximations.
Another way to find the values of x is to use a graphing calculator or online tools. By graphing the functions 2x and cos(x) within the given interval, you can visually identify their points of intersection.
In this case, it is best to use numerical or graphical methods to find the values of x that satisfy the equation 2x = cos(x) over the given interval.