solve for x in a given interval using guess improvement stategy. express to the nearest tenths. cos x=1/x, -4<=x<=0
there are lots of different iterative solution methods:
bisection
Newton's method,
secants,
etc.
give us a hint where you want to go.
for these kind of problems, it is often a good idea to get a ball-park answer by graphing both the left side and the right side of the equation in the given domain.
A good webpage for this is
http://rechneronline.de/function-graphs/
In the first graph box enter
cos(x)
in the second graph box enter
1/x
It will show that there is an answer near -2 and -4.5
the latter being outside the domain.
You can then fine-tune the solution by zooming in.
After a few intermediate changes, I changed
"range x-axis from" -2.08 to -2.06
"range y-axis from" -.5 to -.45
to get x = appr. -2.074
check:
cos(-2.074) = -.48223
1/-2.074 = -.48216
not bad
To solve the equation cos(x) = 1/x in the given interval -4 <= x <= 0 using the guess improvement strategy, you can follow these steps:
1. Start by choosing an initial guess value within the interval -4 <= x <= 0. Let's say x = -4 as our initial guess.
2. Substitute the initial guess value into the equation cos(x) = 1/x and calculate the result.
cos(-4) = 1/(-4)
3. Calculate the difference between the result and the desired value, which is 0 in this case.
Difference = calculated value - desired value = cos(-4) - 1/(-4)
4. Based on the sign of the difference, adjust the guess value to get closer to the solution.
Since the difference is negative, we need to increase our guess value. Let's update our guess to x = -3.5 and repeat step 2.
5. Repeat steps 2-4 until the difference becomes very close to zero. Keep updating the guess value to get closer to the solution.
Repeat step 2 with x = -3.5:
cos(-3.5) = 1/(-3.5)
Difference = cos(-3.5) - 1/(-3.5)
Repeat step 2 with updated guess value until the difference approaches zero:
x = -3.25, -3.125, -3.0625, -3.03125, ...
6. Continue this process until the difference becomes very close to zero, which indicates that you have found an approximate solution for x in the given interval.
By repeating these steps and making incremental updates to the guess value, you will eventually find a value of x that satisfies the equation cos(x) = 1/x within the given interval.