Find all the roots of the given function on the given interval. Use preliminary analysis and graphing to find good initial approximations.
f(x) = x/6 - 6 sec (x) on [0,40]
1. The function has root(s) when x =
At the roots,
x/6 - 6sec x = 0
x/6 = 6secx
x = 36secx
There is no simple algebraic way to solve an equation like this, so
we need to use "technology"
We can get an approximate answer by graphing
y = x and y = 36secx
I used
www.desmos.com/calculator
but had to zoom out a bit to see y = 36sec x
It showed the first at x = 37.423 and a 2nd at x = 38.027
the next one is beyond our domain.
well, just looking at the graph, f(x) has two roots on [0,40].
Since f(x) has asymptotes at odd multiples of π/2, I'd start just to the right of 23(π/2) and to the left of 25(π/2) or, at, say 37 and 39.
Then, Newton's method will be sure to converge on the roots (why?).
Starting with x = 37, iterations produce
37, 37.2475, 37.3829, 37.4199, 37.4225
Now see what you can do with x=39
To find the root(s) of the given function f(x) = x/6 - 6 sec(x) on the interval [0, 40], we need to identify values of x for which f(x) = 0.
To start, let's perform some preliminary analysis and graphing to find good initial approximations for the roots.
1. Preliminary Analysis:
First, let's analyze the function f(x) = x/6 - 6 sec(x). The term "sec(x)" represents the secant function, which is defined as 1/cos(x). Therefore, we can rewrite the function as:
f(x) = x/6 - 6/cos(x)
Now, let's analyze the function on the given interval [0, 40]:
- At x = 0, the term "cos(0)" in the denominator makes the function undefined. So, we exclude x = 0 from our analysis.
- As x approaches 0 from the right, cos(x) approaches 1, making the denominator of the function 6.
- As x increases, the term "x/6" becomes more significant compared to the other term. Therefore, f(x) will approach positive infinity.
- The secant function (cosine reciprocal) has periodic behavior, repeating every 2π radians or 360 degrees.
Based on this preliminary analysis, we expect to find one or more roots for the function f(x) within the interval [0, 40]. Now let's graph the function to get an initial approximation for the root(s).
2. Graphing:
Since the root(s) will occur at the point(s) where f(x) = 0, we can plot the graph of f(x) and look for the x-intercept(s).
Using a graphing calculator or software, plot the function f(x) = x/6 - 6 sec(x) on the interval [0, 40]. Observe the graph and identify any x-values where the function crosses the x-axis or comes close to it.
Once we have visually identified the potential root(s) on the graph, we can use that as an initial approximation. You can estimate the x-values of the roots by reading off the x-axis values where the curve crosses or is close to zero.
If there are multiple roots, repeat the graphical analysis to find additional approximations.
Remember, these initial approximations are not exact solutions but can guide us in the subsequent steps for finding the accurate roots of the function.