1. What is one period of the function f(x) = 120 sin(120πx).
2. When does f(x) = 25 − x − sin(x) cross the x-axis?
sin(kx) has period 2π/k
the other requires some numeric or graphical techniques:
But you know it will be somewhere near x=25, since |sinx| <= 1 and 25-x=0 at x=25
http://www.wolframalpha.com/input/?i=25+%E2%88%92+x+%E2%88%92+sin(x)
To find the period of a function, we need to look at the coefficient in front of the x in the argument of the sine function.
In the first function, f(x) = 120 sin(120πx), the coefficient is 120π. The period of a sine function is given by 2π divided by the coefficient in front of the x.
So, the period of f(x) = 120 sin(120πx) is 2π / (120π) = 1/60.
Now, let's move on to the second question.
To find when a function crosses the x-axis, we need to find the values of x for which the function evaluates to 0. In the given function, f(x) = 25 − x − sin(x), we need to solve the equation:
25 − x − sin(x) = 0
To solve this equation, we can use numerical methods or approximations. One method is to use a graphing calculator or software to plot the function and find the x-values where it intersects the x-axis.
Another method is to use iterative numerical techniques such as the Newton-Raphson method or the bisection method. These methods involve repeatedly refining an initial guess for the root until a desired level of accuracy is achieved.
Alternatively, we can use algebraic manipulations to transform the equation into a form that is easier to solve. However, in this case, the presence of the sine function makes it difficult to find a closed-form solution.
So, in summary, to find when f(x) = 25 − x − sin(x) crosses the x-axis, we can use graphing software or numerical methods to approximate the x-values, or we can try to transform the equation into a form that is easier to solve algebraically.