Suggest a cubic that would best approximate a sine wave (f(x) = sin(x)) over the interval 0 less than or equal to x less than or equal to 2pi
(i) What intervals would you be concerned with. What amplitude would you be working with
(ii)Find a cubic that best approximates sin(x)
(iii) FInd a member of the family of such cubics that passes through the point (pi divided by 2, 0.969)
To approximate a sine wave (f(x) = sin(x)) over the interval 0 ≤ x ≤ 2π using a cubic function, we can follow the steps below:
(i) Intervals and Amplitude:
To best approximate the sine wave, we should choose a cubic that matches the overall shape of the sine wave within the given interval. Since the function has a period of 2π (one complete cycle), we only need to consider the interval 0 ≤ x ≤ 2π.
The amplitude of the sine wave is 1, as the maximum value of sin(x) is 1 and the minimum value is -1. This information will help us determine how to scale our cubic function to match the vertical behavior of the sine wave.
(ii) Finding the Cubic:
To find a cubic that best approximates sin(x), we can use the concept of curve fitting. We need to find a cubic function of the form f(x) = ax³ + bx² + cx + d that closely resembles sin(x) within the interval 0 ≤ x ≤ 2π.
One approach is to use polynomial regression with the least squares method. This method minimizes the sum of the squared differences between the values of sin(x) and the cubic function over the interval.
Using a computer software or programming language capable of performing regression analysis, you can obtain the coefficients a, b, c, and d that provide the best fit for the cubic function. The result will be an equation for the cubic that approximates sin(x) within the interval.
(iii) Passing through a Given Point:
To find a specific member of the family of such cubics that passes through the point (π/2, 0.969), we need to substitute the x-value (π/2) and y-value (0.969) into the equation of the cubic obtained in step (ii). Then, we can solve for the specific coefficients that give us the desired cubic equation.