Show how the function y = e^(-x) sin x could represent damped oscillation.
a) Determine the local extreme for a sequence of wavelengths.
b) Show that the local maxima represent exponential decay.
(a) there is a local extreme every time sinx = cosx: x = π/4 + kπ
(b) since the maxima occur when sinx=1, the envelope of these maxima is the curve y=e^-x.
To demonstrate how the function y = e^(-x) sin x represents damped oscillation and determine its local extremes, we need to find the critical points where the derivative of the function equals zero.
a) Finding the local extremes (peaks and valleys) for a sequence of wavelengths:
1) Start by finding the derivative of the given function: y' = -e^(-x) sin x + e^(-x) cos x.
2) Set y' equal to zero and solve for x:
-e^(-x) sin x + e^(-x) cos x = 0.
sin x = cos x.
This equation is satisfied when x = π/4 + πk, where k is an integer.
3) These values of x correspond to the peaks and valleys of the damped oscillation.
Plug these x-values into the original function, y = e^(-x) sin x, to obtain the corresponding y-values.
b) Demonstrating that the local maxima represent exponential decay:
1) At the peaks, where the derivative equals zero, the amplitude of the oscillation is at its maximum.
2) As x increases, the exponential term e^(-x) approaches zero, which causes the amplitude of the oscillation to decrease.
3) This decrease in amplitude represents exponential decay.
4) You can observe this by comparing the y-values of the peaks and noticing how they decrease as x increases.
In summary, finding the critical points and evaluating the function at those points will determine the local extremes, representing the peaks and valleys of the damped oscillation. Additionally, as x increases, the local maxima (peaks) of the function demonstrate exponential decay.