Calculus Logarithmic Functions help please

[Moonflower]

Calculus Logarithmic Functions help please!

The question is:
A particle moves alonge the x-axis with position at time t given by x(t) = e^(-t) sin t for 0 ≤ t ≤ 2π.

1. Find the time t at which the particle is farthest to the left. Justify your answer.
2. Find the value of the constant A for which x(t) satisfies the equation
Ax" (t) + x' (t) + x(t) = 0 for 0 < t < 2π.

For no.1, I think its when t=0, because within the interval 0 ≤ t ≤ 2pi, 0 is when x has the least value, therefore most to the left.

For no.2, x(t)= e^-t sin t, x'(t)= e^-t (cos t - sin t), and x''(t)= -2e^-t cos t. Factorizing, x(t)+x'(t)+x''(t) gives me e^-t (sin t +cos t -sin t -2A cos t). To make the sums inside parentheses zero, A would have fit the condition 0= cos t - 2A cos t. A= \frac{1}{2} fits, it seems.

Am I on the right track? Thanks

 

[Filip Larsen]

For 1: Is there a way you can check if x(t) is a minimum for t = 0? How would you normally find a minimum of x when you have access to the derivatives x' and x''?

For 2: Looks correct.

 

[Moonflower]

aah, i didn't think about that..thanks!