Calculus
A particle moves along the x-axis with position at time t given by x(t)=e^(-t)sin(t) for 0 is less than or equal to t which is less than or equal to 2 pi.
a) Find the time t at which the particle is farthest to the left. Justify your answer
I think you have to find the prime of this equation and then see when it is negative.
b) Find the value of the constant A for which x(t) satisfies the equation Ax"(t)+x'(t)+x(t)=0 for 0 is less than t which is less than 2 pi.
I have no idea how to even start this problem.
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Answers
To solve this problem, first use implicit differentiation to find the second derivative of x(t). This can be done by taking the derivative of both sides of the equation x(t)=e^(-t)sin(t) with respect to t. This will give x'(t)=-e^(-t)sin(t)+e^(-t)cos(t). Then, differentiate the equation again to find the second derivative x"(t). Substituting this value of x"(t) into the equation Ax"(t)+x'(t)+x(t)=0 and using the values of x(t) and x'(t) found previously, you can solve for A.
