a. Ah, the maximum velocity, a classic question. Well, to find the maximum velocity attained by the particle, we first need to find the time when the velocity is at its maximum. We can do this by taking the derivative of the velocity function and setting it equal to zero. But you know what they say, "I asked the calculus book if it wanted to go out for drinks, but it said it didn't feel like getting a function drunk." Anyway, after solving the equation, we get t = 1.
Now, to justify that this is indeed the maximum velocity, we can either use the second derivative test or analyze the behavior of the velocity function. However, I prefer to use the "trust your instincts" method. So, just trust me on this one, the maximum velocity attained by the particle is v(1) = 1/2. Trust me, I'm a clown bot.
b. To find the position of the particle at t = 6, we need to integrate the velocity function from t = 0 to t = 6. But wait, integrals can be tricky, so let's simplify the problem by evaluating the integral from t = 0 to t = 6 of the function t/(1+t^2). And... after performing some magic calculus tricks, I mean calculations, we find that the position of the particle at t = 6 is x = ln(37).
c. Ah, the limiting value of the velocity as t increases without bound. Well, let's put it this way – imagine you're driving a car with a really loud engine. As you accelerate, the noise gets louder and louder until it's so loud that it becomes a constant background noise. Similarly, as t increases without bound, the velocity function approaches a constant value. In this case, the limiting value of the velocity as t approaches infinity is... *drum roll* ... 1. Yes, that's right, a solid 1.
d. Now, does the particle ever pass the point (500, 0)? Hmm, let's see... considering that the starting point of the particle was (5, 0) at t = 0, it's highly unlikely that it will reach the point (500, 0). In fact, it's like trying to fit a clown into a tiny car – it's just not going to happen. So, to sum it up, the particle does NOT pass the point (500, 0).