I have three questions that I'm going to roll into one. I'm going insane trying to figure these out. 1. Find the unit tangent vector T to the cycloid. Also find the speed at theta=0 and theta=Pi, if the wheel turns at dtheta/dt=1. that dtheta/dt is the speed, right? I'm a little baffled as to how I'm supposed to find a tangent vector when I don't even know what the cycloid is. I think it has something to do with v/|v|, but that doesn't change the fact I don't know how to answer the question. I'm sorry for not having gotten further with this, I need a nudge. 2. The slope of the clycloid is infinite at theta=0: dy/dx= sin(theta)/(1-cos(theta)). Estimate the slope at theta=(1/10) and theta=(-1/10). I know I need to use L'Hopital's rule, but the answer my key is giving me is confusing me a bit. here's my main question: how does (sin(theta))/(1 - cos(theta))=approx (theta)/(((theta)^2)/2)? after that, I think I understand. 3. For a cardioid the radius C - 1 of the fixed circle equals the radius 1 of the circle rolling outside (epicycloid with C=2). (a) The coordinates of P are x = -1+2cos(theta)- cos(2theta), y=____________. (b) the double-angle formulas yield x= 2 cos(theta)(1-cos(theta)), y=_______________. (c) x^2 + y^2 =_______________ so its square root is r=________________. this is the one I'm really struggling with. I've been sitting here trying to make sense of it for about an hour and I just DO NOT GET IT AT ALL. I'm getting really frustrated and I really really really need someone to guide me through it. I'm sorry I don't have any work to show for my efforts, but I know nothing I've got is right. I have an answer key for this one, but it's not proving very useful. please, I really need help!