Symbols. R stands for the set of real numbers; R for the set of ordered pairs of real numbers, aka the Cartesian plane. Functions are indicated by their name, ":", then an expression like S - T. This means S is the domain of the function and the function's values lie in T. Exercise 1. Find the center of mass of the mass distribution given by the density f(x) = a over the interval [0, 1). This is the same as asking: if the area under the curvey = a between r = 0 and r = 1 were made of a homogeneous material, where would its balance point on the r axis be? Exercise 2. Plot the image of each of the following parametric curves. Give it a shot by hand first, then check with something like Wolfram Alpha or R. Figure out what you missed and explain. Attach both plots. 1. y : [-1, 1] - R2 given by y(t) = (12, t?) 2. a : [0,3] - R2 given by a(t) = (cos(at), t) 3. 3 : [-1, 1] - R3 given by 3(t) = (#3, 15) 4. A : [-1, 2#] - R3 given by A(t) = (2 cos(t), sin(t) ) 5. 7 : [1,5] - R' given by n(t) = (} cos(t), ? sin(t)) 6. p : [0, ] - R' given by p(t) = (cos(t), sin(2t)) 7. 1 : [-1, ]] - R' given by w(t) = (cosh(t), sinh(t)) Exercise 3. Velocity vectors. It's very common to use the moving-particle picture where a paramet- ric curve represents the trajectory (aka position function) of a moving particle. The term "velocity vector" of a curve appeals to this picture: it just means that if you think of a parametric curve as the position function of a moving particle, then the particle's velocity vector at a given time is the derivative of the parametric curve at that time. Find the velocity vector of the curve as a function of t, for each curve above. Find the speed funct n of each curve (speed as a function of