2. Find a parametrization of the curve formed by the intersection of the surfaces y" - z? = x - 2 and y'+ 2" =9 with z 2 0 in two different ways. (In both cases, specify the domain - the allowable values of t.) (a) By using cost and sint for two of the three variables (you figure out which two). (b) Solving for z in terms of y, and then substituting that into the first equation. 3. We will show that the curve with the following parametrization lies in a plane. r(t) = (+2 - 1,t - 2t2, 4 - 6t) (a) Show that the points on the curve at t = 0, t = 1, and t = 2 do not lie on a line. Then find an equation of the plane that they determine. (b) Show that for all t, the points on C satisfy the equation of the plane from part (a). 4. For 0 S t S 47, the path of a particle is parametrization(t) sin(t), (sin(t))2, sin(t)). (a) Show that the path of the particle is a closed loop. (b) Let C be the curve on which the particle travels. How many times does the particle traverse C from t = 0 to t = 4#? Justify your answer. (c) Write down, but don't evaluate, an integral whose value is the length of C. (d) Write down, but don't evaluate, an integral whose value is the distance traveled by the particle. 5. Consider the curve C in R3 given by r(t) = (e' cost) i + 2j + (e* sint) k (a) Draw a sketch of C. (b) Calculate the arc length function s(t), which gives the length of the segment of C between r(0) and r(t) as a function of the time t for all t 2 0. Check your answer with the instructor. (c) Now invert this function to find the inverse function t(s). This gives time as a function of arclength, that is, tells how long you must travel to go a certain distance. (d) Suppose h : R -+ R is a function. We can get another parameterization of C by considering the composition f(s) = r(h(s)) This is called a reparametrization. Find a choice of h so that . f(0) = r(0) . The length of the segment of C between f(0) and f(s) is s. (This is called parametrizat length.) Check your answer with the instructor. (e) Without calculating anything, what is If'(s)|? 6. Consider the curve C given by the parametrization r : R - R3 where r(t) = (sint, cost, sin? t). (a) Show that C is in the intersection of the surfaces z = x2 and a + y? = 1. (b) Use (a) to help you sketch the curve C. 7. (a) Sketch the top half of the sphere a? +y? + 2? = 5. Check that P = (1, 1, v3) is on this sphere and add this point to your picture. (b) Find a function f(z, y) whose graph is the top-half of the sphere. Hint: solve for z. (c) Imagine an ant walking along the surface of the sphere. It walks down the sphere along the path C that passes through the point P in the direction parallel to the yz-plane. Draw this path in your picture. (d) Find a parametrization r(t) of the ant's path along the portion of the sphere shown in your picture. Specify the domain for r, i.e. the initial time when the ant is at P and the final time when it hits the ry-plane. 8. As in 4(d), consider a reparametrization f(s) = r(h(s)) of an arbitrary vector-valued function r : R - R3. Use the chain rule to calculate [f'(s)| in terms of r' and h