Shawn Simard Assignment Section 10.7 due 10/28/2016 at 11:59pm MST 1. (1 point) Find the domain of the vector functions, r(t), listed below. You may use \"-INF\" for and use \"INF\" for as necessary, and use \"U\" D for a union symbol if E a union of intervals is needed. 1 a) r(t) = ln(3t), t + 6, 9t 6. (1 point) Find a vector function that represents the curve of intersection of the paraboloid z = 2x2 + 3y2 and the cylinder y = 2x2 . Use the variable t for the parameter. r(t) = ht, t 4 \u0012 2 \u0013 t 16 2. (1 point) Let r(t) = ( t + 3) i + j + sin(t) k. t +4 Then i+ j+ k. lim r(t)= Hint: The curve which is cut lies above a circle in the xyplane which you should parametrize as a function of the variable t so that the circle is traversed counterclockwise exactly once as t goes from 0 to 2*pi, and the paramterization starts at the point on the circle with largest x coordinate. Using that as your starting point, give the parametrization of the curve on the surface. t1 3. (1 point) Find the limit: \u001c t0 h , , e5t 1 t3 5 , 4 3, t t t 15 + t \u001d c(t) = (x(t), y(t), z(t)), where x(t) = y(t) = z(t) = i 4. (1 point) The curve c(t) = hcost, sint,ti lies on which of the following surfaces. Enter T or F depending on whether the statement is true or false. (You must enter T or F - True and False will not work.) 1. 2. 3. 4. 8. (1 point) Find the derivative of the vector function r(t) = ln(19 t 2 )i + 5 + tj + 4e6t k r0 (t) = h a circular cylinder an ellipsoid a plane a sphere B , i , 9. (1 point) For the given position vectors r(t) compute the unit tangent vector T(t) for the given value of t . A) Let r(t) = hcos 5t, sin 5ti. Then T( 4 )h , 5. (1 point) Match the parametric equations with the graphs labeled A - F. As always, you may click on the thumbnail image to produce a larger image in a new window (sometimes exactly on top of the old one). 1. x = cost, y = sint, z = lnt 2. x = cost, y = sint, z = sin 5t 3. x = sin 3t cost, y = sin 3t sint, z = t 4. x = cos 4t, y = t, z = sin 4t 5. x = t 2 2, y = t 3 , z = t 4 + 1 6. x = t, y = 1/(1 + t 2 ), z = t 2 A i , 7. (1 point) Consider the paraboloid z = x2 + y2 . The plane 5x 3y + z 3 = 0 cuts the paraboloid, its intersection being a curve. Find \"the natural\" parametrization of this curve. t 4, sin(2t),t 2 \u001c \u001d t 4t 1/3 c) r(t) = e , 2 ,t b) r(t) = lim Kim MAT 267 ONLINE B Fall 2016 C B) Let r(t) = ht 2 ,t 3 i. Then T(2) = h i i , C) Let r(t) = e5t i + e2t j + tk. Then T(3)= k. i+ j+ 10. (1 point) Find parametric equations for the tangent line at the point \u0001 \u0001 \u0001 cos 56 , sin 65 , 56 on the curve x = cost, y = sint, z = t x(t) = y(t)= z(t)= E F D 1 (Your line should be parametrized so that it passes through the given point at t=0). 16. (1 point) Find a vector parametric equation ~r(t) for the line through the points P = (0, 5, 2) and Q = (1, 6, 5) for each of the given conditions on the parameter t. 11. (1 point) Find the parametric equations for the tangent line to the curve (a) If~r(0) = h0, 5, 2i and~r(4) = h1, 6, 5i, then ~r(t) = x = t 5 1, y = t 3 + 1, z = t 2 (b) If~r(7) = P and~r(10) = Q, then ~r(t) = at the point (242, 28, 9). Use the variable t for your parameter. x= , y= , z= 12. (1 point) Evaluate \u0001 ti + t 2 j + t 3 k dt = (c) If the points P and Q correspond to the parameter values t = 0 and t = 4, respectively, then ~r(t) = Z 3 i+ j+ 17. (1 point) The function r (t) traces a circle. Determine the radius, center, and plane containing the circle r (t) = 7i + (6 cos(t)) j + (6 sin(t)) k k. 0 13. (1 point) If r(t) = cos(4t)i + sin(4t)j 10tk compute r0 (t)= i+ j+ k R i+ and r(t) dt= with C a constant vector. j+ Plane : x = Circle's Center : ( Radius : k+C 14. (1 point) Find a vector parametrization of the curve x = 3z2 in the xz-plane. Use t as the parameter in your answer. , , ) 18. (1 point) Use cos(t) and sin(t), with positive coefficients, to parametrize the intersection of the surfaces x2 + y2 = 49 and z = 2x4 . i r(t) = h , , ~r(t) = 19. (1 point) Find a parametrization, using cos(t) and sin(t), of the following curve: The intersection of the plane y = 6 with the sphere x2 +y2 +z2 = 117 r(t) = h , , i 15. (1 point) Are the following statements true or false? ? 1. The parametric curve x = (3t + 4)2 , y = 5(3t + 4)2 9, for 0 t 3 is a line segment. ? 2. The line parametrized by x = 7, y = 5t, z = 6 + t is parallel to the x-axis. ? 3. A parametrization of the graph of y = ln(x) for x > 0 is given by x = et , y = t for < t < . 20. (1 point) Find the solution r(t) of the differential equation with the given initial condition: r0 (t) = hsin 2t, sin 5t, 7ti , r(0) = h8, 3, 4i r(t) = h c Generated by WeBWorK, http://webwork.maa.org, Mathematical Association of America 2 , , i \f\f\f\f\f\f\f\f