1. (1 point) For the given position vectors r(t), compute the (tangent) velocity vector r0 (t) for the given value of t . A) Let r(t) = (cost, sint). , )? Then r0 ( 4 )= ( B) Let r(t) = (t 2 ,t 3 ). , )? Then r0 (2)= ( C) Let r(t) = et i + e2t j + tk. Then r0 (4)= i+ j+ 1 4. (1 point) The derivative of r(t) = tan(t) 3, 2t sin(t) , 2t . is r0 (t) = Answer(s) submitted: (incorrect) 5. (1 point) Evaluate the integral: k? Answer(s) submitted: Z t i+ Answer : j+ k Answer(s) submitted: (incorrect) 2. (1 point) Given that the acceleration vector is a (t) = (25 cos (5t)) i + (25 sin (5t)) j + (4t) k, the initial velocity is v (0) = i + k, and the initial position vector is r (0) = i + j + k, compute: A. The velocity vector v (t) = i+ j+ k B. The position vector r (t) = i+ j+ k Note: the coefficients in your answers must be entered in the form of expressions in the variable t; e.g. \"5 cos(2t)\" (incorrect) 6. (1 point) Find a vector function r(t) that satisfies the indicated conditions: r0 (t) = hsin 3t, sin 2t, 3ti , r(0) = h4, 6, 3i Answer(s) submitted: r(t) = h , , i Answer(s) submitted: (incorrect) (incorrect) 3. (1 point) A gun has a muzzle speed of 80 meters per second. What angle of elevation [0, /4] should be used to hit an object 190 meters away? Neglect air resistance and use g = 9.8 m/sec2 as the acceleration of gravity. Answer: (9si + 30s2 j + 14k) ds 0 7. (1 point) Consider the parametric curve r(t) = t 3 , t 2 . radians Answer(s) submitted: Determine which of the following graphs is the correct image of this curve: (incorrect) 1 initial velocity v(0) = h1, 2i , and initial position r(0) = h0, 0i . v(t) = r(t) = Usage: To enter a vector, for example hx, y, zi, type \"< x, y, z >\" Answer(s) submitted: (incorrect) 10. (1 point) Consider (cos(2t), sin(2t), 1t). Compute, at t = A. The unit tangent vector T = ( B. The unit normal vector N = ( C. The unit binormal vector B = ( B D. The curvature = A the helix r(t) = 6: , , , , , ) ) , ) Answer(s) submitted: Also, compute the velocity and acceleration vectors at t = 1, and use this information to determine the direction of motion for the curve. The correct graph is ? . v(1) = . a(1) = . The direction of motion is ? . Usage: To enter a vector, for example hx, y, zi, type \"< x, y, z >\" Answer(s) submitted: (incorrect) 11. (1 point) Find the curvature of y = sin (1x) at x = 4 . Answer(s) submitted: (incorrect) (incorrect) 8. (1 point) Calculate the velocity and acceleration vectors, and speed for 12. (1 point) Find the curvature (t) of the curve r(t) = (4 sint) i + (4 sint) j + (3 cost) k r(t) = hsin(t) , sin(4t) , cos(t)i when t = 7 4 . Velocity: Acceleration: Speed: Usage: To enter a vector, for example hx, y, zi, type \"< x, y, z >\" Answer(s) submitted: (incorrect) 13. (1 point) The parabola given by the equation x = y2 + 8y + 13. The curvature of this parabola attains its maximum value at (h, k) for: h= and k= Answer(s) submitted: (incorrect) Answer(s) submitted: 9. (1 point) Find r(t) and v(t) given acceleration a(t) = ht, 5i , (incorrect) 2 C 3 \f{1 point) Consider the parametric curve r(t) = {1:3, t2) . Determine which of the following graphs is the correct image of this curve: Also, compute the velocity and acceleration vectors at t = 1, and use this information to determine the direction of motion for the curve. The correct graph is| (Select One) V | v(1) = i i . 11(1) 2 i i . The direction of motion is| (Select One) ' | - Usage: To enter a vector, for example (3, y, 2), type "i x, y, z >"