A curve c is given by the vector equation rt2 - t 3 2t 3t

Question

1) A curve C is given by the vector equation r(t)=?2 - t, 3 + 2t, 3t?.

(a) Find the length of the curve in the interval 0 ≤ t ≤ 1.                 

(b) Find the arc length function for the curve C measured from point P(1, 5, 3) in the direction of increasing t, then reparametrize the curve with respect to arc length.

(c) Find the point 2 units along the curve (in the direction of increasing t) from P.

2) Let C be the curve with parametric equations x = 2t, y = 3sin2t, z = 3cos2t.

(a) Find the curvature of C at a general point.                                                    

(b) Find the equation of the normal plane at the point P(π, 0, -3).                            

3) Let the acceleration of a particle be given by a(t)=ti + 2j + 3k. Find the velocity, position vectors of the particle if the initial velocity is v(0)=3i + 2k and the initial position is r(0)= i - j.                                             

4) A projectile is launched at an angle of 45^o to the horizontal at an initial speed of 43 m/s.

(a) How high does the shot go?                                                                                

(b) Where does the shot land?                                                                                 

5) Let f(x, y)=√(9 - x² - y²).

(a) Find and sketch the domain of f.                                                                       

(b) Sketch the graph of the function.                                                                     

6) Find the limit lim_{(x,y)→(0,0)}(4x²)/(x² + 5y²) if it exists, or show that the limit does not exist.

7) Given that cos(xyz) = 1 + x²y² + z², use the Chain Rule to find ∂z/∂x.                   

8) Show that the function u = sin(kx)sin(akt) is a solution of the wave equation u_tt=a² u_xx.                                             

9) Let g be a differentiable function and suppose that g(2, 1) = -2, g_x(2, 1) = 3 and g(2, 1) = 5. Use linear approximation to estimate g(2.1, 0.9).                                                       

10) Find the equation of the tangent plane to the surface z = 3x² - y² + 2x at the point (1, -2, 1). 

11) Suppose z = f(x, y), where x = g(s, t), y = h(s,t), g(1, 2) = 3, g_s (1, 2) =- 1, g_t (1, 2) = 4, h(1, 2) = 6, h_s (1 ,2) = -5, h_t (1, 2) = 10, f_x (3, 6)=7, f_y (3, 6)=8. Find ∂z/∂s when s=1 and t=2.                                                        

12) If z = sin(x + sint), show that (∂z/∂x) (∂² z/∂x∂t) = (∂z/∂t) (∂²z/∂x²).                                                

13) Let f(x, y)=x² e^-y.

(a) Find the directional derivative of f at the point(-2, 0) in the direction toward the point (2, -3).               

(b) In what direction does f have maximum rate of change at (-2, 0)? What is the maximum rate of change?