Math 1172 Sample Test #3 1. Find parametric equations for the line segment joining the points (1, –3, ) and (5, 2) 2. For the curve given by the parametric equations x = 2sint , y = 5cost , 0 ≤ t ≤ 2π dy π find dx at t = 4 3. Replace the polar equation r = 3cosθ by an equivalent equation in Cartesian coordinates 4. The Cartesian coordinates of a point are (–1, –1). Find the polar coordinates (r, θ) with 0 ≤ θ < 2π and with (a) r > 0 (b) r < 0 5. 6. π Find the slope of the tangent line to the curve r = cos 2θ at θ = 3 Use integration in polar coordinates to find the area of the region bounded by the curve r = e 2θ and the lines θ = 0 and θ = ln 2 7. Find the foci and the vertices of the conic sections given by the equations. Classify each conic section as a parabola, ellipse, or hyperbola. If the conic section is € a parabola, find the directrix. If the conic section is a hyperbola, find the asymptotes. (a) x2 + 4y2 = 16 (b) y2 – 20x = 0 (c) 2x2 – 3y2 = –12 8. Find the eccentricity and the equation of the directrix of the conic section. Identify the conic section as a parabola, ellipse, or hyperbola. (a) (b) (c) 9. 10 r = 5 + 2cosθ 7 r = 2 – 2cosθ 12 r = 2 – 3sinθ Find the center and radius of the sphere x 2 + y 2 + z 2 = 4 x − 6y + 2z + 2 € Math 1172 Sample Test #3 Page 2 10. For the vectors u = 3i – 3j + k and v = 2i + 3j – 4k, find (a) u (b) v € (c) u+ v € (d) u−v € (e) u⋅ v € (f) u×v (g) v×u € (h) The angle θ between u and v € (i) projv u (j) scalv u € (k) u − projv u € (l) The angle between v and u − projv u € € A boat is steered in an easterly direction at a speed (in still water) of 15 km/h. The 11. boat encounters a current of 4 km/h flowing in a direction 60˚ north of east. € (a) Write velocity vectors for the boat and the current. (b) The true velocity of the boat is the sum of the velocity vectors of the boat and the current. Find the true velocity. (c) Find the magnitude and direction of the true velocity. 12. Find the components of the horizontal force < 20, 0 > in the directions parallel to and normal to a plane that makes an angle of π /6 with the positive x-axis. 13. Find the area of the triangle with vertices P (1, 2, –1), Q (2, 2, 1), and R (3,€1, 2). € 14. Find the area of the parallelogram with vertices P (1, 2, 0), Q (3, 2, 4), R (3, 1, 3), and S (5, 1, 7) Math 1172 Sample Test #3 Page 3 15. Find vector equations for each line. (a) The line through the point (3, –1, 0) parallel to the vector 2i + 3j + k (b) The line through the point (–1, 5, 2) parallel to the vector i + j (c) The line through the points (1, –3, 2) and (5, 2, –3) 16. Find parametric equations for the line segment joining the points (1, –3, 2) and (5, 2, –3) 17. The position of a moving particle is given by r(t) = t3 i + (2t2 + 1)j + k (a) Find the velocity v(t) (b) Find the acceleration a(t) (c) Find the speed when t = 1 (d) Find the unit tangent vector T when t = 1 (e) Find the tangential and normal components of the acceleration when t = 1 (f) Use the formula κ = v×a v 3 to find the curvature when t = 1. 18. The position of a moving particle is given by r(t) = sint i + cost j + tk € (a) Find the unit tangent vector T when t = 0 (b) Find the principal unit normal vector N when t = 0 (c) Find the unit binormal vector B when t = 0 19. Find the length of the curve r(t) = < 5cos t, 5sin t,12t >, 0 ≤ t ≤ 2π 20. Use integration in polar coordinates to find the length of the curve r = e2θ , 0 ≤ θ ≤ ln 2 €