MATH 251 Fall 2008 Sample Test Solutions, Exams of Calculus

Download MATH 251 Fall 2008 Sample Test Solutions and more Exams Calculus in PDF only on Docsity! MATH 251 Instr. K. Ciesielski Fall 2008 SAMPLE TEST # 2 Solve the following exercises. Show your work. Ex. 1. Find the parametric equations of the line that passes through the point P (11, 13,−7) and is perpendicular to the plane with the equation: x− 2z = 17. Ex. 2. Find the unit normal vector to the curve r(t) = 〈et, t, cos πt〉 at the point (1, 0, 1). Ex. 3. Find the volume of the pyramid with the vertices: P (3, 2,−1), Q(−2, 5, 1), R(2, 1, 5), and the origin O(0, 0, 0). The volume of a pyramid id equal 1/6th of the volume of paral- lelepiped spanned by the same vectors. Ex. 4. Find an equation of the plane passing through point (1, 11,−13) and parallel to the plane with equation 2x− 17z + π = 0. Ex. 5. Describe and sketch the graphs of the surfaces given by the following equations. Name each surface. Give specific informations, like center and radius in the case of a sphere. (a) 2x2 + 2y2 + 2z2 = 7x + 9y + 11z. (b) 4y = x2 + z2 (c) 4y = z2 Ex. 6. Find the curvature κ of the curve with position vector r(t) = i cos t + j sin t + 2t k. Ex. 7. Let v(t) = i(t + 1)−1 + kt3 be a velocity of a particle. Find the acceleration vector a(t) of the particle and its position vector r(t), where its initial position was r0 = 3i. Ex. 8. Find the the arc length, s, of the curve with position vector r(t) = 2et i+2t j+e−t k from t = 0 to t = 1. Ex. 9. Sketch and fully describe the graph of a function f(x, y) = √ 1 + x2 + y2. 1