EXAMINATION 1 CALCULUS III REVIEW PROBLEMS prepared by Antony Foster Department of Mathematics (office: NAC 6-73) The City College of The City University of New York Convent Avenue At 138th Street New York, NY 10031 afoster00@ccny.cuny.edu afoster1955@gmail.com
Contents i
Introduction ii
Section 1 Review Problems for Up Coming First Examination Exercise 1.1 Use your knowledge of vectors to show that the midpoint M of the line segment joining points P 1 (x 1, y 1, z 1 ) and P (x, y, z ) in space is the point M in space given by ( x1 + x M, This formula should be familiar to you at least in R. y 1 + y, ) z 1 + z. Exercise 1. A sphere with center at P (x 0, y 0, z 0 ) and radius R > 0 is the set of all points Q(x, y, z) such that PQ = R. Show that a sphere of radius R and center at (x 0, y 0, z 0 ) has the equation (x x 0 ) + (y y 0 ) + (z z 0 ) = R. This equation is called the Standard equation of a sphere. Exercise 1.3 Find the center and radius of the sphere x + y + z + 3x 4z + 1 = 0 Hint: Use completing the square. Exercise 1.4 Find the angle θ between u = i j k and v = 6 i + 3 j + k Exercise 1.5 A force F = i + j 3 k N (newtons) is applied to a spacecraft with velocity vector v = 3 i j. Express F as sum of a vector parallel to v and a vector orthogonal to v. Exercise 1.6 and R( 1, 1, ). Find a unit vector ˆn perpendicular to the plane containing the points P (1, 1, 0), Q(, 1, 1), 1
Muliple Integration Section 1: Review Problems for Up Coming First Examination Exercise 1.7 Consider any nonzero vector A = A 1 i + A j + A 3 k in R 3. The direction angles α, β, and γ of A are defined as follows: α is the angle between A and the positive x-axis (0 α π). β is the angle between A and the positive y-axis (0 β π). γ is the angle between A and the positive z-axis (0 γ π). a) Show that cos α = A 1 A, cos β = A A, cos γ = A 3 A, and cos α + cos β + cos γ = 1. These cosines are called the direction cosines of A. b) Show that if V is a unit vector, then A 1, A, A 3 are the direction cosines of A. Exercise 1.8 Consider a bolt at the origin in the xy-plane. A wrench 8 inches long is placed around the bolt in the usual way so that its handle lies horizontally along the positive x-axis. A downward force F with magnitude 30-lbs and making an angle of 60 with the positive x-axis is applied to the end of the wrench. Determine the direction and magnitude of the torque τ, that drives the bolt. Exercise 1.9 Find parametric equations for the line through (, 0, 4) parallel to v =, 4,. Exercise 1.10 A helicopter is to fly directly from a helipad at the origin in space toward the point at a speed of 60 ft/sec. What is the position of the helicopter after 10 sec? Hint: r = r(t) = r 0 + t v = r 0 + t v v where v t is time and v is Speed, r 0 is the initial position and is the direction of its straight-line motion and r(t) is the position at time t. v v Exercise 1.11 Find an equation for the plane through P 0 ( 3, 0, 7) perpendicular to n = 5 i + j k. Exercise 1.1 x + y z = 5. Find a vector parallel to the line of intersection of the planes 3x 6y z = 15 and Exercise 1.13 Find the point where the line with parametric equations x = 8 3 + t, y = t, z = 1 + t; t R intersect the plane whose equation is 3x + y + 6z = 6.
Muliple Integration Section 1: Review Problems for Up Coming First Examination Exercise 1.14 Follow these steps to find the distance from a point S in space to a line l that passes through a point P and parallel to a vector v. (Draw a diagram). Step #1. Show that the length of the component of PS normal to the line is PS sin θ. Step #. Show that the distance Dist(l, S) from S to the line l through P and parallel to v is Dist(l, S) = PS v v Exercise 1.15 Use the result of the previous exercise to find the distance d from the point (0, 0, 0) to the line x = 5 + 3t, y = 5 + 4t, z = 3 5t; t R. Exercise 1.16 Follow these steps to find the distance from a point S in space to a plane Ax + By + Cz + D = 0. (Draw a diagram). Step #1. Find a point P on the plane. Step #. Find PS. Step #3. Show that the distance is where n = A i + B j + C k. d = PS n n, Exercise 1.17 Use the result of this previous exercise to find the distance from the point (, 3, 4) to the plane x + y + z = 13. Exercise 1.18 Suppose you are an air-traffic controller in charge of three airplanes in space whose flight-paths are lines in space given by L 1 : x = 3 + t, y = 1 + 4t, z = t; t R L : x = 1 + 4s, y = 1 + s, z = 3 + 4s; s R L 3 : x = 3 + r, y = + r, z = + r; r R Determine whether the paths (or lines), taken two at a time, are parallel, intersect, or are skew. If they intersect, find the point of intersection. Exercise 1.19 Find the volume of the box (parallelepiped) determined by the non-coplanar (not all lying in the same plane) vectors u = i + j k, v = i + 3 k, and w = 7 j 4 k Exercise 1.0 Use determinants to State a condition that you could use to detect when three nonzero vectors u, v, and w in space are coplanar (all lying in the same plane). 3