Math 230 MidTerm Exam February 16 2004 Name Section There are 10 questions on this exam Each is worth 10 points In questions with multiple parts the points are divided equally among the parts unless indicated otherwise Please note that throughout this exam bold font is used for symbols representing vectors and regular font is used for symbols representing scalars Show all your work Partial credit may be given Use of books notes calculators and cell phones is not permitted during this exam Please check that your cell phone is off before starting your exam Problem 1 2 3 4 5 6 7 8 9 10 Total Score 1 Match each of the following equations with the sketch below labeled I XII that best represents its graph a x2 y z 2 0 d x2 y 2 z 2 1 b x2 y 2 z 2 1 2 e x2 y 2 z 2 1 2 c x2 y 2 z 0 Click here to see the surfaces 2 Suppose that a b c are nonzero vectors that 0 denotes the zero vector and 0 denotes the zero scalar Place an N in the space provided if an expression has no meaning and a Y if it is defined Do not try to evaluate it or simplify it i a b c ii a b iii b c a iv a b c v a b c vi 0 0a vii 0 0a viii a a ix compc b a x projc b a 3 a 1pt Find the distance from the point P 3 2 1 to the xy plane b 3pt The endpoints of the diameter of a sphere are located at P 0 2 4 and Q 2 6 8 Write down the equation of the sphere c 3pt Consider the surface whose equation in cylindrical coordinates is r z What is its equation in spherical coordinates d 3pt The rectangular coordinates of a point in space are 0 2 2 What are its spherical coordinates 4 Find the shortest distance from the point P 1 2 3 to the plane 2x 3y 6z 6 5 Find an equation for the plane which contains the point P 1 2 3 and is parallel to the plane 2x y z 6 6 Find an equation for the line of intersection of the planes x y z 1 and x 2y 2z 2 7 Consider the space curve r t 4 sin t 3 cos t 7 cos t a 2pt Find its velocity vector b 2pt Find the arc length of r t from t 0 to t 2 c 2pt Find the unit tangent vector T t d 4pt Find the curvature t 8 Assume that the speed of a car on a circular track of radius 13 km is sin 5t 2 km min a 3pt Find aT the tangential component of the car s acceleration at t 0 b 3pt Find aN the normal component of the car s acceleration at t 0 c 2pt Find a the magnitude of the car s acceleration at t 0 d 2pt What is cosine of the angle between the acceleration vector a and T at t 0 9 Suppose that the position of an object at time t is given by r t Suppose that the acceleration of this object is always parallel to r t a What is r r00 b Compute the following derivative and simplify your answer as much as possible d r r0 dt c Compute the following derivative d r r dt d If r is always perpendicular to r0 then what does your answer to Part c say about r e If r is always perpendicular to r0 then using your answers to Part d you say about r0 10 Find the shortest distance from the point P 1 2 3 to the line r 1 1 1 t 1 2 2 and Part b what can