APPM 2350 Summer 2006 Exam 1 June 21, 2006 1
INSTRUCTIONS: Computers, calculators, books, notes, flying monkeys, etc. are not permitted.
Some (possibly) useful formulae are attached. Write your name, your instructor’s name, and the
color of your exam sheet on the front of your bluebook. Work all problems. Start each problem on
anew page. Show your work clearly and box your final answer. A correct answer with incorrect
or no supporting work may receive no credit, while an incorrect answer with relevant work may
receive partial credit.
1. (a) Calculate the equation for the plane through the points A(−1,1,1), B(1,−1,1), C (1,1,−1).
(b) Calculate the distance between the point P(−1,−1,0) and the plane from (a).
(c) Sketch and describe (in words) the curve of intersection between the plane x+y= 0
and the surface x2+y2−z= 1.
(d) Find a parametrization of the curve in (c).
2. Prove or disprove the following statements (be sure to state clearly whether you believe them
to be TRUE or FALSE):
(a) If A·B=C·Band B6=0, then A=C.
(b) (A×B)·A= 0.
(c) If an ob ject moves with constant speed, its acceleration has no tangential component
and a constant normal component.
3. A particle moves along the ellipse, r(t) = cos(t)i+ 2 sin(t)j.
(a) Find the minimum and maximum magnitudes of its velocity and acceleration, i.e. the
minimum and maximum values of |v|and |a|.
(b) Find the curvature as a function of t.
(c) Find the minimum and maximum values of the curvature. What is the position of the
particle at these values?
4. A curve is given by r(t) = cos(t)i+ 2tj+ sin(t)k.
(a) Calculate the tangential, normal and binormal directions of this curve, i.e. calculate
T,Nand B.
(b) Calculate the curvature of the curve.
(c) Calculate the normal and tangential components of the acceleration.
— Over —
