Multivariable Calculus and Mathematica. Exam 1.

Copyright: Kevin Coombes, Paul Green, and Ronald Lipsman, 1998.

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General Instructions

All questions must be answered in the Notebook. Before beginning, look over the entire exam. There are three questions, each containing several parts. Do not spend too much time on any one part.

When answering the questions, make all your comments in text cells. Any input and output you create should appear in the corresponding type of cell. Please delete any extraneous and incorrect cells. Anything that appears will be graded!

Question 1

Input the coordinates of the points (Subscript[A, 1]), (Subscript[A, 2]), (Subscript[B, 1]), (Subscript[B, 2]), and (Subscript[B, 3]) by evaluating the following input cell:

\(A1\  = \ {1, \ 0, \ 1}; 
A2\  = \ {2, \ 3, \ \(-5\)}; 

    B1\  = \ {1, \ 1, \ 2}; 
B2\  = \ {3, \ 2, \ 1}; 

    B3\  = \ {1, \ 0, \ 3}; \)

You should also evaluate the following cell, which defines a new command for computing the lengths of vectors:

norm[v_]  :=  Sqrt[v.v]

(a) Find the length of the line segment (Subscript[A, 1]) (Subscript[A, 2]).

(b) Find the area of triangle (Subscript[B, 1]) (Subscript[B, 2]) (Subscript[B, 3]).

(c) Find an equation for the plane through (Subscript[B, 1]), (Subscript[B, 2]), and (Subscript[B, 3]).

(d) Find parametric equations for the line through (Subscript[A, 1]) and (Subscript[A, 2]).

(e) Find the coordinates of the point where the line through (Subscript[A, 1]) and (Subscript[A, 2]) meets the plane through (Subscript[B, 1]), (Subscript[B, 2]), and (Subscript[B, 3]).

Question 2

(a) Consider the curve given parametrically in the following input cell:

curve[t_] :=  {Cos[t], Sqrt[Sin[t]]}

Verify that curve[t], for 0  ≤  t   ≤  π,  is a parametrization of the quartic curve
	(x^2)+ (y^4) = 1, y  ≥  0.

(b) Next, consider the parabola (((2  y -  1))^( 2))= x.   Parametrize this parabola; that is, fill in coordinates in the next input cell.

parabola[t_] := {    ,    }

(c) Plot the two curves, curve[t] and parabola[t], on the same graph. You should see more than one point of intersection.

(d) Use FindRoot to determine the coordinates of all points of intersection on the two curves.

Question 3

Consider the curve given parametrically on the interval -5  ≤  t  ≤  5 by the equations in the following input cell:

newcurve  = 
       {ArcSinh[t],  -1 + Sqrt[t^2 + 1],  
        3  Sqrt[1 + t^2] + 2  ArcSinh[t]}

(a) Find the velocity, speed, and acceleration of the curve. (Simplify your answers.)

(b) Find a numerical estimate of the arclength of the curve.

(c) Graph the curve. Based only on the graph, decide if the curve lies entirely in a single plane.

(c) Recall the following formulas for the Frenet frame and the geometric invariants of a curve:
	T  =  v /  || v || ,
	B  =  v×a /  || v×a || ,
	N  =  B×T,
	\(TraditionalForm
      \`κ\  = \ 
        \( || v×a\) || \ \(/\ \( || v\( || \^3\)\)\)\),
	τ = (v×a).a'/ ||v×a\( || \^2\).
Compute as many of these quantities as you need in order to verify your answer to part (b). Explain your answer.

(d) Compute the curvature of the curve. Is the curvature increasing, decreasing, or neither when t = 0?