SCIENCE ONE: MATHEMATICS ASSIGNMENT 10 (Due Feb. 3, 2012)
There are two parts to this assignment. The first part is online and is due by 12:00 noon on Fri. Feb. 3;
you will need to login at www.mathxl.com. The second part consists of the questions on this page and is also
due by 12:00 noon on Fri. Feb. 3. You are expected to provide full solutions with complete arguments and
justifications. You will be graded primarily on the correctness, clarity and elegance of your solutions. Your
answers must be typeset or very neatly written. They must be stapled, with your name and student number
at the top of the first page.
1. Successive approximations of the solution of an initial value problem. Let k be a constant, and suppose
y(t) is the solution of the initial value problem
y 0 = ky,
y(0) = 1.
(1)
Let t > 0 (for convenience; t < 0 would work just as well), and find an integral expression for the net
change in y over the interval [0, t], where the integrand is expressed in terms of y only (not y 0 ). Then
solve for the “future value” y(t), obtaining an expression of the form (called an integral equation)
y(t) = 1 + integral involving y.
(2)
(Notice that differentiating equation (2) gives the differential equation, and evaluating equation (2) at
t = 0 gives the initial condition, so the integral equation (2) is equivalent to the initial value problem
(1).) Let y(0) (t) = 1, a constant function, and think of it as an order-0 approximation of y(t) (not too
bad if t is very close to 0, but probably not so good otherwise).
(a) Let y(1) (t) be the result of substituting y(0) into the integral in the right hand side of equation (2) in
place of y, i.e.
y(1) (t) = 1 + integral involving y(0) .
Evaluate the integral to find y(1) (t) explicitly, and think of it as an order-1 approximation of y(t)
(hopefully a better approximation than y(0) (t) is).
(b) For n = 1, 2, 3, . . . , let y(n) (t) be the result of substituting y(n−1) into the integral in the right hand
side of equation (2) in place of y, i.e.
y(n) (t) = 1 + integral involving y(n−1) .
Find y(2) (t), y(3) (t), etc. explicitly until you think you can recognize a pattern and guess an explicit
formula for y(n) (t). Then use induction to prove that your formula for y(n) (t) is true. Think of y(n) (t)
as an order-n approximation of y(t).
(c) What is the relationship between y(n) (t) (explicit expression) and the exact solution y(t) (explicit
expression) of the initial value problem?
2. Find the area of the bounded region in the plane between the curves x = |y − 1| and x = y 2 (2 − y) + 1,
0 ≤ y ≤ 3.
3. A cylindrical glass tumbler of radius r and height h is filled with water and then tilted.
(a) If the glass is tilted until the water remaining in the glass exactly covers its base, find the volume of
the water in the glass:
i. from purely geometric considerations;
ii. by “slicing” the water into parallel cross-sections and then setting up and evaluating a definite
integral for the volume of water in the glass.
(b) If the glass is tilted until the water remaining in the glass exactly covers half the base, find the volume
of the water in the glass by integration, as in part (a) ii.