Numerical Methods for Calculus and Differential Equations
MA 3457

Fall 2009
MA 3457/CS 4033
Homework 9
December 4, 2009
1. (10 points) Recall from Homework 8 that the Bessel function of order zero is dened
by
1
cos(x sin ) d.
J0 (x) =
0
.
Use MATLABs quad to approximate J0 (1) = .765197686557967. First, apply quad
with the defaul
Numerical Methods for Calculus and Differential Equations
MA 3457

Fall 2009
MA 3457/CS 4033
Homework 0
October 27, 2009
This simple computing exercise provides an opportunity to get acquainted with
MATLAB and also introduces an important concept. Dont hand it in.
In a oating point number system, there is an important number calle
Numerical Methods for Calculus and Differential Equations
MA 3457

Fall 2009
MA 3457/CS 4033
Homework 2
November 3, 2009
1. (10 points) For the data
0
2
x
y
1
3
2
6
3
11
construct a divideddierence table
x0
x1
x2
x3
c00
c10
c20
c30
c01
c11
c21
c02
c12 c03
Use your results to write the interpolating polynomial in the Newton form
p
Numerical Methods for Calculus and Differential Equations
MA 3457

Fall 2009
MA 3457/CS 4033
Homework 5
November 13, 2009
1. (10 points) Given knots a = x0 < x1 < . . . < xn = b, we have the following
general denition: S (x) is a spline of degree k if
i. S (x) is (k 1)times continuously dierentiable on [a, b],
ii. for i = 0, . .
Numerical Methods for Calculus and Differential Equations
MA 3457

Fall 2009
MA 3457/CS 4033
Homework 1
October 29, 2009
1. (5 points) Given y0 , y1 , y2 , y3 and distinct x0 , x1 , x2 , x3 , we know that there exists
a unique polynomial p(x) of degree less than or equal to three such that p(xi ) = yi
for i = 0, 1, 2, 3. What are
Numerical Methods for Calculus and Differential Equations
MA 3457

Fall 2009
MA 3457/CS 4033
Homework 4
November 12, 2009
1. (10 points) Suppose P (x) is a piecewiselinear interpolating function determined
by a function f (x) at n + 1 equally spaced interpolating points a = x0 < x1 < . . . <
xn = b. In class, we showed how to use
Numerical Methods for Calculus and Differential Equations
MA 3457

Fall 2009
MA 3457/CS 4033
Homework 6
November 23, 2009
1. (10 points) Assuming f is twice continuously dierentiable near a point x, we
have the Taylor series expansion
1
f (x + h) = f (x) + f (x)h + f (x)h2 + O (h3 ),
2!
which yields the forwarddierence derivative
Numerical Methods for Calculus and Differential Equations
MA 3457

Fall 2009
MA 3457/CS 4033
Homework 8
1. (10 points) Consider approximating I =
and Simpson rules.
1
0
December 3, 2009
sin x2 dx with the composite trapezoid
a. For each rule, obtain an upper bound on the absolute error when 101 equally
spaced points are used.
b. F
Numerical Methods for Calculus and Differential Equations
MA 3457

Fall 2009
MA 3457/CS 4033
Homework 12
December 11, 2009
1. (10 points) For our ODE initialvalue problem x = f (t, x), x(t0 ) = x0 , the k th
step of the rstorder Euler method is given by
xk+1 = xk + hf (tk , xk ),
and the k th step of the secondorder midpoint me
Numerical Methods for Calculus and Differential Equations
MA 3457

Fall 2009
MA 3457/CS 4033
Homework 11
December 10, 2009
1. (10 points) Using Matlab or the language/environment of your choice, apply the
rstorder forward Euler method to the initialvalue problem
x = x + et cos t,
x(0) = 0
over the interval [0, ]. (The exact solu
Numerical Methods for Calculus and Differential Equations
MA 3457

Fall 2009
MA 3457/CS 4033
Homework 7
November 24, 2009
1. (10 points) Assuming f is suciently dierentiable, the Taylor series
f (x + h) = f (x) + f (x)h +
f (x) 2 f (x) 3 f (4) (x) 4
h+
h+
h + .
2
6
24
gives
f (x) =
f (x + h) f (x) f (x)
f (x) 2 f (4) (x) 3
h
h
h .
Numerical Methods for Calculus and Differential Equations
MA 3457

Fall 2009
MA 3457/CS 4033
Homework 10
December 8, 2009
1. (10 points) The ODE x = ax b(x )3 kx was derived by Lord Rayleigh to model
the motion of the reed in a clarinet. Suppose the initial data are x(0) = x (0) = 1.
Set up an equivalent initialvalue problem for
Numerical Methods for Calculus and Differential Equations
MA 3457

Fall 2009
MA 3457/CS 4033
Homework 3
November 9, 2009
1. (10 points) Find the Hermite cubic polynomial p(x) such that
p(0) = 1,
p (0) = 0,
p(1) = 0,
p (1) = 1
()
in two dierent ways, as follows:
(i) Construct the divided dierence table from the conditions () and de