Math 442 Assignment 1 - Spring 2015
Due Friday, January 30th
1. Use MATLAB to evaluate the following expressions correct to four decimal places.
Print out a diary session of your MATLAB commands.
π (a) cos
− 1
3
√
2
(b) e − 1
(c) e3 ln(5)
√
2 π
+ tan−1 ( 3)
(d) sin
4
(e) sin2 (20◦ ) + cos2 (75◦ )
2. Use MATLAB to solve the following problems. Print out a diary session of your
MATLAB commands.
(a) Find the magnitude of the vector ~v = [−1, 2, −3, 5].
(b) Find the magnitude of the vector w
~ = [1 + i, 3i, 2, 1 + 2i].
(c) Consider the matrices


1 2 3
A = 2 3 1
3 2 1
and


2 4 6
B = 6 2 4 .
1 5 7
i. Compute AB and BA.
ii. Define a 3 × 3 matrix, C, whose (m, n) entry is the product of the (m, n)
entries of A and B.
(d) Consider the following system of equations:
x + 2y + 3z = 12
−4x + y + 2z = 13
2x + 7y − 3z = 2.
i. What is the determinant of the coefficient matrix?
ii. Use matrices to find a solution of this set of equations.
3. Use the solve command in MATLAB to solve the following equations. (Hint: You may
need to use the eval or double commands to simplify the MATLAB output). Print out
a diary session of your MATLAB commands.
(a) x3 + 2x2 + 5x − 1 = 0
(b)
x3 − 3x2 + 2x = 1
4x − y + 5z = 2
3x + y − 3z = 4.
1
4. Use MATLAB to evaluate the following expressions. Print out a diary session of your
MATLAB commands.
√
x − 3x − 2
(a) lim
x→2
x2 − 4
!
2
d
ex − 1
√
(b)
dx
x2 + 1
d2
(x2 + y 2 + tan xy)
dy 2
Z
(d)
x tan xdx
(c)
Z
(e)
5
e3x cos xdx
0
5. Write a single MATLAB script M-file to generate the following plots. Use the plot
command rather than ezplot or fplot. Print out your script and a separate plot for each
part. (Hint: Use the figure command before each plot command to generate the figures
in different windows.)
(a) Plot the function f (x) = x + sin(x) for x ∈ [0, 10]. Label the x and y axes and
add a title to the plot.
(b) Plot the functions g(x) = e−x and h(x) = x2 on a single figure for 0 ≤ x ≤ 1.
The graph of g(x) should be a solid blue line and the graph of h(x) should be a
dashed red line. Label the x and y axes and include a legend in your plot.
(c) Consider the parametric curve in R2 defined by x(t) = e−t sin t, y(t) = e−t cos t
for 0 ≤ t ≤ 10. Generate a single figure with two subplots. In the first subplot,
graph x(t) and y(t) as functions of t. In the second plot, graph the curve in the
xy-plane. Label the x and y axes in each subplot and include a legend in the first
subplot.
6. (Radioactive Decay) Let y(t) denote the mass of a radioactive substance at time
t ≥ 0. The Law of Radioactive Decay states that the substance decays at a rate
proportional to its mass. That is,
dy
= −ky,
dt
where k > 0 is the decay rate for the substance. The solution of this differential
equation is y(t) = y(0)e−kt .
(a) Write a function M-file which takes as input the decay rate k, the initial mass
y(0), a final time T > 0, and returns the following:
i. The half-life of the substance.
ii. A plot of the mass y(t) = y(0)e−kt for t ∈ [0, T ]. Be sure to label your axes!
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(b) Suppose that a substance has an initial mass of 500 mg and decay rate k = 0.0001
day−1 . Use your function to determine the half-life of this substance (in days)
and plot the mass over a period of 10 days.
Include your M-file, a diary session of its implementation, and the figure your function
generates.
7. (Bilinear Forms) Consider the bilinear form
BA (~v , w)
~ = ~v T Aw,
~
where A is any n × n matrix and ~v and w
~ are n × 1 column vectors.
(a) Write a MATLAB function M-file that takes as input an n × n matrix A, two
n × 1 vectors ~v and w,
~ and returns the value BA (~v , w).
~
(b) Use your function to compute BA (~v , w)
~ for


 
1 2 3 4
−1
5 6 7 8
−1


A=
~v = 
 9 10 11 12
1
13 14 15 16
0
 
0
1

w
~ =
0 .
1
Include both the M-file and a diary session of its implementation.
3