Math 332
Final Exam
1. Root Finders
a) State the Newton’s method recursion relation needed
to find the cube root of 12.
c) Test the fixed point for stability. Show your work
below and indicate the result of your test.
2. Numerical ODE Solvers
a) The ODEIVP dy/dx = xX where y(1) = 1 is not solvable
except by numerical means. Show by hand three
iterations of the Euler Method, including
x
y
the initial value.
Name
b) Use your calculator to graph the recursion relation you
found in part a) along with y = x. Sketch the graphs
below, and find and label the fixed point in your diagram.
d) What does the result of your stability test guarantee
about a seed value Xo sufficiently close to the fixed
point? Illustrate with a diagram below.
b) Use the Euler program on your calculator to solve the
ODEIVP dy/dx = xX where y(1) = 1, for 20 steps with step
size = .1. Sketch a graph and state the y-value of the 20th
point.
Math 332
Final Exam
3. Numerical Integrators
a) On your calculator graph the function f(x) = xX over the
interval (1,3) and sketch the graph below. Estimate the
area under the curve on the interval (1,3) using the
trapezoid rule with two steps. Show the calculations and
how they relate to trapezoids on your sketch below.
4. The dynamics of the logistic equation f(x) = rx(1-x)
a) Sketch a graph of f(x) and the second generation
function f(f(x)) below on the interval (0,1) for r = 3.2.
Indicate on your graph the coordinates of the parent
fixed point and the coordinates of any of its children.
c) At what growth rate will the logistic function begin to
produce “grandchildren”?
Name
b) Use your calculator’s integration tool to find the area
under the curve f(x) = xX over the interval (1,3).
Specifically, what accounts for the error of the trapezoid
estimate, compared to that of your calculator? Be clear
here!
b) Test the parent fixed point and children for stability,
stating how you reached your conclusions.
d) Sketch a graph of composition of f (not f itself) along
with y = x, based on your choice of r, which produces
grandchildren of the original fixed point. Label the
coordinates of the fixed point, children, and
grandchildren, and state the value of r.
Math 332
Final Exam
Name
5. Discrete Linear Model: Use your calculator to find the solution to the linear traffic flow problem below.
a. Write the Matrix equation 𝑀𝑥⃗ = 𝑏⃗⃗ here.
b. Write the vector solution to the matrix equation here.
c. Describe the physical significance of the Null Space in
the your solution.
6. Given either a skinny or fat fractal, determine its area measure.
Suppose for cube with dimensions 1 unit x 1 unit x 1 unit we remove a corner with volume ½ unit x ½ unit x ½ unit,
and continue in a self-similar fashion to proportionally remove corners of what’s left. Calculate the volume remaining
after removing all smaller sub-cubes in this manner. Show the calculations.
Math 332
Final Exam
7. Number Bases and Digital Arithmetic
a) Convert the base 2 number below to a base 10
number. Show reasoning.
Name
b) Given eight bit byte arithmetic, what is the twoscomplement inverse of the base two number
101011? Show reasoning.
1011.100100100…
c) Add the two base 8 numbers, showing your reasoning.
4 5 6 7 7 7
+ 7 7 7 6 4
d) Subtract the two base 16 numbers showing your
reasoning.
1 C D C 0
- F C 8 3
8. Given that you live in a state that values computer literacy and the ability to program, but still values the Common
Core standards, choose one way you could infuse some of what you learned in this class into a course you will teach in
high school. (This is done in many countries, but sadly, not enough in the US.) Write an essay and turn it in with your
final exam on test day.