Thursday, Sept 26, 2013
STUDENT NAME:
EXAM 1 – MATH 1330, FALL 2013
READ EACH PROBLEM CAREFULLY! SHOW ALL WORK! NO WORK, NO CREDIT!
Make sure you turn in ALL pages. No graphing calculators allowed!
• Problem 1 [14 pts] Consider the data in the following table describing the number of wasps that
can develop inside caterpillars of different weights.
Weight of Caterpillar
0.5
1.0
1.5
2.0
2.5
Number of Wasps
80
115
150
180
220
(a) Graph the data. Which point does not lie on the same line?
(b) Find the equation of the line connecting the first two data points.
(c) How many wasps does the function predict would develop a caterpillar weighting 0.7 g?
• Problem 2 [12 pts]
Consider the discrete-time dynamical system with updating function
vt+1 = 0.6vt
where vt is the population of viruses in a body at time t, measured in thousands.
(i) Find the (explicit) solution vt to the discrete-time dynamical system, given the initial condition
v0 = 10 (thousands). Express your solution in terms of an exponential function in base e.
(ii) Find the half-life of the virus population.
(iii) At what time will the virus population reach 3 thousands?
• Problem 3 [14 pts] A population of bacteria doubles every hour, but 10 thousand bacteria are removed before reproduction each hour to be converted into biological by-products. Initially, assume
there were b0 = 11 thousand bacteria.
(i) Write the discrete time dynamical system that describes the growth.
bt+1 =
(ii) Find the population after 1, 2, 3 hours and then cobweb the dynamical system.
(iii) If the initial population was b0 = 8 thousand bacteria, redo part (ii) and explain the behavior of
the solution.
(iv) [Extra credit] Write down the explicit solution of the DTDS. Explain how your formula matches
the graph of bt .
bt =
• Problem 4 [10 pts]
(i) Graph the following function, indicating the average, maximum, minimum, amplitude, period and
phase, and mark them on the graph.
hπ
i
g(t) = −1 + 3 cos (t + 1)
2
(ii) The graph below represents the volume of an organism changing over time. Identify the average,
max, min, amplitude, period and phase and write down an expression for the volume as a function
of time.
V (t) =