Nov 4, 2014
Worksheet 16: Riemann sum approximations (SOLUTIONS)
1. Below is a sketch of the function f (x) = 1 x2 .
f x
x
0.
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.
(a) On top of this sketch, draw in the rectangles that would represent a left endpoint
Rie
October 20, 2014
Worksheet 15: Ebola, area and derivatives
In Mini-Project 3 we will see that, if D(t) denotes the total, or cumulative, number of deaths
(at time t, with t measured in days) since the start of a certain Ebola outbreak, then D(t)
can be mo
October 9, 2014
Worksheet 11: The second derivative (SOLUTIONS)
On the axes below is the graph of a certain function f (x).
y
5
6
4
2
2
4
x
5
10
15
20
25
30
1. By inspection (just eyeball it), mark the points where the graph of f (x) is steepest. Do
not i
Math 408R:UT Worksheet 5: Secant and tangent lines; the derivative 9/16/2014
On the axes below is the graph of the function f (x) = 4x x2 .
y
4
3
2
1
x
1
x
x
4
x
1. Where (for which values of x) is the graph of f (x) increasing? Where is it decreasing?
In
October 14, 2014
Worksheet 12: Exponential growth/decay
1. Carbon 14 is an isotope of carbon that is formed when radiation from the sun strikes
ordinary carbon dioxide in the atmosphere. Thus plants such as trees, which get their
carbon dioxide from the a
September 30, 2014
Worksheet 8: dierentiation rules; the microscope equation
1. (Geometric interpretation of the Microscope Equation.) The graph of a function f (x) appears below (solid), along with the graph of the line tangent to f (x) at
the point (1,
Math 408R:
Worksheet 2: Functions
September 4, 2014
1. This exercise will involve graphing using MATLAB.
(a) One of the things MATLAB can do is graph functions given a formula. To plot the
function f (x) = x2 on the domain [5, 5], type the following into
Math 408R: UT
Mini-Project 3: Fermentation
Due November 3
For all three of these exercises you will want to modify the m-le SIREulers.m
1. Exercise 8(a)(b), page 195:
2. Exercise 9, page 196:
3. Exercise 11, pp. 196197:
1
November 11, 2014Worksheet 17: Denite integrals, various ways
Fall 2014
In this worksheet, we practice integration by evaluating the denite integral
/2
cos2 (x) dx
0
in three dierent ways.
We will need the following two important trig identities:
1
cos2 (
October 16, 2014
Worksheet 13: Population Growth
For this tutorial, you will need the MATLAB m-les Unlimited.m and Logistic.m, available
on our MATLAB webpage. You will need to download and save these m-les before you can
use them.
1. Constant per capita
Math 408R: Calculus, Stochastics, and Modeling
Mini-Project 4: due in class Monday, November 17
Please do a nice, neat, complete job. Explain your reasoning at every step. Write in complete
sentences, draw careful graphs with axes labeled, etc. You can us
Oct 21-23, 2014:
Worksheet 14: Genetic Toggle Switch
SOLUTIONS
(Throughout this tutorial, please ll in any blanks you nd.)
Goal: To synthetically create a bistable genetic toggle switch. Explanation: by genetic
toggle switch, we mean a genetic system that
M408R Worksheet 1: Prediction using SIR models
Tuesday, September 2
Goal: To use the SIR model to predict (approximately) the evolution of a disease
Recall: we saw, in class this week that the spread of a disease can, under appropriate assumptions,
be mod
November 18, 2014
initial value)
Worksheet 19: Integration by Substitution (indenite, denite,
1. Find
x+
1
x
1
1
x2
dx
in two ways:
(a) by a substitution;
x+
1
x
=
u du =
=
1
1
x+
2
x
1
1
dx
x2
u=x+
u2
+C
2
1
x
du
1
=1 2
dx
x
2
+ C.
du =
1
1
dx
x2
(b) by
Nov 13, 2014
Worksheet 18: indenite and denite integrals
SOLUTIONS
1. Find each of the following indenite integrals. Check your answers by dierentiation.
4x
+C
ln(4)
(a)
4x dx =
(b)
(5x + x5 + 55 ) dx =
5x
x6
+
+ 55 x + C
ln(5)
6
1
Nov 13, 2014
(c)
(d)
Wo
October 2, 2014
Worksheet 9: The product rule
GOAL: To explore the product rule in various ways
Todays question: is the derivative of a product equal to the corresponding product of the
derivatives? That is: is it true, in general, that
d
[f (t)g(t)] = f
October 7, 2014
Worksheet 10: Order of operations in Calculus
GOAL: To explore dierentiation rules and formulas through an order of operations perspective
1. State whether each of the given statements is true or false. If the statement is true,
identify i
Math 408R: UT
Fall 2014
Mini-Project 2: SIR using Matlab, Due October 10
In this assignment, you will be asked to run, modify, discuss, and print output from the
SIREulers program you have on Matlab (which may be found on our 408R MATLAB webpage, as SIREu
Mini-Project 1
Assignment: pp 20-22, problems 8-14
Mark Twains Mississippi, due September 12
The Lower Mississippi River meanders over its at valley, forming broad loops called oxbows. In a ood, the river can jump its banks and cut o one of these loops, g
Worksheet 4: SIR continued
ANSWERS IN RED
September 11, 2014
Goal: To explore more ideas about modeling with rate equations and SIR.
1. A town of population 100,000 is hit with a measles epidemic, which evolves according to
the usual SIR equations
S = a S
September 18, 2014
Worksheet 6: Functions and derivatives
SOLUTIONS
Goal: To reinforce relationships among functions, their derivatives, rates of change, and
slopes of tangent lines
1. (a) Sketch the graph of a function f such that (i) f (x) exists everyw
Oct 28, 2014
Project: Monomers, Dimers, Trimers, and Rates of Change
A beaker contains three types of molecules, called monomers, dimers, and trimers. We use
M , D, and T to stand for the quantities of each of the three respective types. Suppose these
qua
Math 408R: UT
Worksheet 3: Functions, continued
September 9, 2014
Here we explore further features of MATLAB. To begin, please open up MATLAB.
Please note: For each exercise below, enter the requested code into a new line in the
Command Window; of course,
Math 408R: UT
Fall 2014
Individual Homework #9: Due November 24
Please READ Section 11.2, and Section 11.4 up to, but not including, the subsection Partial
Fractions on page 725.
Please DO exercises 1 (all parts EXCEPT for (p) and (r), 2ag, 4, 5, 7, 8b, S
Math 408R: UT
Fall 2014
Individual Homework #8: Due November 12, 2014
Please READ section 6.3, skipping the subsection Error Bounds on pp. 385392; also
READ sections 6.4 and 11.1.
Please DO exercises 1, 2, 5bd, 17, Section 6.3 (pp. 394399); exercises 10,
Math 408R: UT
Fall 2014
Individual Homework #7: Due November 5, 2014
Please read Section 6.1, starting with the subsection Electrical Energy on
page 342; also read Section 6.2, skipping the following parts: (a) everything
from the subsection Calculating D
Math 408R: UT
Fall 2014
Individual Homework #6: Due October 29, 2014
Please read Section 4.4.
A. Please do exercises exercises 3, 4, 5, 6, 10, 11, 15, and 16, in Section 4.4 (pp. 257260).
Some hints and notes on these exercises:
In general, where populat
Math 408R: UT
Fall 2014
Individual Homework #5: Due October 22, 2014
Please read Section 4.3 (you can skim the proofs on pp. 233236), and do exercises 3, 5, 8,
9, 10, and 11, Section 4.3 (pp. 238240).
Some hints and notes on the exercises:
In exercise 8(
Math 408R: UT
Fall 2014
Individual Homework #3: Due October 1, 2014
Please read the subsection The circular functions starting at the bottom of page 40; also
read Sections 3.3 and 3.5.
Please do exercises 10 and 11, page 42; as well as exercises 2ab, 10ab