Numerical Integration Techniques
Today we will compare the dierent numerical integration techniques you have learned so far. There
are ve basic methods of approximating an integral:
1. Left-hand sums
b
f (x) dx
a
ba
[f (x0 ) + f (x1 ) + . . . + f (xn1 )]
Denite Integrals and Sums
Recall that the denition of the denite integral is:
n
b
f (x) dx = lim
x 0
a
f (ci )x
i=1
Where
ba
n
Lets keep things (relatively) simple by choosing the right-hand endpoints of each interval to nd the height
of the rectangle. So
Indenite Integrals and Particular Solutions
This lab will examine nding indenite integrals with Maple, as well as particular solutions.
Begin by trying some basic integrals. Type
Int(sin(x),x);
Maple outputs the symbol
sin(x) dx,
the indenite integral of
Curve Sketching
In this lab we will analyze the behavior of the function
f (x) =
x4 + 2x2 4x + 4 x2
Begin by dening the function:
f:=x->sqrt(x^4+2*x^2-4*x+4)-x^2;
To make things simple, this function has been chosen so that its domain is all of the real n
Derivatives and the Chain Rule in Maple
Dene the functions f (x) = cos 3x and g (x) =
x2 + 3 in Maple:
f:=x->cos(3*x);
g:=x->sqrt(x^2+3);
There is a function that is useful for evaluating derivatives at a point, it is the operator D.
Plug in
D(f);
D(g);
A
Implicit Dierentiation
Todays lab is a preview of plotting and dierentiating implicit functions. We begin with simple
example, the circle:
x2 + y 2 = 4.
In order to plot this function without solving for it, we can use the Maple command implicitplot.
Howe
Secant Lines and Tangent Lines
This lab will walk you through visualizing some secant lines and tangent lines to curves. The ultimate
goal is to nd and plot the derivative of the tangent line to the function
f (x) = x3 x
at the point x = 1/2.
Now, I want
Piecewise Functions and Limits
Piecewise dened functions.
The Maple command piecewise denes a function which is dierent over dierent intervals. The syntax
is piecewise(range, function, range, function, ., function dened elsewhere). In other words, the
sta
Some basic Maple commands
This lab will introduce you to some of the basic commands of Maple and show you how to start using
it to manipulate functions.
Dening functions.
Lets start things o simply, looking at the function f (x) = x2 . We can dene this in
MTH 2401 - Calculus I
Your Name
Final Exam
Your Signature
Problem
Total Points
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
Total
Fall 2010
80
Score
This exam is closed book.
Calculators are not allowed.
In order to receive credit, you must show your work.
MTH 2401 - Calculus I
Your Name
Final Exam
Your Signature
Problem
Total Points
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
Total
Fall 2009
80
Score
This exam is closed book.
Calculators are not allowed.
In order to receive credit, you must show your work.
MTH 2401 - Calculus I
Your Name
Final Exam
Your Signature
Problem
Total Points
1
13
2
13
3
12
4
12
5
12
6
12
7
13
8
13
Total
Fall 2008
100
Score
This exam is closed book.
Calculators are not allowed.
In order to receive credit, you must show your work.
MTH 2401 - Calculus I
Your Name
Exam 1
Your Signature
Problem
Total Points
1
10
2
10
3
10
4
10
5
10
Total
Fall 2010
50
Score
This exam is closed book. You may use a single handwritten 35 note card for reference.
Calculators are not allowed.
In order to
MTH 2401 - Calculus I
Your Name
Exam 1
Your Signature
Problem
Total Points
1
6
2
8
3
5
4
5
5
10
6
8
7
8
Total
Fall 2009
50
Score
This exam is closed book.
Calculators are not allowed.
In order to receive credit, you must show your work. You must also j
MTH 2401 - Calculus I
Your Name
Exam 1
Your Signature
Problem
Total Points
1
6
2
4
3
8
4
6
5
8
6
10
7
8
Total
Fall 2008
50
Score
This exam is closed book.
Calculators are not allowed.
In order to receive credit, you must show your work. You must also j