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
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
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
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
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
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
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 f
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
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, l
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.
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.
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
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 fo
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 r
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 r