Chapter 1
The fundamental theorem of
Calculus
In Dierential Calculus, we have learned about derivatives of functions, and their uses. Integral Calculus,
as we shall see, uses the reverse concept, i.e.
72
CHAPTER 2. TOOLS FOR INTEGRATION
2.4
Integration by parts
Integration by parts is the next most used technique for integration after substitution. Just as the substitution method was a consequence
92
CHAPTER 2. TOOLS FOR INTEGRATION
2.7
2.7.1
Improper integrals
Case Study: Revisiting probability distributions
In many of the Case Studies we have looked at so far, we encountered probability distr
87
2.6
2.6.1
Integrals of functions with radicals, and the length of curves.
Case Study: The Golden Gate Bridge
The Golden Gate Bridge is a remarkable engineering feat and one of the most beautiful br
Chapter 3
Advanced tools for the solution of
ODEs
3.1
3.1.1
Numerical methods
Case Study: The SIR epidemic model
One of the main applications of Calculus in biology and health sciences is in epidemiol
AMS 15B, Midterm 1
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AMS 15B, Midterm 2
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AMS 15B, Midterm 2
Name:
Electronic items are not allowed until 15 minutes before the end of the test.
Read all the questions before you start working on any of them. Start with the ones you are most
AMS'15B, Midterm 1 5%
Name: W o A M-
Electronic items are not allowed until 20 minutes before the end of the test.
Read all the questions before you start working on any of them. Start with the
80
2.5
CHAPTER 2. TOOLS FOR INTEGRATION
Integrals of trigonometric functions
It is very common in various problems in physics and engineering to come across integrals of trigonometric
functions. In th
64
2.3
CHAPTER 2. TOOLS FOR INTEGRATION
Integrating rational functions
In the Logistic Equation Case Study, we ended up having to integrate a rational function. Many problems,
in fact, lead to integra
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Homework 6: Improper integrals and more ODE problems
In ALL questions, you MUST justify your answers step by step. Missing steps will not count for full
credit.
1 Integrals of o
13
In this Case Study, we will continue to look at Antiderivatives as solutions of dierential equations.
In this case, however, we will see a dierent class of dierential equations which are called aut
33
Example: What is
a
0
x2 dx?
Note that, in practice, integrals are RARELY calculated using Riemann sums. (just as derivatives
were rarely ever calculated using the formal denition of derivatives). M
23
1.2
Integrals
In this Section, we change tack completely and look at an apparently unrelated problem, which deals with
probabilities. We will learn about discrete probabilities, sums, and learn abo
37
1.3
The fundamental theorem of Calculus
In the nal section of this Chapter, we will study the relationship between antiderivatives and integrals,
to discover that they are intimately related throug
58
CHAPTER 2. TOOLS FOR INTEGRATION
2.2
Application of the method of substitution: ODEs
In this Section, we will re-visit ordinary dierential equations and re-interpret the method of solution
using in
53
2.1.8
Mathematical corner: The method of substitution (part II)
The substitution method learned above can be generalized to transform variables x into u even when u is
x
not a linear function of x.
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Chapter 2
Tools for integration
In the last Chapter, we learned a number of general concepts related to integration and saw that integrals
and derivatives are inverse operation of one-another. Thanks
AMS 15B, Practice Final
Name:
Electronic items are not allowed until 40 minutes before the end of the test.
Read all the questions before you start working on any of them. Start with the ones you are
AMS 15B, Midterm 1
Name:
Electronic items are not allowed until 20 minutes before the end of the test.
Read all the questions before you start working on any of them. Start with the ones you are most
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Homework 3: The Fundamental Theorem of Calculus, Method of
Substitution
1
Calculus problems [75 points]
In ALL questions, you MUST justify your answers step by step. Missing steps will not count for f
Homework 4: ODEs, Rational functions and integration by parts.
1
Calculus problems [75 points]
In ALL questions, you MUST justify your answers step by step. Missing steps will not count for full
credi
Homework 5: Miscellaneous integration techniques
In ALL questions, you MUST justify your answers step by step. Missing steps will not count for full
credit.
1 Miscellaneous integration techniques
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AMS 1513 Final 2012
Name: __L_.M,,,,,,L_
Electronic items are not allowed until 40 minutes before the end of the test.
Read all the questions before you start working on any of them. Start with the