2
CHAPTER 0. A SHORT MATHEMATICAL REVIEW
The exponential function exp () = and natural logarithm ln are inverse
functions satisfying
ln = , ln = .
The usual rules of exponents apply:
= + ,
/ = ,
( ) = .
The corresponding rules for the logarithmic functi
4
CHAPTER 0. A SHORT MATHEMATICAL REVIEW
Riemann Sum definition is extended to all values of and and for all values
of () (positive and negative). Accordingly,
() =
( () =
and
().
()
Also, if < < , then
() =
() +
(),
which states (when () > 0) tha
Chapter 0
A short mathematical
review
A basic understanding of calculus is required to undertake a study of differential
equations. This zero chapter presents a short review.
0.1
The trigonometric functions
The Pythagorean trigonometric identity is
sin2 +
Contents
0 A short mathematical review
0.1 The trigonometric functions . . . . . . . . . . . . . .
0.2 The exponential function and the natural logarithm
0.3 Definition of the derivative . . . . . . . . . . . . . .
0.4 Differentiating a combination of fun
The Hong Kong University of Science and Technology
Department of Mathematics
Clear Water Bay, Kowloon
Hong Kong
c
Copyright 20092016 by Jeffrey Robert Chasnov
This work is licensed under the Creative Commons Attribution 3.0 Hong Kong License.
To view a co
MATH 109 SYLLABUS
Instructor: Maxim Arap
Course: MATH 109
E-mail: [email protected]
Oce: 311 Krieger Hall
Course description
Sections of the book: 7.1-7.4, 9.1-9.5, 10.1-10.4, 7.8, 11.1-11.11
Textbook
Single Variable Calculus: Early Transcendentals, 7th
0.5. DIFFERENTIATING ELEMENTARY FUNCTIONS
0.4.4
3
The chain rule
The derivative of the composition of () and () is
( (
)
(
)
()
= () (),
and should be memorized as the derivative of the outside times the derivative
of the inside.
0.5
0.5.1
Differentiatin