Transcript File

CALCULUS I
Enea Sacco
2
WELCOME TO CALCULUS I
Topics/Contents
Before Calculus
Functions. New functions from the old. Inverse Functions. Trigonometric Functions. Inverse
Trigonometric Functions. Exponential and Logarithmic Functions
Limits and continuity
Limits, an Intuitive Approach. Computing Limits. Limits more Rigorously. Continuity.
Continuity of Trigonometric, Exponential and Inverse Functions
The derivative
Tangent Lines and Rate of Change. The Introduction to the Techniques of Differentiation
The Product and the Quotient Rule. Derivatives of Trigonometric Functions. The Chain Rule
The derivative in graphing and applications
Increasing, Decreasing and Concave Functions. Relative Extrema. Graphing Polynomials.
Absolute Maxima and Minima. Graphing function. Applied Maximum and Minimum Problems
Integration
The indefinite Integral. Integration by Substitution. Integration by Parts. The Definite
Integral. Applications of definite integral. The Fundamental Theorem of Calculus. Integrating
Trigonometric Functions. Trigonometric Substitutions. Area Between Two Curves
3
BOOK
CALCULUS EARLY TRANSCENDENTALS 9th
edition by HOWARD ANTON, IRL BIVENS,
STEPHEN DAVIS.
4
EVALUATION
Assiduity and attendance
Homework assignments (1 every 2 weeks)
Midterm
Final
Total
10%
30%
30%
30%
100%
5
WHAT IS A FUNCTION?
If a variable y depends on a variable x in such a
way that each value of x determines exactly one
value of y, then we say that y is a function of x.
6
COMMON WAYS OF REPRESENTING
FUNCTIONS
Numerically by tables
 Geometrically by graphs
 Algebraically by formulas
 Verbally

7
DENOTING FUNCTIONS BY LETTERS OF THE
ALPHABET
In the 18th century, a clever chap
by the name of Leonhard Euler
came up with the idea to represent
functions using letters:

A function𝑓is a rule that
associates a unique output with
each input. If the input is
denoted by 𝑥, then the output is
denoted by 𝑓 (𝑥)(read “𝑓of 𝑥”).
8
INDEPENDENT AND DEPENDENT
VARIABLES
Sometimes its useful to denote the output by a
single letter, say 𝑦, and write
Independent variable (or
Dependent variable
argument)
𝑦 = 𝑓(𝑥)
You can have other names like 𝑔 ... or even
𝑚𝑎𝑟𝑚𝑎𝑙𝑎𝑑𝑒 if you want.
9
EXAMPLE OF A FUNCTION
A tree grows 20 cm every year, so the height of the
tree is related to its age using the function ℎ:
ℎ(𝑎𝑔𝑒) = 𝑎𝑔𝑒 × 20
So, if the age is 10 years, the height is:
ℎ(10) = 10 × 20 = 200 𝑐𝑚
age
𝒉(𝒂𝒈𝒆) = 𝒂𝒈𝒆 × 𝟐𝟎
0
0
1
20
3.2
64
15
300
...
...
10
EXAMPLE OF A FUNCTION (2)
The equation
𝑦 = 𝑥 2 + 2𝑥 − 13
is in the form 𝑦 = 𝑓 𝑥 where
𝑓 𝑥 = 𝑥 2 + 2𝑥 − 13
11
GRAPHS OF FUNCTIONS
A very useful way of representing functions is
through graphs.
12
THE VERTICAL LINE TEST
A curve in the 𝑥𝑦-plane is the graph of some
function 𝑓if and only if no vertical line intersects
the curve more than once.
So which one of these is a function?
13
THE ABSOLUTE VALUE FUNCTION
The effect of taking the absolute value of a number
is to strip away the minus sign if the number is
negative and to leave the number unchanged if it is
non-negative.
For example
14
WHAT IS THE GRAPH OF 𝑓 𝑥 = 𝑥 ?
𝑥,
𝑓 𝑥 = 0,
−𝑥,
𝑥≥0
𝑥=0
𝑥<0
15
PROPERTIES OF ABSOLUTE VALUES
If a and b are real numbers, then:


−𝑎 = 𝑎
𝑎𝑏 = 𝑎 𝑏

𝑎
𝑏

𝑎+𝑏 ≤ 𝑎 + 𝑏
=
𝑎
𝑏
,𝑏≠0
16
PIECEWISE FUNCTIONS
The function 𝑓 𝑥 = 𝑥 is an example of a piecewise
function.
A piecewise-defined function (also called a
piecewise function or a hybrid function) is a
function which is defined by multiple sub-functions,
each sub-function applying to a certain interval of
the main function's domain (a sub-domain).
17
PIECEWISE FUNCTIONS
𝑥2,
𝑥<2
6,
𝑥=2
𝑓 𝑥 =
10 − 𝑥,
𝑥>2
10 − 𝑥,
𝑥≤6
18
EQUATION FOR A CIRCLE
The equation for a circle can be re-written as a
piecewise function.
𝑥2 + 𝑦2 = 1
𝑦 = 1 − 𝑥2
→𝑓 𝑥 =
1 − 𝑥2, 𝑦 > 0
− 1 − 𝑥2, 𝑦 < 0
19
DOMAIN AND RANGE OF A FUNCTION
For any function 𝑓 𝑥 ,
 The domainis the set of all the values that 𝑥 can
have.
 The range is the set of all possible values of 𝑓 𝑥 .
For example, if 𝑓 𝑥 = sin(𝑥),
 Domain: 𝑎𝑙𝑙 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠
 Range: −1 ≤ 𝑓 𝑥 ≤ 1
20
DOMAIN AND RANGE OF A FUNCTION
The domain and range of a function 𝑓 can be easily
pictured by projecting the graph of 𝑦 = 𝑓(𝑥) onto
the coordinate axes.
For example, what is
the domain and range of
𝑦 = −𝑥 4 + 4?
21
NEW FUNCTIONS FROM OLD
Arithmetic operations that can be performed on a
function:
 𝑓+𝑔 𝑥 =𝑓 𝑥 +𝑔 𝑥
 𝑓−𝑔 𝑥 =𝑓 𝑥 −𝑔 𝑥
 𝑓𝑔 𝑥 = 𝑓 𝑥 ∙ 𝑔 𝑥

𝑓
𝑔
𝑥 =
𝑓 𝑥
𝑔 𝑥
22
For 𝑓 𝑥 = 1 + 𝑥 − 2 and 𝑔 𝑥 = 𝑥 − 3, find all the
combinations
23
COMPOSITION OF FUNCTIONS
It is possible to composite functions. If 𝑓 𝑥 =
𝑥 2 and 𝑔 𝑥 = 𝑥 + 1are functions then the composite
function can be described by the following equation:
𝑓 𝑔 𝑥
= 𝑔 𝑥
2
= 𝑥+1
2
24
For 𝑓 𝑥 = 𝑥 3 and 𝑔 𝑥 = 𝑥 + 3, find 𝑓 ∘ 𝑔 and 𝑔 ∘ 𝑓
25
INVERSE FUNCTIONS
26
INVERSE FUNCTIONS
Let’s look at an example. The function 𝑓 𝑥 = 2𝑥 +
3 can be represented as a diagram,
The inverse of this function just goes the other way,
So the inverse is 𝑔 𝑦 =
𝑓 −1
𝑦 =
𝑦−3
2
27
For 𝑓 𝑥 = 2𝑥, draw it and its inverse
28
INVERSE TRIGONOMETRIC FUNCTIONS
Inverse trigonometric functions are only valid in
the following domains:
−𝜋
2

sin 𝜃 ,

cos 𝜃 , 0 ≤ 𝜃 ≤ 𝜋

−𝜋
2
tan 𝜃 ,
≤𝜃≤
𝜋
2
≤𝜃≤
Otherwise arcsin
solutions.
1
2
𝜋
2
= 𝑥 has infinitely many
Inverse trigonometric functions should be
represented using arcsin(𝑥) or asin 𝑥 , avoid using
29
EXPONENTS
Exponentiation is a mathematical operation,
written as 𝑏 𝑛 , involving two numbers, the base 𝑏
and the exponent (or power) 𝑛.
𝑏𝑛 = 𝑏 × 𝑏 × 𝑏 × ⋯ 𝑏
𝑛
When 𝑛 is a negative integer and 𝑏 is not zero, 𝑏 𝑛 is
1
naturally defined as 𝑏 −𝑛 = 𝑛
𝑏
30
EXPONENTS WITH FRACTIONS
We can represent roots by
𝑝
𝑏𝑞
=
𝑞
𝑏𝑝
𝑞
=
𝑏
𝑝
When the root is negative we have that
𝑝
−𝑞
𝑏
=
1
𝑝
𝑏𝑞
31
OPERATIONS WITH EXPONENTS
Assuming that 𝑏 > 0 (otherwise we have imaginary
numbers), the following statements are true,

𝑏 𝑝 𝑏 𝑞 = 𝑏 𝑝+𝑞
𝑏𝑝
 𝑞
𝑏

= 𝑏 𝑝−𝑞
𝑏𝑝
𝑞
= 𝑏 𝑝𝑞
32
THE EXPONENTIAL FUNCTION
In general, an exponential function is one of the
form 𝑏 𝑥 , where the base is "𝑏" and the exponent is
"𝑥".
33
THE EXPONENTIAL FUNCTION
However, nowadays the term exponential function
is almost exclusively used as a shortcut for the
natural exponential function 𝑒 𝑥 , where 𝑒 is Euler's
number, calculated from the infinite series
𝑒 is one of the numbers in Euler’s identity
𝑒 𝑖𝜋 + 1 = 0
34
THE EXPONENTIAL FUNCTION
𝑏 = 𝑒 is the only base for which the slope of the
tangent line to the curve 𝑦 = 𝑒 𝑥 at any point 𝑃 on
the curve is equal to the 𝑦-coordinate at 𝑃.
35
LOGARITHMIC FUNCTIONS
Here is the definition of the logarithm function: if 𝑏
is any real number such that 𝑏 > 0, 𝑏 ≠ 1, and 𝑥 >
0 then
𝑓 𝑥 = log 𝑏 𝑥
𝑓 𝑥 = 𝑏 𝑥 and 𝑓 𝑥 = log 𝑏 𝑥 are inverse functions.
In other words
log 𝑏 𝑏 𝑥 = 𝑥 and 𝑏 log𝑏 𝑥 = 𝑥
36
ALGEBRAIC PROPERTIES OF LOGARITHMS
If 𝑎, 𝑏, 𝑟, and 𝑐 are all real numbers,

log 𝑏 𝑎𝑐 = log 𝑏 𝑎 + log 𝑏 𝑐
𝑎
 log 𝑏
= log 𝑏 𝑎 − log 𝑏
𝑐
𝑟
 log 𝑏 (𝑎 ) = 𝑟 log 𝑏 𝑎
1
 log 𝑏
= − log 𝑏 𝑐
𝑐
𝑐
37