c alculus i enea sacco. 2 w elcome to c alculus i 3 topics/contents before calculus functions. new...

CALCULUS IEnea Sacco

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Welcome to Calculus I!

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WELCOME TO CALCULUS I

Topics/Contents

Before CalculusFunctions. 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 derivativeTangent Lines and Rate of Change. The Introduction to the Techniques of DifferentiationThe Product and the Quotient Rule. Derivatives of Trigonometric Functions. The Chain

RuleThe derivative in graphing and applications

Increasing, Decreasing and Concave Functions. Relative Extrema. Graphing Polynomials.Absolute Maxima and Minima. Graphing function. Applied Maximum and Minimum

ProblemsIntegration

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

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BOOK

CALCULUS EARLY TRANSCENDENTALS 9th edition by HOWARD ANTON, IRL BIVENS, STEPHEN DAVIS.

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EVALUATION

Assiduity and attendance 10%Homework assignments (1 every 2 weeks) 30%Midterm 30%Final 30%Total 100%

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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.

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COMMON WAYS OF REPRESENTING FUNCTIONS

Numerically by tables Geometrically by graphs Algebraically by formulas Verbally

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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 functionis a rule that associates a unique output with each input. If the input is denoted by , then the output is denoted by (read “of ”).

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INDEPENDENT AND DEPENDENT VARIABLES

Sometimes its useful to denote the output by a single letter, say , and write

You can have other names like ... or even if you want.

Independent variable (or argument)

Dependent variable

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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 :

So, if the age is 10 years, the height is:

age

0 0

1 20

3.2 64

15 300

... ...

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EXAMPLE OF A FUNCTION (2)

The equation

is in the form where

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GRAPHS OF FUNCTIONS

A very useful way of representing functions is through graphs.

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THE VERTICAL LINE TEST

A curve in the -plane is the graph of some function if and only if no vertical line intersects the curve more than once.

So which one of these is a function?

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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

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WHAT IS THE GRAPH OF ?

𝑓 (𝑥 )={ 𝑥 , 𝑥≥00 ,𝑥=0− 𝑥 ,𝑥<0

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PROPERTIES OF ABSOLUTE VALUES

If a and b are real numbers, then:

,

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PIECEWISE FUNCTIONS

The function 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).

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PIECEWISE FUNCTIONS

𝑓 (𝑥 )={ 𝑥2 , 𝑥<26 ,𝑥=2

10−𝑥 , 𝑥>210−𝑥 , 𝑥≤6

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EQUATION FOR A CIRCLE

The equation for a circle can be re-written as a piecewise function.

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DOMAIN AND RANGE OF A FUNCTION

For any function , The domainis the set of all the values that

can have. The range is the set of all possible values

of .

For example, if , Domain: Range:

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DOMAIN AND RANGE OF A FUNCTION

The domain and range of a function can be easily pictured by projecting the graph of onto the coordinate axes.

For example, what is the domain and range of ?

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NEW FUNCTIONS FROM OLD

Arithmetic operations that can be performed on a function:

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For and , find all the combinations

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COMPOSITION OF FUNCTIONS

It is possible to composite functions. If and are functions then the composite function can be described by the following equation:

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For and , find and

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INVERSE FUNCTIONS

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INVERSE FUNCTIONS

Let’s look at an example. The function can be represented as a diagram,

The inverse of this function just goes the other way,

So the inverse is

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For , draw it and its inverse

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INVERSE TRIGONOMETRIC FUNCTIONS

Inverse trigonometric functions are only valid in the following domains:

Otherwise has infinitely many solutions.

Inverse trigonometric functions should be represented using or , avoid using .

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EXPONENTS

Exponentiation is a mathematical operation, written as , involving two numbers, the base and the exponent (or power) .

When is a negative integer and is not zero, is naturally defined as

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EXPONENTS WITH FRACTIONS

We can represent roots by

When the root is negative we have that

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OPERATIONS WITH EXPONENTS

Assuming that (otherwise we have imaginary numbers), the following statements are true,

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THE EXPONENTIAL FUNCTION

In general, an exponential function is one of the form , where the base is "" and the exponent is "".

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THE EXPONENTIAL FUNCTION

However, nowadays the term exponential function is almost exclusively used as a shortcut for the natural exponential function , where is Euler's number, calculated from the infinite series

is one of the numbers in Euler’s identity

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THE EXPONENTIAL FUNCTION

is the only base for which the slope of the tangent line to the curve at any point on the curve is equal to the -coordinate at .

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LOGARITHMIC FUNCTIONS

Here is the definition of the logarithm function: if is any real number such that , , and then

and are inverse functions. In other words

and

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ALGEBRAIC PROPERTIES OF LOGARITHMS

If , , , and are all real numbers,