Calculus With Tech ICalculus With Tech IInstructor:Instructor:Dr. Chekad SaramiDr. Chekad Sarami

What is calculus? What is calculus? Calculus is a branch of mathematics Calculus is a branch of mathematics

that deals with rates of change. that deals with rates of change. Modern Calculus began with Newton Modern Calculus began with Newton and Leibnitz in the 17th century. and Leibnitz in the 17th century.

Today it is used extensively in many Today it is used extensively in many areas of science. Basic ideas of areas of science. Basic ideas of calculus include the idea of calculus include the idea of limit limit , , derivative derivative , and , and integral integral . .

ExamplesExamples The derivative of a function is its The derivative of a function is its

instantaneous rate of change, with instantaneous rate of change, with respect to something else. Thus, respect to something else. Thus,

the derivative of the derivative of height ,height , (with (with respect to position) is respect to position) is slope slope ; ;

the derivative of the derivative of positionposition , (with , (with respect to time) is respect to time) is velocity velocity ; and ; and

the derivative of the derivative of velocity velocity (with (with respect to time) is respect to time) is accelerationacceleration. .

Why is calculus Why is calculus Extremely important? Extremely important? In the sciences, many processes involving In the sciences, many processes involving

change, or related variables, are studied.change, or related variables, are studied. If these variables are linked in a way that If these variables are linked in a way that

involves chance, and significant random involves chance, and significant random variation, statistics is one of the main tools variation, statistics is one of the main tools used to study the connections. used to study the connections.

In cases where a deterministic model is at In cases where a deterministic model is at least a good approximation, calculus is a least a good approximation, calculus is a powerful tool to study the ways in which the powerful tool to study the ways in which the variables interact. variables interact.

Situations involving rates of change over Situations involving rates of change over time, or rates of change from place to place, time, or rates of change from place to place, are particularly important examples.are particularly important examples.

ApplicationsApplications Physics, astronomy, mathematics, and Physics, astronomy, mathematics, and

engineering make particularly heavy use of engineering make particularly heavy use of calculus; calculus;

it is difficult to see how any of those disciplines it is difficult to see how any of those disciplines could exist in anything like its modern form could exist in anything like its modern form without calculus. without calculus.

Biology, chemistry, economics, computing Biology, chemistry, economics, computing science, and other sciences use calculus too. science, and other sciences use calculus too.

Many faculties of science therefore require a Many faculties of science therefore require a calculus course from all their students; in other calculus course from all their students; in other cases you may be able to choose between, say, cases you may be able to choose between, say, calculus, statistics, and computer programming. calculus, statistics, and computer programming.

ResourcesResources http://www.stewartcalculus.com/mediahttp://www.stewartcalculus.com/media/3_home.php/3_home.php http://archives.math.utk.edu/visual.calhttp://archives.math.utk.edu/visual.calculus/index.htmlculus/index.html http://web.rollins.edu/~dchild/calcAssishttp://web.rollins.edu/~dchild/calcAssist/t/

http://www.brookscole.com/cgi-http://www.brookscole.com/cgi-wadsworth/course_products_wp.pl?wadsworth/course_products_wp.pl?fid=M20bI&product_isbn_issn=053440fid=M20bI&product_isbn_issn=0534409865&discipline_number=19865&discipline_number=1

Section 1.1Section 1.1

Functions Functions and and

ModelsModels

Let X and Y be two nonempty sets of real numbers. A function from X into Y is a relation that associates with each element of X a unique element of Y.

The set X is called the domain of the function.

For each element x in X, the corresponding element y in Y is called the image of x. The set of all images of the elements of the domain is called the range of the function.

DOMAIN RANGE

X Y

f

x

x

x

y

y

Example: Which of the following relations are function?

{(1, 1), (2, 4), (3, 9), (-3, 9)}

{(1, 1), (1, -1), (2, 4), (4, 9)}

A Function

Not A Function

Functions are often denoted by letters such as f, F, g, G, and others. The symbol f(x), read “f of x” or “f at x”, is the number that results when x is given and the function f is applied.Elements of the domain, x, can be though of as input and the result obtained when the function is applied can be though of as output.

Restrictions on this input/output machine:

1. It only accepts numbers from the domain of the function.

2. For each input, there is exactly one output (which may be repeated for different inputs).

For a function y = f(x), the variable x is called the independent variable, because it can be assigned any of the permissible numbers from the domain.

The variable y is called the dependent variable, because its value depends on x.

The independent variable is also called the argument of the function.

Example: Given the function f x x( ) 2 52

Find: )3(f

f ( )3 2 3 5 232( )

f (x) is the number that results when the number x is applied to the rule for f.

Find: )( hxf

5)(2)( 2 hxhxf 2 2 52 2( )x xh h 2 4 2 52 2x xh h

The domain of a function f is the set of real numbers such that the rule of the function makes sense.

Domain can also be thought of as the set of all possible input for the function machine.Example: Find the domain of the following function:

g x x x( ) 3 5 13

Domain: All real numbers

Example: Find the domain of the following function:

s tt

( ) 4

1Domain of s is tt|1.

Example: Find the domain of the following function:

h z z( ) 2z 2 0z 2

Domain of h is zz|2.

Example: Express the area of a circle as a function of its radius.

2)( rrA The dependent variable is A and the independent variable is r.

The domain of the function is }0|{ rr

Four ways to represent Four ways to represent a Functiona Function verbal verbal numerical numerical visual visual algebraic algebraic http://archives.math.utk.edu/visual.http://archives.math.utk.edu/visual.calculus/0/functions.11/index.htmlcalculus/0/functions.11/index.html

The graph of f(x) is given below.

4

0

-4(0, -3)

(2, 3)

(4, 0) (10, 0)

(1, 0) x

y

What is the domain and range of f ?Domain: [0,10]

Range: [-3,3]

Find f(0), f(4), and f(12)

f(0) = -3 f(4) = 0

f(12) does not exist since 12 isn’t in the domain of f

A function f is even if for every number x in its domain the number -x is also in the domain and f(x) = f(-x).

A function is even if and only if its graph is symmetric with respect to the y-axis.

A function f is odd if for every number x in its domain the number -x is also in the domain and -f(x) = f(-x).

A function is odd if and only if its graph is symmetric with respect to the origin.

Example of an Even Function. It is symmetric about the y-axis x

y

(0,0)

x

y

(0,0)

Example of an Odd Function. It is symmetric about the origin

Determine whether each of the following functions is even, odd, or neither. Then determine whether the graph is symmetric with respect to the y-axis or with respect to the origin.

2)( 2 zzga.)

22)()( 22 zzzgg z g z( ) ( )

Even function, graph symmetric with respect to the y-axis.

xxxf 34)( 5 b.)

f x x x x x( ) ( ) ( ) 4 3 4 35 5

f x f x( ) ( )

Not an even function.

f x x x x x( ) ( )4 3 4 35 5

f x x x x x( ) ( ) ( ) 4 3 4 35 5

f x f x( ) ( )

Odd function, and the graph is symmetric with respect to the origin.

A function f is increasing on an open interval I if, for any choice of x1 and x2 in I, with x1 < x2, we have f(x1) < f(x2).

A function f is decreasing on an open interval I if, for any choice of x1 and x2 in I, with x1 < x2, we have f(x1) > f(x2).

A function f is constant on an open interval I if, for any choice of x in I, the values of f(x) are equal.

Determine where the following graph is increasing, decreasing and constant.

4

0

-4(0, -3)

(2, 3)

(4, 0)

(10, -3)

(1, 0) x

y

(7, -3)

Increasing on (0,2)

Decreasing on (2,7)

Constant on (7,10)

Section 1.2Section 1.2

Mathematical Mathematical Models:Models:

A Catalog ofA Catalog ofEssentialEssential

FunctionsFunctions

The following library of functions will be used throughout the text. Be able to recognize the shape of each graph and associate that shape with the given function.

The Constant Functioncxf )( x

y (0,c)

The Identity Function

xxf )(x

y

(0,0)

The Square Function

x

y

(0,0)

The Cube Function

2)( xxf

3)( xxf x

y

(0,0)

The Square Root Function

x

y

(0,0)xxf )(

x

y

(1,1)

(-1,-1)

The Reciprocal Function

xxf 1)(

(0,0) x

yThe Absolute Value Function

xxf )(

The Cube Root Function

x

y

3)( xxf

When functions are defined by more than one equation, they are called piecewise- defined functions.

Example: The function f is defined as:

1 31 3

12- 3)(

xxx

xxxf

a.) Find f (1) = 3Find f (-1) = (-1) + 3 = 2Find f (4) = - (4) + 3 = -1

1 31 3

12- 3)(

xxx

xxxf

b.) Determine the domain of fDomain: in interval notation),2[

or in set builder notation}2|{ xx

c.) Graph f

x

y

1 2 3

321

d.) Find the range of f from the graph found in part c.

Range: in interval notation)4,(

or in set builder notation}4|{ yy

x

y

1 2 3

321

Power FunctionsPower Functions

A polynomial function is a function of the form

f x a x a x a x ann

nn( )

11

1 0

where an , an-1 ,…, a1 , a0 are real numbers and n is a nonnegative integer. The domain consists of all real numbers. The degree of the polynomial is the largest power of x that appears.

Example: Determine which of the following are polynomials. For those that are, state the degree.

(a) f x x x( ) 3 4 52

Polynomial of degree 2

(b) h x x( ) 3 5

Not a polynomial

(c) F x xx

( ) 3

5 2

5

Not a polynomial