What is Calculus?
• Study of continuous changes
• High-school Math (geometry and algebra)
with one additional concept:
Limit
Intervals
Intervals
Certain sets of real numbers, called intervals, occur frequently in calculus
and correspond geometrically to line segments. For example, if a < b, the
open interval from a to b consists of all numbers between a and b and is
denoted by the symbol (a, b). Using set-builder notation, we can write
.
Notice that the endpoints of the interval - namely, a and b - are excluded.
This is indicated by the round brackets ( ) and by the open dots in Figure.
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Intervals
Definition 1
The closed interval from a to b is the set
.
Here the endpoints of the interval are included. This is indicated by the
square brackets [ ] and by the solid dots in Figure.
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Intervals
We also need to consider infinite intervals such as
This does not mean that ∞ (“infinity”) is a number. The notation (a, ∞)
stands for the set of all numbers that are greater than a, so the symbol
∞ simply indicates that the interval extends indefinitely far in the positive
direction.
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Inequalities
Inequalities
Rules for Inequalities
a<
a<
a<
a<
b then a + c < b + c,
b and c < d then a + c < b + d,
b and c > 0 then ac < bc,
b and c < 0 then ac > bc,
4
If
If
If
If
5
If 0 < a < b or a < b < 0 then
1
2
3
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Inequalities
Example 2
Solve the inequality x 2 − 5x + 6 ≤ 0.
Solution.
First we factor the left side:
(x − 2)(x − 3) ≤ 0
We know that the corresponding equation ( x − 2)(x − 3) = 0 has the
solutions 2 and 3. The numbers 2 and 3 divide the real line into three
intervals:
(−∞, 2)
(2, 3)
(3, ∞)
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Inequalities
Solution (cont.)
On each of these intervals we determine the signs of the factors.
Then we read from the chart that ( x − 2)(x − 3) is negative when
2 < x < 3. Thus, the solution of the inequality ( x − 2)(x − 3) ≤ 0 is
Notice that we have included the endpoints 2 and 3 because we are looking for values
of such x that the product is either negative or zero.
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Inequalities
Example 3
Solve x 3 + 3x 2 > 4x.
Solution.
First we take all nonzero terms to one side of the inequality sign and factor
the resulting expression:
x 3 + 3x 2 − 4x > 0
x (x − 1)(x + 4) > 0
As in previous example we solve the corresponding equation
x 3 + 3x 2 − 4x = 0 and use the solutions x = 0, x = −4 and x = 1 to
divide the real line into four intervals (−∞, −4), (−4, 0), (0, 1) and (1, ∞).
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Inequalities
Solution (cont.)
On each of these intervals we determine the signs of the factors.
Then we read from the chart that the solution set is
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Four Ways to Represent a Function
Functions and Their Graphs
Functions arise whenever one quantity depends on another.
Example:
The area A of a circle depends on the radius r of the circle. The rule
that connects r and A is given by the equation A = πr 2 . With each
positive number r there is associated one value of A, and we say that
A is a function of r.
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Function
Function
The set D is called the domain of the function
The number f (x ) is called the value of f at x .
The range of f is the set of all possible values of f as x varies throughout
the domain.
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Function
The set D is called the domain of the function
The number f (x ) is called the value of f at x .
The range of f is the set of all possible values of f as x varies throughout
the domain.
A symbol that represents an arbitrary number in the domain of a function
f is called an independent variable.
A symbol that represents a number in the range of is called a
dependent variable.
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Function
The most common method for visualizing a function is its graph. If f is a
function with domain A, then its graph is the set of ordered pairs
y
{𝑥,ƒ }
In other words, the graph of f consists of all points (x, y) in the coordinate plane such that y = f (x )
and x is in the domain of f .
ƒ
f (2)
f (1)
0
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2
𝑥
𝑥
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Function
The graph of f also allows us to picture the domain of f on the x−axis
and its range on the y−axis as in Figure.
y
range
y = ƒ(x)
𝑥
0
domain
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Function
Example 5
y
1
0
𝑥
1
The graph of a function f is shown in Figure.
a) Find the values of f (1) and f (5).
b) What are the domain and range of f ?
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Function
Solution.
a) We see from Figure that the point (1, 3) lies on the graph of f , so the
value of f at 1 is f (1) = 3. (In other words, the point on the graph
that lies above x = 1 is 3 units above the x-axis.)
When x = 5, the graph lies about 0.7 unit below the x−axis, so we
estimate that f (5) ≈ −0.7.
b) We see that f (x) is defined when 0 ≤ x ≤ 7, so the domain of f is
the closed interval [0, 7]. Notice that f takes on all values from −2 to
4, so the range of f is
{ y| − 2 ≤ y ≤ 4} = [−2, 4].
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Representations of Functions
Function
Representations of Functions
Representations of Functions
• verbally (by a description in words)
• numerically (by a table of values)
• visually (by a graph)
• algebraically (by an explicit formula)
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Vertical Line Test
Function
Vertical Line Test
The graph of a function is a curve in the xy−plane. But the question
arises: Which curves in the xy−plane are graphs of functions? This is
answered by the following test.
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Function
Vertical Line Test
A curve in the xy−plane is the graph of a function of x if and only if no
vertical line intersects the curve more than once.
If each vertical line x = a intersects a curve only once, at (a, b), then
exactly one functional value is defined by f (a) = b. But if a line x = a
intersects the curve twice, at (a, b) and (a, c), then the curve can’t
represent a function because a function can’t assign two different values to
a.
y
y
𝑥=a
(a, c)
𝑥=a
(a, b)
(a, b)
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Piecewise Defined Function
Function
Piecewise Defined Function
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Function
Piecewise Defined Function
Solution.
If x ≤ 1 then the value of f (x) is 1 − x. On the other hand, if x > 1, then
the value of f (x) is x 2 .
y
1
1
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Symmetric Function
Function
Symmetric Function
Definition 10
If a function f satisfies f (−x) = f (x) for every number x in its domain,
then is f called an even function.
For instance, the function f (x) = x 2 is even because
f (−x ) = (−x ) 2 = x 2 = f (x ).
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Function
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Symmetric Function
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Function
Symmetric Function
Definition 11
If f satisfies f (−x) = −f (x) for every number x in its domain, then f is
called an odd function.
For example, the function f (x ) = x 3 is odd because
f (−x ) = (−x ) 3 = −x 3 = −f (x ).
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Function
Symmetric Function
Example 12
Determine whether each of the following functions is even, odd or neither
even nor odd.
(a) f (x ) = x 5 + x
(b) g(x ) = 1 − x 4
(c) h(x ) = 2x − x 2
Solution.
5
5 5
(a) f (−x) = (−x) + (−x) = (−1) x + (−x)
5
5
= −x − x = −(x + x )
= −f (x )
Therefore, f is an odd function.
(b) g(−x) = 1 − (−x) 4 = 1 − x 4 = g(x). So g is even.
(c) h(−x) = 2(−x) − (−x) 2 = −2x − x 2 . Since h(−x) / = h(x) and
h(−x) / = −h(x), we conclude that h is neither even nor odd.
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Function
Symmetric Function
Solution (cont.)
y
1
y
1
f
y
g
h
1
1
_1
1
𝑥
𝑥
1
𝑥
_1
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Linear Models
Mathematical Models
Linear Function
When we say that y is a linear function of x, we mean that the graph of
the function is a line.
Therefore we can use the slope-intercept form of the equation of a line to
write a formula for the function as
y = f (x ) = m x + b
where m is the slope of the line and b is the y−intercept.
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Mathematical Models
Linear Models
A characteristic feature of linear functions is that they grow at a
constant rate.
For instance, Figure shows a graph of the linear function f (x) = 3x − 2
and a table of sample values. Notice that whenever x increases by 0.1, the
value of increases by 0.3. So f (x) increases three times as fast as x.
Thus, the slope of the graph y = 3x − 2, namely 3, can be interpreted as
the rate of change of y with respect to x.
y
y=3𝑥-2
0
𝑥
_2
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x
f (x) = 3x — 2
1.0
1.1
1.2
1.3
1.4
1.5
1.0
1.3
1.6
1.9
2.2
2.5
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Polynomials
Mathematical Models
Polynomials
Definition 14
A function P is called a polynomial if
P (x) = a n x n + a n − 1 x n − 1 + . . . + a 2 x 2 + a 1 x + a0
where n is a nonnegative integer and the numbers a0 , a1 , a2 , . . . , a n are
constants, which are called the coefficients of the polynomial.
The domain of any polynomial is R = (−∞, ∞).
If the leading coefficient a n =/ 0, then the degree of the polynomial is n.
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Mathematical Models
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Polynomials
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Mathematical Models
Polynomials
The graph of P is always a parabola obtained by shifting the parabola
y = ax 2 , as we will see in the next section. The parabola opens upward if
a > 0 and downward if a < 0.
y
y
2
2
0
1
𝑥
Figure 2 : y = x 2 + x + 1
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𝑥
y = −2x2 + 3x + 1
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Mathematical Models
Polynomials
A polynomial of degree 3 is of the form
ax 3 + bx2 + cx + d
and is called a cubic function.
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Power Functions
Mathematical Models
Power Functions
Definition 15
A function of the form
f (x ) = x a
where a is a constant, is called a power function.
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Rational Functions
Mathematical Models
Rational Functions
Definition 16
A rational function f is a ratio of two polynomials:
R (x ) =
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Q (x )
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Mathematical Models
Rational Functions
The function
f (x ) =
2x 4 − x 2 + 1
x2 − 4
is a rational function with domain { x |x =/ ±2} .
y
20
0
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𝑥
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Algebraic Functions
Mathematical Models
Algebraic Functions
Definition 17
A function is called an algebraic function if it can be constructed using
algebraic operations (such as addition, subtraction, multiplication, division,
and taking roots) starting with polynomials.
Any rational function is automatically an algebraic function.
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New Functions from Old Functions
New Functions from Old Functions
In this section we start with the basic functions we discussed in previous
section and obtain new functions by shifting, stretching, and reflecting
their graphs. We also show how to combine pairs of functions by the
standard arithmetic operations and by composition.
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New Functions from Old Functions
Transformations of Functions
Transformations of Functions
By applying certain transformations to the graph of a given function we
can obtain the graphs of certain related functions.
This will give us the ability to sketch the graphs of many functions quickly
by hand.
It will also enable us to write equations for given graphs.
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New Functions from Old Functions
Transformations of Functions
Let’s first consider translations.
If c is a positive number, then the graph of y = f (x) + c is just the graph
of y = f (x) shifted upward a distance of c units (because each
y−coordinate is increased by the same number c).
Likewise, if g(x) = f ( x − c), where c > 0, then the value of g at x is the
same as the value of f at x − c (c units to the left of x). Therefore, the
graph of y = f (x − c) is just the graph of y = f (x) shifted c units to the
right.
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New Functions from Old Functions
Transformations of Functions
Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph of
• y = f (x) + c, shift the graph y = f (x) a distance c units upward.
• y = f (x) − c, shift the graph y = f (x) a distance c units downward.
• y = f ( x − c), shift the graph y = f (x) a distance c units to the right.
• y = f ( x + c), shift the graph y = f (x) a distance c units to the left.
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New Functions from Old Functions
Transformations of Functions
Suppose c > 0.
y
y = ƒ + c
y=f(𝑥+c)
c
c
0
y=f(𝑥-c)
y= ƒ
c
𝑥
c
y = ƒ - c
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New Functions from Old Functions
Transformations of Functions
Now let’s consider the stretching and reflecting transformations.
If c > 1, then the graph of y = cf (x) is the graph of y = f (x) stretched
by a factor of c in the vertical direction (because each y-coordinate is
multiplied by the same number c).
The graph of y = −f (x) is the graph of y = f (x) reflected about the
x-axis because the point (x, y) is replaced by the point (x, −y).
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New Functions from Old Functions
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Transformations of Functions
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New Functions from Old Functions
Transformations of Functions
Example 19
Sketch the graph of the function f (x ) = x 2 + 6x + 10.
Solution.
Completing the square, we write
the equation of the graph as
y
y = x 2 + 6x + 10 = ( x + 3)2 + 1
1
-3
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x
This means we obtain the desired graph by starting with the
parabola y = x 2 and shifting 3
units to the left and then 1 unit
upward.
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New Functions from Old Functions
Transformations of Functions
Example 19
Sketch the graph of the function f (x ) = x 2 + 6x + 10.
Solution.
Completing the square, we write
the equation of the graph as
y
y = x 2 + 6x + 10 = ( x + 3)2 + 1
1
-3
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x
This means we obtain the desired graph by starting with the
parabola y = x 2 and shifting 3
units to the left and then 1 unit
upward.
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New Functions from Old Functions
Transformations of Functions
Example 20
Sketch the graph of the function y = |x 2 − 1|.
Solution.
y
1
-1
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1
x
We first graph the parabola y =
x 2 −1 by shifting the parabola y =
x 2 downward 1 unit. We see that
the graph lies below the x−axis
when −1 < x < 1, so we reflect
that part of the graph about the
x−axis to obtain the graph of y =
|x 2 − 1|.
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New Functions from Old Functions
Transformations of Functions
Example 20
Sketch the graph of the function y = |x 2 − 1|.
Solution.
y
1
-1
MAT 1001
0
1
x
We first graph the parabola y =
x 2 −1 by shifting the parabola y =
x 2 downward 1 unit. We see that
the graph lies below the x−axis
when −1 < x < 1, so we reflect
that part of the graph about the
x−axis to obtain the graph of y =
|x 2 − 1|.
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New Functions from Old Functions
Transformations of Functions
Example 20
Sketch the graph of the function y = |x 2 − 1|.
Solution.
y
1
-1
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0
1
x
We first graph the parabola y =
x 2 −1 by shifting the parabola y =
x 2 downward 1 unit. We see that
the graph lies below the x−axis
when −1 < x < 1, so we reflect
that part of the graph about the
x−axis to obtain the graph of y =
|x 2 − 1|.
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Algebra of Functions
New Functions from Old Functions
Algebra of Functions
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Trigonometry
sin x
cos x