(Section 1.5: Piecewise-Defined Functions; Limits and Continuity in Calculus) 1.5.9
A Right-Hand, One-Sided Limit
()
Evaluate lim+ f x , which is read:
x
2
()
the limit of f x as x approaches 2 from the right.
()
We want the real number that f x approach
(Section 1.4: Transformations) 1.4.14
(
)(
)
2
2
Answer: x + 2 + y 1 = 9 .
We will use this technique in Section 2.2 and Chapter 10 on conic sections.
Equivalence of Translation Methods for Functions
()
Consider the graph of y = f x . A coordinate shift
(Section 0.11: Solving Equations) 0.11.11
PART F: SOLVING ABSOLUTE VALUE EQUATIONS
Solving Absolute Value Equations
If d > 0 , then x = d
x = d .
If d = 0 , then x = d
x =0
x = 0.
If d < 0 , then x = d has no solutions.
For example, x = 3
x = 3 , while x
(Section 1.4: Transformations) 1.4.13
PART F: TRANSLATIONS THROUGH COORDINATE SHIFTS
Translations through Coordinate Shifts
A graph G in the xy-plane is shifted h units horizontally
and k units vertically.
If h < 0 , then G is shifted left by h units.
I
(Section 0.11: Solving Equations) 0.11.10
We must check our tentative solutions.
x = 5 checks out:
(5) + 4 = (5) + (5)
2
21
TIP 3: If we trust that we have not made any
mechanical errors, we only need to check that
radicands of even roots are nonnegative.
(Section 0.11: Solving Equations) 0.11.9
PART E: SOLVING RADICAL EQUATIONS
We often solve a radical equation by isolating radicals on one or both sides of the
equation and then raising both sides to the appropriate positive integer power.
WARNING 12: If w
(Section 0.11: Solving Equations) 0.11.8
PART D: SOLVING RATIONAL EQUATIONS
We often solve a rational equation by first multiplying both sides by the LCD.
WARNING 10: Indicate restrictions that are hidden by this step.
Example 6 (Solving a Rational Equati
(Section 1.4: Transformations) 1.4.12
Strategy 3 (Switches the order in Strategy 2, but this fails!)
Basic graph: y = x
()
Begin with: f1 x = x
Effect: Shifts graph down by 1 unit
Transformation: f 2 x = f1 x 1
Effect: Reflects graph about x-axis
Transfor
(Section 0.11: Solving Equations) 0.11.7
Completing the Square (CTS) Method
This method creates a perfect square trinomial (PST), which can be
factored as the square of a binomial. That square is then isolated, and the
Square Root Method is applied.
This
(Section 1.4: Transformations) 1.4.11
There are different strategies that can lead to a correct equation.
Strategy 1 (Raise, then reflect)
Effect: Shifts graph up by 1 unit
Transformation: f 2 x = f1 x + 1
Effect: Reflects graph about x-axis
Transformatio
(Section 1.4: Transformations) 1.4.10
WARNING 4: We are expected to carefully trace the movements of any
key points on the developing graphs. Here, we want to at least trace the
movements of the endpoint. We may want to identify intercepts, as well.
Why i
(Section 0.11: Solving Equations) 0.11.6
Square Root Method
If d > 0 , then x 2 = d
x= d.
If d = 0 , then x 2 = d
x2 = 0
x = 0.
If d < 0 , then x 2 = d has no real solutions.
For example, x 2 = 3
x = 3 , while x 2 = 3 has no real solutions.
This method
(Section 1.4: Transformations) 1.4.9
PART E: SEQUENCES OF TRANSFORMATIONS
Example 9 (Graphing a Transformed Function)
Graph y = 2
x+3.
Solution
We may want to rewrite the equation as y =
indicate the vertical shift.
x + 3 + 2 to more clearly
We will bu
(Section 0.11: Solving Equations) 0.11.5
Observe that the radicand b2
of ax 2 + bx + c .
4 ac in the Quadratic Formula is the discriminant
In Example 2, we saw that the discriminant of 2 x 2 7 x 15 was 169, a perfect
square, which allowed us to eliminate
(Section 1.11: Limits and Derivatives in Calculus) 1.11.16
FOOTNOTES
1. Etymology. The word secant comes from the Latin secare (to cut). The word tangent
comes from the Latin tangere (to touch). A secant line to the graph of f must intersect it in
at leas
(Section 0.12: Solving Inequalities) 0.12.1
SECTION 0.12: SOLVING INEQUALITIES
LEARNING OBJECTIVES
Know how to solve linear inequalities and absolute value inequalities.
PART A: DISCUSSION
We will solve inequalities when we perform sign analyses and fin
(Section 1.5: Piecewise-Defined Functions; Limits and Continuity in Calculus) 1.5.1
SECTION 1.5: PIECEWISE-DEFINED FUNCTIONS;
LIMITS AND CONTINUITY IN CALCULUS
LEARNING OBJECTIVES
Know how to evaluate and graph piecewise-defined functions.
Know how to e
(Section 0.12: Solving Inequalities) 0.12.2
Example 1 (Solving a Linear Inequality)
Solve
3x > x + 8 .
Solution Method 1
3x > x + 8
Now subtract x from both sides.
4x > 8
Now divide both sides by
4.
We must then reverse the direction of the inequality
si
(Section 0.13: The Cartesian Plane and Circles) 0.13.5
Example 2 (Finding the Equation of a Circle)
Find an equation of the circle in the xy-plane with center
(
)
2, 1 and
radius 3.
Solution
( ( ) + ( y 1) = (3) , which we usually
We obtain the equation
(Section 1.5: Piecewise-Defined Functions; Limits and Continuity in Calculus) 1.5.8
PART F: LIMITS IN CALCULUS
Example 7 (Limits and the Greatest Integer or Floor Function)
()
Let f x = x or
x.
A Left-Hand, One-Sided Limit
()
Evaluate lim f x , which is r
(Section 0.13: The Cartesian Plane and Circles) 0.13.4
Example 1 (The Graph of an Equation; A Circle Centered at the Origin)
()
The graph of x 2 + y 2 = 9 is the circle below with center 0, 0 and
radius 3.
( )(
The ordered pairs 3, 0 ,
)( )
(
)
3, 0 , 0,
(Section 1.5: Piecewise-Defined Functions; Limits and Continuity in Calculus) 1.5.7
We may think of these as piecewise constant functions.
For example, the greatest integer (or floor) function is given by:
x=
2 x< 1
1 x<0
0,
0
1,
2,
()
f x = x or
2,
1,
1
(Section 0.13: The Cartesian Plane and Circles) 0.13.3
PART D: THE GRAPH OF AN EQUATION and CIRCLES
The Graph of an Equation; the Basic Principle of Graphing
The graph of an equation consists of all points whose coordinates satisfy
the equation. The point
(Section 1.5: Piecewise-Defined Functions; Limits and Continuity in Calculus) 1.5.6
PART E: THE GREATEST INTEGER (OR FLOOR) FUNCTION
()
The greatest integer (or floor) function is defined by f x = x or x ,
the greatest integer that is not greater than x.
(Section 0.13: The Cartesian Plane and Circles) 0.13.2
PART C: DISTANCE AND MIDPOINT FORMULAS
Distance Formula
(
)
(
)
The distance between points P x1 , y1 and Q x2 , y2 in the Cartesian
plane is given by:
d=
( x2
x1 ) + ( y2
2
y1 )
2
or, equivalently,
(
(Section 1.5: Piecewise-Defined Functions; Limits and Continuity in Calculus) 1.5.5
Example 3 (Graphing a Piecewise-Defined Function with a Removable
Discontinuity)
()
Graph f , if f x =
x + 3, x 3
.
7,
x=3
Solution
()
We graph the line y = x + 3 , exce
(Section 0.13: The Cartesian Plane and Circles) 0.13.1
SECTION 0.13: THE CARTESIAN PLANE and CIRCLES
LEARNING OBJECTIVES
Understand the Cartesian plane and associated terminology.
Know how to plot points and graph equations in the Cartesian plane.
Know
(Section 1.5: Piecewise-Defined Functions; Limits and Continuity in Calculus) 1.5.4
)
2 is included in the subdomain
(
2, 4 is included in this piece, and we plot it as a filled-in circle.
)
()
2, 1 . Therefore, the endpoint
()
2
1 = 1 , so the right end
(Section 0.12: Solving Inequalities) 0.12.4
Example 2 (Solving an Absolute Value Inequality;
Related to Section 0.11, Example 8)
Solve x 1 < 2 .
Solution
x 1 <2
2 < x 1< 2
We can add 1 to all three parts of this
compound inequality.
1< x < 3
The solution
(Section 1.5: Piecewise-Defined Functions; Limits and Continuity in Calculus) 1.5.3
PART C: EVALUATING PIECEWISE-DEFINED FUNCTIONS
Example 1 (Evaluating a Piecewise-Defined Function)
Let the function f be defined by:
()
x2 ,
fx=
x + 1,
To evaluate f
2 x