(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 t
(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 Function
(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 =
(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 vertic
(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
(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 t
(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 hi
(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
(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
(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
(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 endpo
(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 =
(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 th
(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 wa
(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 sec
(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 s
(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 evaluat
(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.
(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
( ( ) +
(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
(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
radi
(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 give
(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 p
(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 o
(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 giv
(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
.
(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 p
(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 .
(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 p
(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