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Topics for Test #1 (Chap 1 Sec 1.1-1.3 and Chap2 Sec 2.1-2.6)
CHAPTER 1 FUNCTIONS
The opening chapter of the text focuses on functions: their properties, their graphs, and their use
in applications.
It can also be viewed as an overview of the prerequisite knowledge from algebra and
trigonometry that is necessary for success in a calculus course.
Additional review material appears in Appendix A.
Since the material of this chapter is basic skills that you must have, you must spend extra time
reviewing this material. The main reason for failure in calculus usually stems from difficulties in
algebra and trigonometry. Without knowing algebra and trigonometry, you cannot do calculus.
The calculus curriculum starts with Chapter 2.
1.1 Review of Functions
Overview
A function is defined and its properties are developed.
Lecture
Review the definition of a function, its geometric interpretation (the vertical line test), and the
concepts of domain and range (both the domain of definition and the domain in the context of an
application) Do Examples1-3 in the e-book.
Watch the Video Presentation (Functions)
See an animation of domain and range at:
http://mathdemos.gcsu.edu/mathdemos/domainrange/domainrange.html
Watch the video how to find the Domain of a Function at:
http://video.google.com/videoplay?docid=2378867002563806676#docid=58023305625975124
Review composition of functions. Composite functions are featured prominently in calculus, and
do Examples 4–7 in the e-book.
Watch the Video Composition of Functions at http://www.brightstorm.com/math/algebra2/functions/composition-of-functions
Review the notion of symmetry in graphs, and give definitions of even and odd functions, and do
Example 8 in the e-book..
Watch the Video Even and Odd Functions at http://www.youtube.com/watch?v=2Tnvwai5cqc
and at http://www.youtube.com/watch?v=oKKcIK_PgEk&feature=related
Watch the Video Presentation (Even and Odd Functions) in the 1.1 Video Lecture link in the
Interactive e-book.
1.2 Representing Functions
Overview
We introduce the full catalog of functions that will be encountered in calculus, and present four
ways to represent a function: through formulas, graphs, tables, and words.
Lecture
Review the standard functions and provide representative graphs for each family of functions.
Examples 1–3 in the e-book.
Review piecewise functions, which are used repeatedly in the next chapter (limits). Included is
the piecewise definition of the absolute value function, another fact that will be used frequently
in upcoming material. Example 4 in the e-book.
Review rational functions and the Area function Examples 5, 7 in the e-book.
Watch the Video Presentation (Graphs of Basic Functions)
Watch the Video Piecewise Functions at http://www.youtube.com/watch?v=-gwffMEr8i8
You need to review the basic shapes of the standard functions.
Watch the Video Presentation (Graphing Techniques)
Reviewing transformations of graphs is important, as this topic gives students the tools needed to
quickly visualize more complicated functions. Examples 8-9 in the e-book.
Watch the Video Horizontal and Vertical Graphs Transformation at
http://www.youtube.com/watch?v=3Q5Sy034fok
1.3 Trigonometric Functions
Overview
The trigonometric functions are defined, and the graphs and properties of these functions are
examined.
Lecture
 Review the six trigonometric functions, using both the right-triangle definition and by
treating them as circular functions
 You are required to evaluate the trigonometric functions at special angles.
You can either memorize the unit circle, or you can evaluate the trigonometric functions by
generating the table below for angles in the first quadrant, and with the signs of the trig functions
in the other quadrants, you can obtain the other values. Observe the pattern in building the table.

0
30
45
60

0
 /6
 /4
 /3
 /2
sin 
0/2
cos
4/2
1/ 2
2/2
3/2
3/2
2/2
1/ 2
tan 
0
3
1/
1
3
Und.
4/2
0/2
The values are obtained when the fractions are simplified. The tangent is obtained by dividing
sine by cosine, and the values of the other trigonometric functions are obtained by using the
reciprocal relations.
90
To find the trigonometric functions of non-acute angles:
1) Draw the angle in standard position (from the positive x-axis).
2) Find the reference angle (angle with the x-axis)
3) Since the reference angle is acute, use the values of the trigonometric functions in the
table above with the sign of the trigonometric function in that quadrant.
Eg. Evaluate cos(5 / 6) .
Since 5 / 6 is in quadrant II (cosine negative) with reference angle
of  / 6 , cos(5 / 6)   cos( / 6)   3 / 2
Eg. Evaluate tan(5 / 3) .
Since 5 / 3 is in quadrant IV (tangent negative) with reference angle
of  / 3 , tan(5 / 3)   tan( / 3)   3
Eg. Evaluate sin( 3 / 4) .
Since 3 / 4 is in quadrant III (sin negative) with reference angle
of  / 4 , sin(3 / 4)   sin( / 4)   2 / 2
Eg. Evaluate sec(11 / 6) .
Since 11 / 6 is in quadrant IV (secant positive) with reference angle
of  / 6 , sec(11 / 6)  sec( / 6) 
1
1
2


cos( / 6)
3/2
3
Quadrantal Angles
To find quadrantal angles, select the point on the terminal side of the angle with distance one
from the origin. The coordinate of that point on a unit circle will be ( x, y )  (cos  ,sin  )
Eg. Evaluate sin(3 / 2) and cos(3 / 2)
Since the terminal side is in the negative y-axis, the point on the unit circle will be:
(0, 1)   cos3 / 2,sin 3 / 2 , so cos3 / 2  0 and sin3 / 2  1
--- (0,-1)



Review the trigonometric identities. These identities are the most frequently used in
calculus.
Solve trigonometric equations. See Example 2 in the e-book.
Review the graphs of the trigonometric functions.
Video Presentation (Other Trigonometric Functions)
Your Turn (Sections 1.1-1.3) with answers:
𝑓(𝑥+ℎ)−𝑓(𝑥)
1
−1
1. Find the difference quotient
for
the
function
𝑓(𝑥)
=
.
ANS:
ℎ
𝑥+1
(𝑥+ℎ+1)(𝑥+1)
2. Simplify 𝑔(𝑥) =
𝑥−|𝑥|
𝑥−1
0 𝑖𝑓 𝑥 ≥ 0
as a piecewise function. ANS: { 2𝑥
}
𝑖𝑓 𝑥 < 0
𝑥−1
𝑥
3. The function ℎ(𝑥) = 𝑥 3 −𝑥 is an even function, odd function or neither. ANS: f(-x)= f(x), even.
4. Evaluate cot (−7π/3) =
−1
√3
𝜋
𝜋
5. Solve 2𝑠𝑖𝑛2 𝑥 − 1 = 0 for all x. ANS: 4 + 𝑘 2 , 𝑘 = 0, ±1, ±2 …
6. If f (x) = √𝑥 and g(x) = 1/ (x −1), find f ∘ g (x), and give its domain. ANS: D: x > 1
CHAPTER 2 LIMITS
Limits provide the foundation for all the key ideas of calculus (differentiation, integration, and
infinite series, to name a few). This chapter supplies the tools that your students will use to
understand the important concepts of calculus.
2.1 The Idea of Limits
Overview
In this opening section, we introduce the idea of a limit through an investigation of the
relationship between instantaneous velocity and tangent lines. The intent is to provide the
motivation for the entire chapter and to give an intuitive sense of how limits work.
Video Presentation Limits and Graphs Video Link:
Lecture
Limits arise naturally when we define instantaneous velocity and the line tangent to a curve.
Click on the Average Velocity link in your Interactive e-book (see Example 1). To see the
solution of Example 1, click on solution.
We can see that average velocities are just slopes of secant lines on a position curve (Figures
2.1–2.3).
Click on the Instantaneous Velocity link (see Example 2). Notice that average velocity can be
used to approximate instantaneous velocity. We can see that shrinking the time interval leads to
better approximations of instantaneous velocity. We introduce the idea of a limit in moving from
average to instantaneous velocity.
Click on the Slope of Tangent Lines link.
In this paragraph, we introduce the notion of a tangent line as the limit of approaching secant
lines.
2.2 Definitions of Limits
Overview
This section gives a standard treatment of left-hand, right-hand, and two-sided limits, and shows
how to compute them (informally) with graphs and numerical methods.
Lecture
Read the preliminary (informal) definition of a limit.
Computing the value of a limit is most easily carried out with a graph (see Example 1).
Notice that the limit of a function f at the point a (if it exists) does not depend upon the value of
f (a).
The value of a limit at a point can also be investigated by tabulating function values near that
point (see Example 2).
Video Presentation (One Sided Limits)
Example 3 shows the left-and right-hand limits and their relationship to the corresponding twosided limit.
The limits may fail to exist, either because the left- and right-hand limits do not agree (see
Example 4), or because the values of a function do not approach a single number (Example 5).
2.3 Techniques for Computing Limits
Overview
Analytical methods for evaluating limits are presented.
Lecture
Limits of polynomial, rational, and algebraic functions can usually be evaluated by direct
substitution, provided the function is defined at the limit point (exceptions include piecewise
functions; see Example 5).
Limit laws and algebraic manipulation provide tools for evaluating more challenging limits.
Start by studying Theorem 2.3 (Limit laws), and apply them to Example2.
Video Presentation: (Limits Evaluated Algebraically)
As you saw in the video, limits that cannot be evaluated by direct substitution can often be
transformed into limits that yield to direct substitution via factoring and multiplication by the
conjugate.
For additional examples, see Examples 3 though 6
Example 7 uses the Squeeze Theorem to help you find some limits.
Basic Concepts of Limits Review Handout
Your Turn (Sections 2.1-2.3):
1. The position of an object moving along a line is given by the function s(t) = t 2 − 2t.
a) What is the average velocity over the interval [1, 2]? ANS: 1
b) What is the instantaneous velocity at the point t = 1? ANS: 0
2. Find the slope of the tangent line to 𝑓(𝑥) = 𝑥 2 + 1 at 𝑥 = 2. ANS: 4
3. For 𝑔(𝑥) defined below, approximate lim+ 𝑔(𝑥) in the table below. If the limit Does Not
𝑥→0
Exist (DNE), explain why. ANS: 3
x
0.1
0.01
0.001
0.0001
0.00001
g(x)
2.9
2.99
2.999
2.9999
2.9999
4. For ℎ(𝑥) find two (2) of the most appropriate values of slopes of secant lines to make a
conjecture about the slope of the tangent line at x =1. ANS: 3.99, 4.01, Average 4
x
h(x)
0.97
6.8809
0.99
6.9601
1.00
7
1.01
7.0401
1.03
7.1209
5. Evaluate: If the limit DNE explain:
a) 𝑓(1) = 1
b) lim+ 𝑓(𝑥) = 1
c) lim− 𝑓(𝑥) =0
d) lim 𝑓(𝑥) = DNE
e) 𝑓(2) = 2
f) lim− 𝑓(𝑥) = 1
g) lim 𝑓(𝑥) = 1
h) lim 𝑓(𝑥) = 2
i) 𝑓(4) = 0.4
j) lim+ 𝑓(𝑥) = DNE
k) lim− 𝑓(𝑥) = 1
l) lim 𝑓(𝑥) =DNE
𝑥→1
𝑥→1
𝑥→2
𝑥→3
𝑥→4
𝑥→4
6. lim
(1+𝑥)2 − 1
ℎ
ℎ→0
1
−
7. lim 2+ℎℎ
1
2
ℎ→0
√𝑥− 2
𝑥→4 𝑥−4
8. lim
9. lim
|𝑥−1|
𝑥→1 𝑥−1
10. lim
𝑥→4
𝑥→1
𝑥→2
𝑥→4
=2
= -1/4
= 1/4
= DNE
𝑥 2 − 2𝑥−3
𝑥−3
= 5
11. For x real, find lim √𝑥. ANS: DNE
𝑥→0
𝑓(𝑥)
12. True/ False; the limit lim 𝑔(𝑥) DNE if 𝑔(𝑎) = 0. Explain. ANS: false; see #6,7,8 above.
𝑥→a
2.4 Infinite Limits
Overview
Infinite limits are introduced (initially alongside a limit at infinity to help students distinguish
between the two scenarios), and their connection to vertical asymptotes is explained.
Lecture
Start reading ‘An Overview’ to distinguish between Infinite Limits and Limit at Infinity.
It is important that you understand that when an infinite limit does not exist, we use the symbol
∞ (or −∞) as a convenience to indicate that the function attains arbitrarily large values. See the
definition: Infinite Limits.
We say a two-sided limit is ∞ (or −∞) only when the left- and right-hand limits “agree,” despite
the fact that neither the left nor right-hand limit exists (see Example 1).
Remember that the limits may fail to exist, either because the left- and right-hand limits do not
agree (see Example 4 in section 2.2), because the values of a function do not approach a single
number (Example 5 in section 2.2), or because the limit is ∞ (or −∞) .
The graph of f has the vertical asymptote x = a whenever the limit of f (left, right, or two-sided)
is infinite in magnitude at a. See Example 2.
Video Presentation (Infinite Limits and Asymptotes)
For additional examples, see Examples 3 though 5
2.5 Limits at Infinity
Overview
Limits at infinity determine the end behavior of a function, detect the presence of horizontal
asymptotes, and reveal whether a system attains a steady state.
Lecture
When they exist, limits at infinity indicate a horizontal asymptote. See the definition of Limits at
Infinity and Horizontal Asymptote.
Example1 shows you how to compute a simple limit.
Infinite limits at infinity do not exist because the limit is ∞ (or −∞) (see section 2.4), whereas
lim cos(𝑥) because the limit does not give a number (fluctuate between -1 and 1)
𝑥→∞
Review Theorem 2.6, and do Example 2 for some infinite limits at infinity. Example 3 deals
with the end behavior of rational functions.
Video Presentation (Limits at Infinity)
Theorem 2.7 helps you work limits in Example 4.
Infinite limits at infinity do not exist because the limit is ∞ (or −∞) (see section 2.4), whereas
lim sin(𝑥) and lim cos(𝑥) do not exist because the limit does not give a number (oscillate
𝑥→∞
𝑥→∞
between -1 and 1)
2.6 Continuity
Overview
A standard treatment of continuity is offered, with the important Intermediate Value Theorem
given at the end of the section.
Lecture
We are going to avoid defining the phrase “continuous function,” which is usually taken to mean
a function continuous on its domain. Rather, we are careful to claim that a function is continuous
either at a point, or on an interval (occasionally specified only as the domain of the function in
question, which of course could be a collection of intervals). The reason behind this decision:
It is correct to say that f (x) = 1/ x is a continuous function, and yet it has a discontinuity at x = 0.
Avoiding this apparent inconsistency in terminology is easier when you are encountering
continuity for the first time.
Video Presentation (Continuity)
Read the definition of Continuity at a Point. The definition of continuity allows the use of direct
substitution when evaluating lim 𝑓(𝑥), provided f is continuous at a. It is important that you
𝑥→𝑎
read the Continuity Checklist, because for continuity at a point all 3 conditions must hold. You
must memorize this definition.
Do Example 1 to learn how to distinguish the different types of discontinuities.
As you will see, the standard families of functions (polynomial, rational, trigonometric, etc.) are
continuous on their domains. Do Examples 2 and 3.
Theorem 2.10 is used to evaluate limits of composition of functions. See Example 4
Study the definition of Continuity at Endpoints and Continuity on an Interval along with
Example 5. Study Functions Involving Roots with Example 6. Read The Intermediate Value
Theorem with Example 8.
See the Limits and Graphs Summary Video Link
Infinite Limits and Limits at Infinity Review Handout
Your Turn (Sections 2.4-2.6):
1. Find the Following limits. If the limit does not exist (DNE) explain why.
lim ℎ(𝑥) = ∞
lim ℎ(𝑥) = − ∞
𝑥→3−
2. lim
(𝑥−2)(𝑥+1)
(𝑥−1)2
𝑥→1
lim ℎ(𝑥) = 𝐷𝑁𝐸
𝑥→3+
= −∞
3. lim
𝑥→3
(𝑥−1)(𝑥+1)
(𝑥−1)2
𝑥→1
= 𝐷𝑁𝐸
𝑓(𝑥)
4. If lim 𝑓(𝑥) = 1 and lim 𝑔(𝑥) = 0, find lim [𝑔(𝑥)]2 = ∞.
𝑥→1
5.
x→1
lim + tan(𝑥) = −∞
𝑥→−π/2
𝑥+1
7. lim− 𝑥 2 −1 = −∞
𝑥→1
9. lim
𝑥→−∞
𝑥 2 +1
3𝑥 2 −2
= 1/3
𝑥→1
4
6. lim− 𝑥−2 = −∞
𝑥→2
8. lim
𝑥 2 −5𝑥+6
𝑥→−1 𝑥 2 −2𝑥−3
10. lim
6𝑥 3
𝑥→−∞ 𝑥 2 +1
= 𝐷𝑁𝐸
= −∞
lim ℎ(𝑥) = − ∞
𝑥→−2
11. lim
3𝑥
𝑥→∞ 2𝑥 2 +1
=0
𝜋𝑥
12. Find the vertical asymptotes of tan( 3 ) ANS: 3/2+3k
13. Evaluate lim
sin(𝑥)
𝑥→∞
14. Evaluate lim
𝑥
=0
𝑥
𝑥→−∞ 𝑥−√𝑥 2 +1
= 1/2
15. Use limits to find the horizontal asymptote(s) of 𝑓(𝑥) =
16. Use limits to find the vertical asymptote(s) of 𝑔(𝑥) =
17. Find the value of ‘a’ that will make ℎ(𝑥) = {1
2
𝑥 4 +𝑥 2
𝑥 3 −2𝑥
𝑥2
𝑥 3 −4𝑥
. ANS: No HA
. ANS: 𝑥 = ±2
2𝑥
𝑖𝑓 𝑥 < 1
𝑎
𝑖𝑓 𝑥 ≥ 1
, continuous at x = 1.
ANS: 𝑎 = 4
18. Find the largest interval on which the function √1 − 𝑥 2 is continuous. ANS [-1,1]
Test #1 Concepts Practice Quiz Link
Chapter 2 Key Terms and Concepts
Average velocity (Section 2.1)
Instantaneous velocity (Section 2.1)
Slope of the tangent line (Section 2.1)
Informal limit definitions (Section 2.2)
Relationship between one-sided and two-sided limits (Theorem 2.1) (Section 2.2)
Limits of linear functions (Theorem 2.2) (Section 2.3)
Limit laws (Theorem 2.3) (Section 2.3)
Limits of polynomials and rational functions (Theorem 2.4) (Section 2.3)
Squeeze Theorem (Theorem 2.5) (Section 2.3)
Definition of various infinite limits (Section 2.4)
Vertical asymptotes (Section 2.4)
Definition of various limits at infinity (Section 2.5)
End behavior of polynomials (Theorem 2.6) (Section 2.5)
End behavior of rational functions (Theorem 2.7) (Section 2.5)
Continuity at a point (Section 2.6)
Continuity check list (Section 2.6)
Continuity of sums, products, quotients (Theorem 2.8) (Section 2.6)
Common continuous functions (Theorem 2.9) (Section 2.6)
Continuity of composite functions (Theorem 2.10) (Section 2.6)
Left-continuous and right-continuous functions (Section 2.6)
Continuity on an interval (Section 2.6)
Continuity of functions with roots (Theorem 2.11) (Section 2.6)
Intermediate Value Theorem (Theorem 2.13) (Section 2.6)