Appendix
A
Answers and Hints
A.1
Chapter I
2 a. The left-side limit: 0, The right-side limit: 1
b. The left-side limit: 1, The right-side limit: -1
4 a. The left-side limit: 1, The right-side limit: 0
b. The left-side limit: 1, The right-side limit: 1
c.
1
Calculus I - Chapter 1
List for Open Response Test
Numbers, equations and/or graphs might be changed in the actual test.
1. For the graph of a function y = f (x) below
(a) nd the function value and the (one-side) limit at/toward x =
4, 3, 2, 1, 0, 1, 2,
1.9. CONTINUITY
1.9
55
Continuity
In Section 1.3 we covered the continuity of basic functions. In this section we will discuss the continuity
of non-basic functions, that is, we will discuss the continuity laws for operations. We also cover the
Intermedia
1.8. LIMIT: FORMAL DEFINITION
1.8
46
Limit: Formal Denition
We recall that lim f (x) = L if and only if f (x) is getting closer to L as the x(= 0) gets closer to a from
xa
both sides. This means that lim f (x) = L if and only if |f (x) L| is getting close
1.7. SPECIAL TRIGONOMETRY LIMIT LAWS
1.7
41
Special Trigonometry Limit Laws
Case 3 in Step B of the Substitution Procedure for Limit in Section 1.6 suggests a special treatment when
0
we have the indeterminate form with trigonometry. For example, there is
1.6. SUBSTITUTION WITH
1.6
34
Substitution with
Because of the limit laws of basic functions toward x = in Subsection: Limits with of Basic
Functions in Section 1.2, we can extend the Substitution for Limit method for the limit of function
toward x = si
1.5. SUBSTITUTION FOR LIMIT
1.5
29
Substitution for Limit
In Section 1.4 we have that the limit of the function at a point is equal to the function value at that
point if the function is continuous at the point. So all we need to get the limit of a contin
1.4. LIMIT LAWS
1.4
21
Limit Laws
Using graphical explanation we introduce limit laws for several function operations. Then we discuss
substitution for limit, the simple way to nd the limit of (non-basic) function.
We will prove these limit laws with the
1.3. CONTINUITY AND LIMIT LAWS OF BASIC FUNCTIONS
1.3
13
Continuity and Limit Laws of Basic Functions
We have checked that the limit, lim f (x), of a function y = f (x) toward a point x = a MAY or MAY
x a
NOT be equal to the function value, f (a), at that
1.2. ONE-SIDE LIMITS WITH
1.2
6
One-Side Limits with
Since you are taking Calculus 1, we believe that you are familiar with the
1
graph of y = . If not, please look at the graph on the right. (But you are
x
1
1
in big trouble!) What is lim
? How about l
Chapter
1
Limits
All of calculus is based on the idea of limit, though limit itself is very important. So
it is very important to develop a solid understanding of limits. We start this chapter
by introducing limit of functions intuitively with graphical a
Calculus I
Preface
Here are a set of practice problems for my Calculus I notes. If you are viewing the pdf version of
this document (as opposed to viewing it on the web) this document contains only the problems
themselves and no solutions are included in
4.4. SKETCHING GRAPHS OF TRIGONOMETRIC FUNCTIONS
4.4
135
Sketching Graphs of Trigonometric Functions
Objectives
- Be able to use the Procedure of sketching the graph of trigonometic function
Because of periodicity of basic trigonometric functions, handlin
4.3. OTHER TRIGONOMETRIC FUNCTIONS AND GRAPHS
4.3
132
Other Trigonometric Functions and Graphs
Objectives
- Be able to apply the properties of Secant, Cosecant and Cotangent functions to sketch the graph
- Be able to use Pythagorean theorem on a (special)
4.2. BASIC TRIGONOMETRIC FUNCTIONS
4.2
123
Basic Trigonometric Functions
Objectives
- Be able to nd the basic trigonometric function values by using a coordinate point
- Be able to use the properties of the basic sine, cosine and tangent functions to sket
2.5. RADICAL FUNCTION
2.5
76
Radical Function
Objectives
- Identify radical functions
- Sketch the graph of radical functions
251 Denition For a positive integer n > 1 the function f (x) = x1/n : X R is called a radical
function. It can be written by f (x
2.4. POWER FUNCTIONS
2.4
72
Power Functions
When a function equation is given by x for the base and a real number for the exponent, such as
y = x(number) , it is called the (basic) power function.
239 Denition The function f (x) = xp : X R is called a pow
2.3. LOGARITHMIC FUNCTION
2.3
70
Logarithmic Function
Objectives
- Identify elogarithmic functions
- Sketch the graph of logarithmic functions
When a function equation comes by the word log with a real number for the base and x for the
insider, such as y
2.2. EXPONENTIAL FUNCTION
2.2
68
Exponential Function
Objectives
- Identify exponential functions
- Sketch the graph of exponential functions
When a function equation is given by a real number for the base and x for the exponent, such as
y = (number base)
2.1. FUNCTION
2.1
60
Function
We often see situations where one variable is somehow linked to the value of another variable, which is
called a relation. That is, a relation is a correspondence between two sets, and a function is a relation
satisfying some
36
CHAPTER 1. LIMITS AND CONTINUITY
1.3
Continuity
Before Calculus became clearly de.ned, continuity meant that one could draw the graph of a function without having to lift the pen and pencil. While this is fairly accurate and explicit, it is not precise