Preliminaries
0.1
A Brief Review of Functions
Exercise Set 0.1. The following questions are review questions from precalculus.
Some terminology is provided. For a more detailed review of this material, please see
Chapter 1 in your textbook.
Definition 0.1 (function). Let A and B be any two sets. A function f from
A to B, written f : A −→ B, is a rule that assigns to each x in A a unique (exactly
one) element y, called f (x), in B.
Comments:
• The set of input values (set A) is called the domain.
• The set of output values is called the range (which is a subset of set B).
• The element f (x) is called the value of the function f at x, and we write y = f (x).
Problem 0.1. Let f (x) = 1 − 2x2 . Evaluate the following:
1. f (3)
2. f (3h)
3. f (3 + h)
f (3 + h) − f (3)
h
Problem 0.2. Find the domain for the following functions. Use interval notation.
4.
1. f (x) = x2 − 4
1
x2 − 4
√
3. h(x) = x − 4
2. g(x) =
Definition 0.2 (absolute value).
1. (geometric definition) The absolute value of a number is the distance a number
is from the origin on the number line. We denote the absolute value of a real
number x by |x|.
Dr. Allen’s MATH 1261 Class Notes and Problem Sets, Fall 2012
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2. (algebraic definition) The absolute value of a real number x, denoted by |x|,
is defined by
x
if x ≥ 0
|x| =
−x if x < 0.

 |x| if −2 ≤ x < 0
−2 if 0 ≤ x < 1
Problem 0.3. Let f (x) =

−x2 if 1 ≤ x < 2.
1. Evaluate the following.
(a)
(b)
(c)
(d)
f (0) =
f (−1) =
f (1) =
f (1261) =
2. Graph f (x).
Definition 0.3 (function composition). Let f and g be two functions. The
composition of the functions f and g, denoted f ◦ g is defined by
(f ◦ g)(x) = f (g(x)).
√
Problem 0.4. Let f (x) = x2 − 2 and let g(x) = x. Find (f ◦ g)(x).
Problem 0.5. Let h(x) = (2x + 5)8 . Express the function h(x) as a composition of
two simpler functions f and g.
Definition 0.4 (inverse function). Two functions f and g are inverses of
each other provided that
1. (f ◦ g)(x) = x for each x in the domain of g and
2. (g ◦ f )(x) = x for each x in the domain of f .
The domain of f (x) becomes the range of g(x), and the range of f (x) becomes the
domain of g(x). We say that g is the inverse of f , and we denote this by g = f −1 .
Some notation: Suppose that f has an inverse function, f −1 . Then f −1 (y) = x if
and only if f (x) = y.
Dr. Allen’s MATH 1261 Class Notes and Problem Sets, Fall 2012
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Exercise Set 0.2.
MATH 1261, Calculus I
Team Members’ Names:
Classwork, Fall 2012
Instructions: Show all work neatly, clearly, and carefully. Remember that one of
the expectations for this class is that students communicate mathematical ideas
with clarity and coherence through both speaking and writing.
Problem 0.6. (Justify why you are a team). List your team’s cards:
Card 1.
Card 2.
1. Call one of the functions f (x), and the other function g(x).
f (x) =
and g(x) =
2. Give the domain and range of each function (use interval notation).
(a) Domain of f (x):
(b) Range of f (x):
(c) Domain of g(x):
(d) Range of g(x):
3. On the same set of axes, graph f (x) and g(x) as well as the line y = x. Observe
that f and g are reflections about the line y = x.
Show that the two functions, f (x) and g(x), are inverses of each other by using the
definition of inverse function. Show all steps.
• (f ◦ g)(x) = f (g(x)) =
Dr. Allen’s MATH 1261 Class Notes and Problem Sets, Fall 2012
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• (g ◦ f )(x) =
Problem 0.7. Do all functions have inverses? If you answer ”no” given an example
of a function with no inverse, and then describe the type of functions that do have
inverses.
Problem 0.8. If the function f (x) = 1261 one-to-one? Justify your answer using
complete sentences.
Problem 0.9. Does {(2, 7), (1, 4)} represent the inverse of {(1, 4), (2, 7)}? Justify
your answer using complete sentences.
1
?
3x
Justify your answer using complete sentences. If g(x) is not the inverse of f (x), then
find f −1 , if possible. Justify.
Problem 0.10. Suppose f (x) = 3x. Is the inverse function given by g(x) =
Problem 0.11. Use f (x) = x + 5 to complete the table for f −1 (x).
x
f (x)
−3
0
3
−1
Dr. Allen’s MATH 1261 Class Notes and Problem Sets, Fall 2012
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0.2
Some Tips on Mathematical Writing
A primary purpose of the journal assignments in this course is to develop your skills in
understanding and communicating calculus concepts. Your journal assignments (and
many of your test and quiz problems) will require responses using complete sentences.
In giving your solutions, you are expected to use precise mathematical language and
to write clear explanations of methods and justifications. Be sure to use the following
guidelines in your mathematical writing.
1. For clear communication, present one idea at a time. Since an idea is expressed by a complete sentence, write in complete sentences. The rules
of grammar (spelling, punctuation, sentence structure, etc) apply to
mathematics!
2. Include a statement of the problem along with your solution. Including the
original problem makes the solution self-contained and easier to follow.
3. In general, be short and eliminate unnecessary words, keeping the following
points in mind:
•
•
•
•
Do not start a sentence with a symbol.
Explain the meaning of every symbol that you introduce.
Try to avoid words such as clearly, obviously, certainly, etc.
Use a variety of the following words that are frequently used: therefore,
thus, hence, consequently, so, it follows that.
• Avoid the use of pronouns such as it, they, or this unless there is a clear
antecedent.
• Strike a balance between words and symbols.
4. In mathematical writing, use the pronoun we (avoid using I, one, etc.). You
may also use let’s.
5. Display important (or lengthy) equations and mathematical expressions. These should be centered and appear on a separate line when possible;
otherwise, you may write them within the text of your proof or explanation as
long as no “wrapping” occurs.
6. Write out numbers as words when they are used as adjectives, when they are
relatively small, when they are easy to describe in words, or when starting a
sentence. Write out numbers numerically when they specify the value of something.
7. Proofread, edit, rewrite, proofread, edit, rewrite, . . . . Try reading your solution
aloud. Does it make sense? Is it clumsy or confusing? A solution is judged by
its clarity, so be sure that it is succinct yet complete.
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Exercise Set 0.3. Refer to the Tips on Mathematical Writing, and determine
what is wrong with each of the following examples. Also provide a revised statement
that follows the guidelines on mathematical writing.
1. x2 − 6x + 8 = 0 has two distinct roots.
2. To solve it, take away 3 from both sides then take it and divide by 4.
3. An even number multiplied by an even number results in an even number.
4. (x + y)2
x2 + 2xy + y 2
5. Problem: If lim+ f (x) = −2, lim− f (x) = 3, and f (0) = −2, what can you say
x→0
x→0
about lim f (x)? Justify.
x→0
a student solution: It does not exist because for a limit to exist it must equal
the same thing as it approaches from both the right and left.
6. Since x is positive.
7. If you expand the expression (a + b)4 , then you obtain (a + b)4 = a4 + 4a3 b +
6a2 b2 + 4ab3 + b4 .
8. Clearly 1 is not a prime number.
9. If you start with any number and subtract one from the square of the number,
then the answer you will get is the same as the answer you will get if you multiply one more than the number by one less than it.
10. There are exactly 2 groups of order 4.
Dr. Allen’s MATH 1261 Class Notes and Problem Sets, Fall 2012
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Dr. Allen’s MATH 1261 Class Notes and Problem Sets, Fall 2012
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Chapter 1
Limits
1.1
An Introduction to Limits
Note: This section corresponds to section 2.2 in your text.
x2 − 1
Example 1.1. Consider the function f (x) =
. Even though the function is
x−1
undefined at 1, what happens to the function values as x gets closer and closer to 1
from both the right and left? Are the functions values approaching a specific value?
1. Complete the following table.
x
f (x) =
x2 − 1
x−1
0
.5
.9
.99
.999
As x gets very close to 1 from the left (denoted x → 1− ), the function values
f (x) are getting close to what number?
x2 − 1
We denote this by writing lim−
=
.
x→1 x − 1
2. Complete the following table.
x
f (x) =
x2 − 1
x−1
2
1.5
1.1
1.01
1.001
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As x gets very close to 1 from the right (denoted x → 1+ ), the function values
f (x) are getting close to what number?
x2 − 1
=
.
We denote this by writing lim+
x→1 x − 1
3. Even though the function f (x) is undefined at 1, what happens to the function
values as x gets closer and closer to 1 from both the right and left? Are the
function values approaching a specific value?
x2 − 1
=
x→1 x − 1
We denote this by writing lim
4. Sketch a graph of f (x) =
.
x2 − 1
.
x−1
5. Some Observations:
Definition 1.1 (limit (informal)). The limit of a function f (x), as x approaches
a, equals L, written
lim f (x) = L,
x→a
means that when x is near but different from a, then f (x) is near L. More precisely,
L is the limit of f as x approaches a if we can make the function values f (x) as close
as we like to the number L by making x sufficiently close (but not equal) to a.
Comment: The function f need not be defined at a.
Dr. Allen’s MATH 1261 Class Notes and Problem Sets, Fall 2012
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Example 1.2.
Below is the graph of a function f .
1. Use the graph to approximate lim f (x).
x→1
2. What happens to the function values as x gets close to −1 from the left? In
other words, find lim − f (x).
x→−1
3. What happens to the function values as x gets close to −1 from the right? In
other words, find lim + f (x).
x→−1
4. What can we conclude about the limit of f (x) as x approaches −1?
Definition 1.2 (right- and left-hand limits). The limit of f (x) as x approaches
a from the right is equal to L, denoted lim+ f (x) = L, means that when x is
x→a
very close to a but x is greater than a (so x is to the right of a), then f (x) is near
L. The limit of f (x) as x approaches a from the left is equal to L, denoted
lim− f (x) = L, means that when x is very close to a but x is less than a (so x is to
x→a
the left of a), then f (x) is near L.
Theorem 1. L is the limit of f (x) as x approaches a if and only if the left-hand and
right-hand limits of f (x) as x approaches a both exist and are equal to L. In limit
notation, lim f (x) = L if and only if lim− f (x) = L and lim+ f (x) = L.
x→a
x→a
x→a
Dr. Allen’s MATH 1261 Class Notes and Problem Sets, Fall 2012
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Exercise Set 1.1.
Problem 1.1. Use the graph of f (x) to evaluate the following.
1. lim f (x)
x→0
2. lim− f (x)
x→3
3. lim+ f (x)
x→3
4. lim f (x) (justify)
x→3
5. f (3)
Problem 1.2. Sketch the graph of an example of a function that satisfies all of the
given conditions: lim f (x) = 3, lim− f (x) = 3, lim+ f (x) = −3, f (1) = 1, f (4) = −1.
x→1
x→4
x→4
sin x
. What is your conclusion? Justify.
x→0 x
Problem 1.3. Investigate lim
1
Problem 1.4. Investigate lim sin . What is your conclusion? Justify. (See problem
x→0
x
21 on page 63).
Problem 1.5. Pages 61-63, Problems 7, 9, 11, 13, 19, 25
Dr. Allen’s MATH 1261 Class Notes and Problem Sets, Fall 2012
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1.2
Limit Theorems
Note: This section corresponds to section 2.3 in your text.
1.2.1
Limit Laws
Theorem 2 (Limit Laws). Let n be a positive integer, c a constant (a real number),
and f and g functions with limits at a; that is, lim f (x) and limx→a g(x) both exist.
x→a
Then
1. lim c =
(constant limit law)
2. lim x =
(identity limit law)
x→a
x→a
3. lim (cf (x)) = c lim f (x) (constant multiple limit law)
x→a
x→a
4. lim [f (x) + g(x)] = lim f (x) + lim g(x) (sum limit law)
x→a
x→a
x→a
5. lim [f (x) − g(x)] = lim f (x) − lim g(x) (difference limit law)
x→a
x→a
x→a
6. lim [f (x)g(x)] = lim f (x) · lim g(x) (product limit law)
x→a
x→a
x→a
f (x)
limx→a f (x)
=
provided
x→a g(x)
limx→a g(x)
law)
7. lim
(quotient limit
8. lim [f (x)]n = [lim f (x)]n (power limit law)
x→a
x→a
q
p
9. lim n f (x) = n lim f (x) provided lim f (x) > 0 when n is even. (root limit law)
x→a
x→a
x→a
Example 1.3.
Suppose lim f (x) = 3, lim g(x) = 0, and lim h(x) = −2. Using the limit laws, evaluate
x→2
x→2
x→2
the following limits showing all steps. State each limit law that you use.
1. lim (f (x) + 2h(x))
x→2
2. lim (h(x))3
x→2
4
x→2 f (x) − h(x)
3. lim
Dr. Allen’s MATH 1261 Class Notes and Problem Sets, Fall 2012
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Exercise Set 1.2.
√
Problem 1.6. Use the limits laws to find lim
x→4
x2 + 9
. Justify each step.
x
Problem 1.7.
1. Sketch a graph of f (x) = tan x cot x. Find lim f (x).
x→0
2. Some first-year calculus students claim that they have a proof that zero equals
one. Their argument follows below. Do you believe the students (in other words
do you accept that zero equals one)? If you doubt the students’ claim, there
must be an error in their reasoning. Find the error.
The students claim that by looking at the graph of y = tan x cot x it is
easily observed that
1 =
lim tan x cot x
x→0
and using limit laws and properties of the real number 0 we have:
lim tan x cot x =
x→0
lim tan x lim cot x
x→0
x→0
= 0 · lim cot x
x→0
= 0
Therefore, 1 = 0. What is wrong with the students’ argument?
Problem 1.8. Suppose that lim f (x) does not exist. Also, suppose that lim g(x)
x→a
x→a
does not exist. Is it possible for lim (f (x) + g(x)) to exist? If so, then give an
x→a
example. If it is not possible, then explain.
Problem 1.9. Page 73, Problems 11, 15, 17, 19, 21, 33, 55
Dr. Allen’s MATH 1261 Class Notes and Problem Sets, Fall 2012
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Exercise Set 1.3.
Team Members’ Names:
List your teams’ cards:
Problem 1.10. Justify why you are a team; that is, evaluate your limit showing all steps.
State each limit law that you use.
Note: A polynomial function is a function of the form
f (x) = cn xn + cn−1 xn−1 + · · · + c2 x2 + c1 x + c0 .
A rational function is the quotient of two polynomial functions in simplest form. Using
the limit laws, you will investigate the limits of polynomial and rational functions. Complete the following problems showing all work.
Problem 1.11. Let f (x) = 3x − 5
1. Evaluate f (2).
2. Evaluate lim f (x) by using the limit laws. Justify each step.
x→2
3. What do you observe about the relationship between the values of f (2) and lim f (x)?
x→2
Problem 1.12. Let f (x) = 2x2 − 3x + 1.
1. Evaluate f (2).
2. Evaluate lim f (x) by using the limit laws. Justify each step.
x→2
3. What do you observe about the relationship between the values of f (2) and lim f (x)?
x→2
Problem 1.13. Suppose f (x) is a polynomial function. What can you say about the relationship between f (a) and lim f (x)?
x→a
Dr. Allen’s MATH 1261 Class Notes and Problem Sets, Fall 2012
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Problem 1.14. Based on your previous answer, fill in the blank:
.
If f (x) is a polynomial function, then lim f (x) =
x→a
Problem 1.15. Let h(x) =
5x3 + 4
.
x−3
1. Evaluate h(2).
2. Evaluate lim h(x) using the limit laws and your observations in the previous probx→2
lems.
3. What do you observe about the relationship between h(2) and lim h(x)?
x→2
p(x)
where p(x) and q(x) are polynomial functions with q(a) 6= 0,
q(x)
then what can you say about the relationship between h(a) and lim h(x)? Justify.
Problem 1.16. If h(x) =
x→a
Problem 1.17. Based on your previous answer, fill in the blank:
p(x)
where p(x) and q(x) are polynomial functions with q(a) 6= 0 (that is, h(x)
q(x)
.
is a rational function), then lim h(x) =
If h(x) =
x→a
Problem 1.18. Complete the following statement of the Substitution Theorem:
If f is a polynomial or a rational function, then
lim f (x) =
x→a
provided f (a) is defined (that is, a is in the domain of f ). In the case of a rational function,
this means that the value of the denominator at a is not zero.
x2 − 1
by using a table of values.
x→1 x − 1
Note that if we try to evaluate this limit using the substitution theorem, we end up with an
indeterminate form 0/0. In other words, we cannot use the substitution theorem to begin
with. We can, however, do some algebra and simplifying, and then use the substitution
theorem. Complete the following:
Problem 1.19. Recall that last week, we evaluated lim
x2 − 1
x→1 x − 1
lim
=
(x + 1)(x − 1)
x→1
x−1
lim
(by factoring)
(1.1)
=
(simplifying)
(1.2)
=
(substitution theorem)
(1.3)
Dr. Allen’s MATH 1261 Class Notes and Problem Sets, Fall 2012
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1.2.2
Limits of Polynomial and Rational Functions
Example 1.4.
x2 + 1
.
x→0 x − 1
Evaluate lim
Example 1.5.
x2 + 3x − 10
.
x→2 x2 + x − 6
Find lim
Example 1.6.
√
x+4−2
Find lim
. (Rationalizing the numerator)
x→0
x
Exercise Set 1.4.
Problem 1.20. Evaluate the following limits.
1. lim 1261
x→151
2. lim (x2 + 3x − 7)
x→−4
√
x−2
x→4 x − 4
2x
8
4. lim
+
x→−4 x + 4
x+4
3. lim
(2 + h)3 − 8
h→0
h
5. lim
Hint: a3 − b3 = (a − b)(a2 + ab + b2 ) and a3 + b3 = (a + b)(a2 − ab + b2 ).
Problem 1.21. Page 73, Problems 1, 3, 5, 7, 9, 23, 25, 37, 39, 45
Dr. Allen’s MATH 1261 Class Notes and Problem Sets, Fall 2012
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1.2.3
Limits of Piecewise Graphs
Example 1.7. Suppose f (x) =
4 − x2 if x ≤ 2
Compute lim f (x), or state that
x − 1 if x > 2.
x→2
the limit does not exist. Justify.
Example 1.8. Evaluate lim −
x→−4
1.2.4
|x + 4|
.
x+4
The Squeeze Theorem
Theorem 3 (Squeeze Theorem). Let f , g, and h be functions satisfying f (x) ≤
g(x) ≤ h(x) for all x near a, except possibly at a. If lim f (x) = lim h(x) = L, then
x→a
lim g(x) =
x→a
x→a
.
Example 1.9. Suppose 3x ≤ g(x) ≤ x3 + 2 for all x satisfying 0 ≤ x ≤ 2. Evaluate
lim g(x).
x→1
Exercise Set 1.5.
Problem 1.22. (piecewise functions) Page 73, Problems 31, 33, 35, 65
Problem 1.23. (squeeze theorem) Page 74, Problems 52, 53
Problem 1.24. Pages 73 - 75, Problems 41, 43, 47, 61, 63, 67, 69, 73
Dr. Allen’s MATH 1261 Class Notes and Problem Sets, Fall 2012
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1.3
Infinite Limits and Vertical Asymptotes
Note: This section corresponds to section 2.4 in your text.
Problem 1.25. Consider f (x) =
1
.
x−1
1. What is the domain of f (x)?
2. What can we say about the function values of f (x) as x gets close to 1 from
the right?
1
x
f (x) =
x−1
2
1.5
1.1
1.01
1.001
3. What can we say about the function values of f (x) as x gets close to 1 from
the left?
1
x
f (x) =
x−1
0
.5
.9
.99
.999
Definition 1.3 (infinite limit).
1. lim− f (x) = ∞ means that f (x) increases without bound as x approaches a
x→a
from the left.
2. lim+ f (x) = ∞ means that f (x) increases without bound as x approaches a
x→a
from the right.
3. lim− f (x) = −∞ means that f (x) decreases without bound as x approaches a
x→a
from the left.
4. lim+ f (x) = −∞ means that f (x) decreases without bound as x approaches a
x→a
from the right.
5. lim f (x) = ∞ if and only if lim− f (x) = ∞ and lim+ f (x) = ∞
x→a
x→a
x→a
6. lim f (x) = −∞ if and only if lim− f (x) = −∞ and lim+ f (x) = −∞
x→a
x→a
x→a
Dr. Allen’s MATH 1261 Class Notes and Problem Sets, Fall 2012
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Definition 1.4 (vertical asymptote). The line x = a is a vertical asymptote of
the graph y = f (x) if at least one of the following 6 statements is true:
lim f (x) = ∞
x→a+
lim f (x) = −∞
lim f (x) = ∞
x→a−
x→a+
lim f (x) = −∞
x→a−
lim f (x) = ∞
x→a
lim f (x) = −∞
x→a
Exercise Set 1.6.
Problem 1.26. Let f (x) =
1
. Find the following, if possible.
(x − 1)2
1. lim+ f (x)
x→1
2. lim− f (x)
x→1
3. lim f (x)
x→1
8272012
. Justify your answer.
x→5
x−5
8272012
?
What does the infinite limit tell you about the graph of f (x) =
x−5
Problem 1.27. Determine the infinite limit of lim−
Problem 1.28. Fill in the blanks.
1. lim− r(θ) = γ means that r(θ) gets near to
when θ approaches β
θ→β
.
from the
2. If
lim
x→−8272012−
r(x) = −∞, then the line
is a
asymptote of the graph of y = r(x).
Problem 1.29. Sketch the graph of a function that satisfies the following conditions:
lim + f (x) = −∞ and lim − f (x) = −∞. What can you deduce about lim f (x)?
x→−1
x→−1
x→−1
Justify.
Problem 1.30. Pages 81-84, Problems 1, 2, 5, 7, 9, 15, 17, 19, 21, 23, 25, 27, 29, 33,
36, 37, 39, 45
Dr. Allen’s MATH 1261 Class Notes and Problem Sets, Fall 2012
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1.4
Limits at Infinity and Horizontal Asymptotes
Note: This section corresponds to section 2.5 in your text.
Example 1.10.
1
.
x
1. Evaluate the following limits if possible.
Consider f (x) =
(a) lim+ f (x)
x→0
(b) lim− f (x)
x→0
(c) lim f (x)
x→∞
(d) lim f (x)
x→−∞
2. Give the equations of the asymptotes.
Definition 1.5 (horizontal asymptote). The line y = b is a horizontal asymptote
of the graph of f (x) if either
lim f (x) = b or
x→∞
lim f (x) = b.
x→−∞
Note: To find the horizontal asymptotes of the graph of f (x), calculate
lim f (x)
x→∞
and
lim f (x).
x→−∞
Problem 1.31. How many horizontal asymptotes can the graph y = f (x) have?
Explain. Sketch graphs to illustrate the possibilities.
Theorem 4 (Limits at Infinity). If r is a positive rational number and c is any real
number, then
c
lim r =
.
x→∞ x
Furthermore, if xr is defined when x < 0, then
c
lim r =
.
x→−∞ x
Dr. Allen’s MATH 1261 Class Notes and Problem Sets, Fall 2012
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Example 1.11.
Find the following limits, if possible.
2
1. lim 5 − 2
x→∞
x
1 − x − x2
x→∞ 2x2 − 7
2. lim
3. lim cos x
x→∞
Example 1.12.
Sketch the graph of a function f that satisfies the following conditions: the domain
of f is (−∞, 2) ∪ (2, ∞), lim f (x) = ∞, lim f (x) = 3, and lim f (x) = −3.
x→−2
x→−∞
x→∞
Exercise Set 1.7.
Problem 1.32. (Fill in the Blanks)
1. If lim f (x) = π, then the line
is a
x→−∞
asymptote of the graph of f (x).
2. If lim f (x) = −∞, then the line
is a
x→π
asymptote of the graph of f (x).
Problem 1.33. Sketch the graph of a function f that has the properties: lim+ f (x) =
x→0
∞, lim− f (x) = −∞, lim f (x) = 2, and lim f (x) = −3. Give the equation for
x→0
x→∞
x→−∞
each asymptote. (Be sure to sketch any asymptotes.)
Problem 1.34. Jim is asked to evaluate the following limit: lim (x7 − 10000x2 ).
x→∞
Jim does the following : lim (x7 − 10000x2 ) = lim x7 − lim 10000x2 = ∞ − ∞ = 0.
x→∞
x→∞
x→∞
Is Jim’s work correct? Justify.
8x2 − 5x + 9
.
x→∞ 6 − 9x − 3x2
Problem 1.35. Evaluate lim
Problem 1.36. Pages 92-93, Problems 1, 3, 7, 9, 11, 15, 17, 19, 21, 23, 25, 27, 31,
33, 35, 37, 39, 43, 45, 48, 50, 51, 53, 57, 59, 63, 65
Dr. Allen’s MATH 1261 Class Notes and Problem Sets, Fall 2012
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1.5
Continuity
Note: This section corresponds to section 2.6 in your text.
Example 1.13. Consider the following 3 graphs.
Find the following if possible. Explain your answers.
1. lim f (x)
x→a
2. lim g(x)
x→a
3. lim h(x)
x→a
Definition 1.6 (continuity at a point). Let f be defined on an open interval containing a. The function f is continuous at the number a if and only if
lim f (x) = f (a).
x→a
The function f is discontinuous at a if f is not continuous at a.
Note: For f to be continuous at a, the following 3 conditions must be satisfied:
1. f (a) exists (a is the domain of f ).
2. lim f (x) exists.
x→a
3. lim f (x) = f (a).
x→a
If any one of these conditions fails, then f is discontinuous at a.
Dr. Allen’s MATH 1261 Class Notes and Problem Sets, Fall 2012
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Definition 1.7. Suppose f (x) is discontinuous at x = a.
1. If lim f (x) exists, then x = a is a removable discontinuity of f .
x→a
2. If lim− f (x) and lim+ f (x) exist, but are not equal, then x = a is a jump disx→a
x→a
continuity of f .
3. If one or both of lim− f (x) or lim+ f (x) is infinite, then x = a is an infinite
x→a
x→a
discontinuity of f .
Example 1.14. Let
f (x) =
−3x + 4 if x ≤ 3
−2
if x > 3.
Is f continuous at 3? If f is discontinuous at 3, what type of discontinuity does it
have? Justify.
Example 1.15.
Sketch a graph of a function that has a jump discontinuity at x = 2 and a removable
discontinuity at x = 4 but is continuous elsewhere.
Definition 1.8 (left and right continuous). A function f is right continuous at
a if lim+ f (x) = f (a). A function f is left continuous at a if lim− f (x) = f (a).
x→a
x→a
Dr. Allen’s MATH 1261 Class Notes and Problem Sets, Fall 2012
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Continuous Functions
Theorem 5 (Some Properties of Continuous Functions). Let c be a constant. Suppose
f and g are continuous at a. Then the following functions are also continuous at a.
1. f + g
2. f − g
3. cf
4. f g
5.
f
(if g(a) 6= 0)
g
Note: Theorem 5 follows from the limit laws and the definition of continuity at a
point.
Theorem 6 (Continuity of Familiar Functions). The following types of functions
are continuous at every number in their domain: polynomial functions, rational
functions, root functions, trigonometric functions, exponential functions,
logarithmic functions, inverse trigonometric functions.
Note: Continuity extends the Substitution Theorem.
Example 1.16.
Use continuity to evaluate the following limits.
√
1. lim
x
x→1261
√
2. lim ( x + x2 )
x→9
3. lim
x→8
√
x+1
4. limπ cos(2x)
x→ 2
Theorem 7 (Intermediate Value Theorem). Suppose f is continuous on the closed
interval [a, b]. If N is a real number satisfying f (a) < N < f (b) (or f (b) < N < f (a)),
then there exists a number c between a and b (a < c < b) such that f (c) = N . (In
other words, if f is continuous on the closed interval [a, b], then f takes on every
value between f (a) and f (b).)
Note: The Intermediate Value Theorem only applies to continuous functions.
Dr. Allen’s MATH 1261 Class Notes and Problem Sets, Fall 2012
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Geometric Interpretation of the Intermediate Value Theorem
Theorem 8 (special case of the Intermediate Value Theorem). If f is continuous on
a closed interval [a, b], then if f (a) and f (b) have opposite signs (one positive and one
.
negative), there exists a point c in the interval (a, b) such that f (c) =
Example 1.17.
Use the Intermediate Value Theorem to show that x5 − x2 − 4 = 0 has a real solution
between 1 and 2.
Exercise Set 1.8.
Problem 1.37. (Fill in the Blank.) By definition of continuity, a function g is
continuous at c if and only if
.
Problem 1.38. (Fill in the Blank) The Intermediate Value Theorem says that
if a function f is continuous on [a, b] and W is a number between f (a) and f (b), then
and
such that
there is a number c between
Problem 1.39. If lim f (x) = 5 and lim g(x) = −7, and if g is continuous at x = 5,
x→5
x→5
find g(5), if possible. Justify your answer.
Problem 1.40.
1. Complete the following definition of infinite discontinuity (which should involve limits). A function f has an infinite discontinuity at x = 2 if
2. Sketch a graph of a function that has a removable discontinuity at x = 2.
Explain why your graph has a removable discontinuity at x = 2.
3. Give an algebraic representation of a function that has a jump discontinuity
at x = 2, and explain why it has the discontinuity.
1
. Is f (x) continuous at 2? If f is discontinuous at
Problem 1.41. Let f (x) =
x−2
2, what type of discontinuous does it have? Explain.
Problem 1.42. Suppose lim− f (x) = 3, lim+ f (x) = 1, and f (2) = 3. Is f (x) conx→2
x→2
tinuous at x = 2? If f (x) is discontinuous at x = 2, what type of discontinuity does
it have? In addition, determine if f is left or right continuous (or neither) at x = 2.
Explain your answers completely.
Problem 1.43. Pages 103-104, Problems 7, 9, 11, 13, 15, 19, 23, 25, 29, 31, 33, 43,
45, 51
Dr. Allen’s MATH 1261 Class Notes and Problem Sets, Fall 2012
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Chapter 2
Differentiation
2.1
Definition and Interpretation of the Derivative
Note: This section corresponds to section 3.1 in your text.
Definition 2.1 (derivative). The derivative of a function f is another function f 0
(read f prime) whose value at any number a, denoted f 0 (a), is defined to be,
f (a + h) − f (a)
,
h→0
h
f 0 (a) = lim
provided the limit exists and is not ∞ or −∞.
Notes:
1. One interpretation of the derivative: f 0 (a) is the slope of the tangent line to the
curve f at x = a.
2. Another interpretation of the derivative: f 0 (a) is the instantaneous rate of
change of f at x = a.
3. A function f is differentiable at a if f 0 (a) exists.
4. A function f is differentiable on an open interval, (a, b), (a, ∞), (−∞, a),
(−∞, ∞) if it is differentiable at every number in the interval.
5. Finding a derivative is called differentiation.
6. Other notations for the derivative function, f 0 (x), are
y0,
df
dy d
,
f (x),
, Df (x), Dx f (x).
dx dx
dx
f (x + h) − f (x)
provided the limit exists and is not infinite.
h→0
h
7. f 0 (x) = lim
Dr. Allen’s MATH 1261 Class Notes and Problem Sets, Fall 2012
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Example 2.1.
Let f (x) = x2 − x + 3. Find f 0 (1).
Example 2.2. Let f (x) =
√
x.
1. Give the domain of f (x).
2. Find f 0 (x).
3. What is the domain of f 0 (x)?
4. Find f 0 (4).
Exercise Set 2.1.
Problem 2.1. Find the derivative of f (x) =
2
. What is f 0 (1)?
x
Problem 2.2. Find the derivative of each of the functions using the definition of the
derivative.
1. f (x) = 1261
2. f (x) = mx + b
3. f (x) = 5x − 9x2
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Geometric Interpretation of the Derivative
A line tangent to a curve at a point P is a line that best approximates the curve
near P .
Example 2.3.
√
√
√ √
2, 1.1, .97, and 1.5 by using the tangent line to the curve
Approximate
√
f (x) = x at the point (1, f (1)).
Definition 2.2 (tangent line). The tangent line to the curve y = f (x) at the point
P (a, f (a)) is the line through P with slope
f (a + h) − f (a)
,
h→0
h
mtan = lim msec = lim
h→0
provided the limit exists.
Note: One interpretation of the derivative: f 0 (a) is the slope of the tangent line
to the curve f at x = a (that is, at the point (a, f (a)). What is the equation of the
tangent line at this point? Justify.
f (x) − f (a)
.
x→a
x−a
Note: An equivalent form of the derivative of f at a is given by f 0 (a) = lim
Why?
Exercise Set 2.2.
Problem 2.3. Find the slope of the tangent line to the curve y = f (x) = x2 at the
point (2, 4). Then write the equation of the tangent line. Graph the function as well
as the tangent line.
Problem 2.4. Each of the following is a derivative, but of what function and at what
point? Justify.
(4 + h)2 − 16
h→0
h
1. lim
2. lim
x→3
2
x
− 32
x−3
cos(π + h) + 1
h→0
h
3. lim
Dr. Allen’s MATH 1261 Class Notes and Problem Sets, Fall 2012
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Problem 2.5. A curve has equation y = f (x),
1. Write an expression for the slope of the secant line through the points P (1261, f (1261))
and Q(x, f (x)).
2. Write an expression for the slope of the tangent line at P .
Problem 2.6. Find an equation of the tangent line to the graph of y = f (x) at x = 5
if f (5) = −3 and f 0 (5) = 4.
Problem 2.7. Sketch the graph of a function f with the domain R that satisfies the
following conditions: f (0) = 0, f 0 (0) = 3, f 0 (1) = 0, and f 0 (2) = 1.
Problem 2.8. Page 132, Problems 11, 15, 17, 27, 29, 33, 41, 65, 67
Interpretation of the Derivative as a Rate of Change
Problem 2.9. Jim accidentally drops his calculus textbook from his dorm room
window which is 100 feet above the ground. The height (in feet) of the book above
the ground t seconds after being dropped is given by the position function
s(t) = −16t2 + 100.
1. Find the average velocity of the book in each of the following time intervals:
(a) [1, 2]
(b) [1, 1.5]
(c) [1, 1.1]
2. Find the instantaneous velocity (or simply the velocity) of the book 1 second
after Jim dropped it.
Step 1 First approximate the velocity at t = 1 by calculuting the average velocity over a small interval, [1, 1 + h].
Step 2 Now take the limit (of the average velocity in step 1) by letting h approach zero, which will give you the velocity at t = 1.
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