Calculus (Math 1A) Lecture 1

Transcription

1 Calculus (Math 1A) Lecture 1 Vivek Shende August 23, 2017

2 Hello and welcome to class! I am Vivek Shende I will be teaching you this semester. My office hours Starting next week: 1-3 pm on tuesdays; 2-3 pm fridays 873 Evans hall. Come ask questions! Your GSIs Kathleen Kirsch Kenneth Hung Kubrat Danilov Izaak Meckler Isabelle Shankar

3 Some administrative details: Enrolling in the class/sections: The book James Stewart, Single Variable Calculus: Early Transcendentals for UC Berkeley, 8th edition Prerequisites Trigonometry, coordinate geometry, plus a satisfactory grade in one of the following: CEEB MAT test, an AP test, the UC/CSU math diagnostic test, or 32.

4 Prerequisites: test yourself

5 Prerequisites: test yourself Compute (x + 4)(x + 3).

6 Prerequisites: test yourself Compute (x + 4)(x + 3). Compute 1 x x 1.

7 Prerequisites: test yourself Compute (x + 4)(x + 3). Compute 1 x x 1. Sketch y = x 2 + 4x + 4.

8 Prerequisites: test yourself Compute (x + 4)(x + 3). Compute 1 x x 1. Sketch y = x 2 + 4x + 4. After class, go outside. Measure the angle to the top of the bell tower (you may need a protractor) and then the distance to it (e.g. walk there and count your steps). Estimate the height.

9 Prerequisites: test yourself Can you compute dx a 2 x 2

10 Prerequisites: test yourself Can you compute dx a 2 x 2 If so, you should take a more advanced class.

11 Grading Your grade is determined by the homework/quizzes (15%), three midterms (15% each), and final (40%). Homework One online homework assignment per week; due one minute before midnight on friday; starting next week. Quizzes Every thursday, in section: one problem from a list of problems from the book. Exams Three in-class midterms (Sep. 18, Oct. 11, Nov. 8), and the final exam (Dec 12).

12 Grade distribution I intend to follow the same grade distribution as this course has historically had; very roughly 40% A s, 30% B s, 20% C s, and 10% D s and F s. You can find detailed statistics at

13 Makeup policy There are no makeups for any reason Instead, The two lowest quiz grades will be dropped. The lowest (curved) midterm grade can be replaced by your (curved) final exam grade.

14 Website The website has a full syllabus, including all of the above I will also post the slides on the website after each class. We will also use bcourses and piazza.

15 What do we study in calculus?

16 What do we study in calculus? Topic Objects What you do

17 What do we study in calculus? Topic Objects What you do Arithmetic Numbers Add, subtract, multiply, divide

18 What do we study in calculus? Topic Objects What you do Arithmetic Numbers Add, subtract, multiply, divide Algebra Indeterminates Solve equations

19 What do we study in calculus? Topic Objects What you do Arithmetic Numbers Add, subtract, multiply, divide Algebra Indeterminates Solve equations Geometry Shapes Draw lines, circles,...

20 What do we study in calculus? Topic Objects What you do Arithmetic Numbers Add, subtract, multiply, divide Algebra Indeterminates Solve equations Geometry Shapes Draw lines, circles,... Calculus Functions Limit, derivative, integral

21 What do we study in calculus? To emphasize: in calculus, the basic objects of study are functions,

22 What do we study in calculus? To emphasize: in calculus, the basic objects of study are functions, which you may not be accustomed to thinking about.

23 What do we study in calculus? To emphasize: in calculus, the basic objects of study are functions, which you may not be accustomed to thinking about. More precisely, you have seen functions everywhere,

24 What do we study in calculus? To emphasize: in calculus, the basic objects of study are functions, which you may not be accustomed to thinking about. More precisely, you have seen functions everywhere, but may not be used to the abstract notion of function,

25 What do we study in calculus? To emphasize: in calculus, the basic objects of study are functions, which you may not be accustomed to thinking about. More precisely, you have seen functions everywhere, but may not be used to the abstract notion of function, or the formal manipulations of them.

26 What do we study in calculus? To emphasize: in calculus, the basic objects of study are functions, which you may not be accustomed to thinking about. More precisely, you have seen functions everywhere, but may not be used to the abstract notion of function, or the formal manipulations of them. Becoming comfortable with functions is one of the largest conceptual steps in learning calculus.

27 What do we study in calculus? To emphasize: in calculus, the basic objects of study are functions, which you may not be accustomed to thinking about. More precisely, you have seen functions everywhere, but may not be used to the abstract notion of function, or the formal manipulations of them. Becoming comfortable with functions is one of the largest conceptual steps in learning calculus. We will spend the first week on this.

28 Functions of time

29 Functions of time At each time t, the earth is at some distance d(t) from the sun.

30 Functions of time At each time t, the earth is at some distance d(t) from the sun.

31 Functions of time At each time t, the earth is at some distance d(t) from the sun. The picture gets more interesting when you look closer:

32 Functions of time At each time t, there is some number of living humans l(t).

33 Functions of time At each time t, there is some number of living humans l(t).

34 Functions of time At each time t, a stock has some price p(t).

35 Functions of time At each time t, a stock has some price p(t).

36 Functions of time A sample of carbon contains a certain amount of the unstable isotope carbon 14, which decays:

37 Functions of time A sample of carbon contains a certain amount of the unstable isotope carbon 14, which decays:

38 Functions of time A sample of carbon contains a certain amount of the unstable isotope carbon 14, which decays, hence takes different values c(t) over time.

39 Functions of space At each point of space there is a temperature, T (x, y, z); you could measure it with a thermometer.

40 Functions of space At each point of space there is a temperature, T (x, y, z); you could measure it with a thermometer. Each point of space has some distance D(x, y, z) to the center of the sun.

41 Functions of space At each point of space there is a temperature, T (x, y, z); you could measure it with a thermometer. Each point of space has some distance D(x, y, z) to the center of the sun. In this class we will not see this kind of function, because we are studying functions of one real variable.

42 Functions of space At each point of space there is a temperature, T (x, y, z); you could measure it with a thermometer. Each point of space has some distance D(x, y, z) to the center of the sun. In this class we will not see this kind of function, because we are studying functions of one real variable.

43 Functions of space At each point of space there is a temperature, T (x, y, z); you could measure it with a thermometer. Each point of space has some distance D(x, y, z) to the center of the sun. In this class we will not see this kind of function, because we are studying functions of one real variable. We will see analogous things in which space is constrained to be one dimensional; e.g. if we throw a ball straight up, or are diving in a well.

44 Another function A function from {A+, A, A, B+, B, B, C+, C, C, D, F } to percents:

45 Another function A function from {A+, A, A, B+, B, B, C+, C, C, D, F } to percents:

46 Another function A function from {A+, A, A, B+, B, B, C+, C, C, D, F } to percents: It is the historical grade distribution for this class.

47 Functions from formulas

48 Functions from formulas f (x) = 0

49 Functions from formulas f (x) = 0

50 Functions from formulas f (x) = x

51 Functions from formulas f (x) = x

52 Functions from formulas f (x) = 2x

53 Functions from formulas f (x) = 2x

54 Functions from formulas f (x) = x/2

55 Functions from formulas f (x) = x/2

56 Functions from formulas f (x) = x 2

57 Functions from formulas f (x) = x 2

58 Functions from formulas f (x) = x 2 + 4x + 4

59 Functions from formulas f (x) = x 2 + 4x + 4

60 Functions from formulas f (x) = 1/x

61 Functions from formulas f (x) = 1/x

62 To make these graphs

63 Functions from formulas f (x) = (x + 3)/(x + 4)

64 Functions from formulas f (x) = (x + 3)/(x + 4)

65 What is a function?

66 What is a function? In your book, you will find:

67 What is a function? In your book, you will find: A function f is a rule that assigns to each element x in a set D exactly one element, called f (x), in a set E.

68 What is a function? It is somewhat better to say less:

69 What is a function? It is somewhat better to say less: A function f is a rule that assigns to each element x in a set D exactly one element, called f (x), in a set E.

70 What is a function The difference between these arises when considering the question of whether two functions are the same.

71 What is a function The difference between these arises when considering the question of whether two functions are the same. For instance, consider the following two rules, which can be applied to positive integers:

72 What is a function The difference between these arises when considering the question of whether two functions are the same. For instance, consider the following two rules, which can be applied to positive integers: Rule 1.

73 What is a function The difference between these arises when considering the question of whether two functions are the same. For instance, consider the following two rules, which can be applied to positive integers: Rule 1. Divide by 9 and consider the remainder.

74 What is a function The difference between these arises when considering the question of whether two functions are the same. For instance, consider the following two rules, which can be applied to positive integers: Rule 1. Divide by 9 and consider the remainder. E.g., 421 = ,

75 What is a function The difference between these arises when considering the question of whether two functions are the same. For instance, consider the following two rules, which can be applied to positive integers: Rule 1. Divide by 9 and consider the remainder. E.g., 421 = , so we get 7.

76 What is a function The difference between these arises when considering the question of whether two functions are the same. For instance, consider the following two rules, which can be applied to positive integers: Rule 1. Divide by 9 and consider the remainder. E.g., 421 = , so we get 7. Rule 2.

77 What is a function The difference between these arises when considering the question of whether two functions are the same. For instance, consider the following two rules, which can be applied to positive integers: Rule 1. Divide by 9 and consider the remainder. E.g., 421 = , so we get 7. Rule 2. Add the digits together. Repeat until the result has fewer than one digit.

78 What is a function The difference between these arises when considering the question of whether two functions are the same. For instance, consider the following two rules, which can be applied to positive integers: Rule 1. Divide by 9 and consider the remainder. E.g., 421 = , so we get 7. Rule 2. Add the digits together. Repeat until the result has fewer than one digit. E.g., = 7.

79 What is a function The difference between these arises when considering the question of whether two functions are the same. For instance, consider the following two rules, which can be applied to positive integers: Rule 1. Divide by 9 and consider the remainder. E.g., 421 = , so we get 7. Rule 2. Add the digits together. Repeat until the result has fewer than one digit. E.g., = 7. In fact, though these look like rather different rules, in fact they always produce the same result. In mathematics, we say they give the same function.

80 What is a function?

81 What is a function? A function f assigns to each element x in a set D exactly one element, called f (x), in a set E.

82 What is a function? A function f assigns to each element x in a set D exactly one element, called f (x), in a set E. The set D is called the domain, and the set E is called the codomain. One says f is a function from D to E.

83 What is a function? A function f assigns to each element x in a set D exactly one element, called f (x), in a set E. The set D is called the domain, and the set E is called the codomain. One says f is a function from D to E. In particular, to precisely specify a function, one is strictly speaking supposed to say what the domain D and codomain E are.

84 Domain

85 Domain In particular, to precisely specify a function, one is strictly speaking supposed to say what the domain D and codomain E are.

86 Domain In particular, to precisely specify a function, one is strictly speaking supposed to say what the domain D and codomain E are. In practice this is rarely done explicitly.

87 Domain In particular, to precisely specify a function, one is strictly speaking supposed to say what the domain D and codomain E are. In practice this is rarely done explicitly. Indeed in all the examples previously, we did not specify either the domain or the codomain.

88 Domain In particular, to precisely specify a function, one is strictly speaking supposed to say what the domain D and codomain E are. In practice this is rarely done explicitly. Indeed in all the examples previously, we did not specify either the domain or the codomain. In fact, you will often encounter questions like: What is the domain of the function x

89 Domain What is the domain of the function x Strictly speaking, this question is somewhat ambiguous.

90 Domain What is the domain of the function x Strictly speaking, this question is somewhat ambiguous. Indeed, to give x as a function,

91 Domain What is the domain of the function x Strictly speaking, this question is somewhat ambiguous. Indeed, to give x as a function, I should have told you what its domain was.

92 Domain What is the domain of the function x Strictly speaking, this question is somewhat ambiguous. Indeed, to give x as a function, I should have told you what its domain was. You should interpret this question as asking what is the largest subset of the real numbers on which the formula x makes sense and defines a function.

93 Domain What is the domain of the function x Strictly speaking, this question is somewhat ambiguous. Indeed, to give x as a function, I should have told you what its domain was. You should interpret this question as asking what is the largest subset of the real numbers on which the formula x makes sense and defines a function. The answer is: [0, ).

94 Domain To belabor the point, I can define a function f (x) from say [1, ) to the real numbers, given by the formula f (x) = x.

95 Domain To belabor the point, I can define a function f (x) from say [1, ) to the real numbers, given by the formula f (x) = x. The domain of this function would be [1, ),

96 Domain To belabor the point, I can define a function f (x) from say [1, ) to the real numbers, given by the formula f (x) = x. The domain of this function would be [1, ), because that s what I said the domain was.

97 Domain To belabor the point, I can define a function f (x) from say [1, ) to the real numbers, given by the formula f (x) = x. The domain of this function would be [1, ), because that s what I said the domain was. The domain is part of the data included in the function.

98 Domain To belabor the point, I can define a function f (x) from say [1, ) to the real numbers, given by the formula f (x) = x. The domain of this function would be [1, ), because that s what I said the domain was. The domain is part of the data included in the function. However, when a function is given by a formula and the domain is not explicitly specified,

99 Domain To belabor the point, I can define a function f (x) from say [1, ) to the real numbers, given by the formula f (x) = x. The domain of this function would be [1, ), because that s what I said the domain was. The domain is part of the data included in the function. However, when a function is given by a formula and the domain is not explicitly specified, we understand the domain to be the largest possible such that the formula makes sense.

100 What is a function? A function f assigns to each element x in a set D exactly one element, called f (x), in a set E. The set D is called the domain, and the set E is called the codomain. One says f is a function from D to E. In particular, to precisely specify a function, one is strictly speaking supposed to say what the domain D and codomain E are.

101 Codomain

102 Codomain What is the codomain is x?

103 Codomain What is the codomain is x? Is it (, )?

104 Codomain What is the codomain is x? Is it (, )? Or [0, )?

105 Codomain What is the codomain is x? Is it (, )? Or [0, )? Or maybe ( 1001, 23) ( , )?

106 Codomain What is the codomain is x? Is it (, )? Or [0, )? Or maybe ( 1001, 23) ( , )? The answer is that I have not given you enough information to know, or in other words, strictly speaking I have not specified x as a function.

107 Codomain What is the codomain is x? Is it (, )? Or [0, )? Or maybe ( 1001, 23) ( , )? The answer is that I have not given you enough information to know, or in other words, strictly speaking I have not specified x as a function. We will avoid thinking about this by making the convention that all functions in this class have codomain (, ), and never using the word codomain again.

108 Range

109 Range The range of a function is the set of values it takes.

110 Range The range of a function is the set of values it takes. The range of f (x) = 0 is

111 Range The range of a function is the set of values it takes. The range of f (x) = 0 is {0}.

112 Range The range of a function is the set of values it takes. The range of f (x) = 0 is {0}. The range of f (x) = x is

113 Range The range of a function is the set of values it takes. The range of f (x) = 0 is {0}. The range of f (x) = x is (, ).

114 Range The range of a function is the set of values it takes. The range of f (x) = 0 is {0}. The range of f (x) = x is (, ). The range of f (x) = 2x is

115 Range The range of a function is the set of values it takes. The range of f (x) = 0 is {0}. The range of f (x) = x is (, ). The range of f (x) = 2x is (, ).

116 Range The range of a function is the set of values it takes. The range of f (x) = 0 is {0}. The range of f (x) = x is (, ). The range of f (x) = 2x is (, ). The range of f (x) = x 2 is

117 Range The range of a function is the set of values it takes. The range of f (x) = 0 is {0}. The range of f (x) = x is (, ). The range of f (x) = 2x is (, ). The range of f (x) = x 2 is [0, ).

118 Range The range of a function is the set of values it takes. The range of f (x) = 0 is {0}. The range of f (x) = x is (, ). The range of f (x) = 2x is (, ). The range of f (x) = x 2 is [0, ). The range of f (x) = x is

119 Range The range of a function is the set of values it takes. The range of f (x) = 0 is {0}. The range of f (x) = x is (, ). The range of f (x) = 2x is (, ). The range of f (x) = x 2 is [0, ). The range of f (x) = x is [2, ).

120 Range The range of a function is the set of values it takes.

121 Range The range of a function is the set of values it takes. The range of f (x) = x is

122 Range The range of a function is the set of values it takes. The range of f (x) = x is [0, ).

123 Range The range of a function is the set of values it takes. The range of f (x) = x is [0, ). Consider the function f (x) with domain [1, ) given by the formula f (x) = x.

124 Range The range of a function is the set of values it takes. The range of f (x) = x is [0, ). Consider the function f (x) with domain [1, ) given by the formula f (x) = x. Its range is

125 Range The range of a function is the set of values it takes. The range of f (x) = x is [0, ). Consider the function f (x) with domain [1, ) given by the formula f (x) = x. Its range is [1, ).

126 Domain and Range

127 Functions and graphs Given a function f, its graph is the collection of pairs (x, f (x)).

128 Functions and graphs Given a function f, its graph is the collection of pairs (x, f (x)). When the domain and codomain are subsets of the real numbers (as will always be the case in this class), this collection can be plotted in the plane.

129 Functions and graphs Given a function f, its graph is the collection of pairs (x, f (x)). When the domain and codomain are subsets of the real numbers (as will always be the case in this class), this collection can be plotted in the plane. We already saw many examples.

130 Functions and graphs Given a function f, its graph is the collection of pairs (x, f (x)). When the domain and codomain are subsets of the real numbers (as will always be the case in this class), this collection can be plotted in the plane. We already saw many examples. One can do this procedure in reverse: a graph defines a function.

131 Functions and graphs Given a function f, its graph is the collection of pairs (x, f (x)). When the domain and codomain are subsets of the real numbers (as will always be the case in this class), this collection can be plotted in the plane. We already saw many examples. One can do this procedure in reverse: a graph defines a function. More precisely, a curve C in the plane is the graph of a function exactly when it intersects each vertical line at most once.

132 Functions and graphs Given a function f, its graph is the collection of pairs (x, f (x)). When the domain and codomain are subsets of the real numbers (as will always be the case in this class), this collection can be plotted in the plane. We already saw many examples. One can do this procedure in reverse: a graph defines a function. More precisely, a curve C in the plane is the graph of a function exactly when it intersects each vertical line at most once. The domain of the function will be the x-values for which the vertical line intersects exactly once.

133 Functions and graphs Given a function f, its graph is the collection of pairs (x, f (x)). When the domain and codomain are subsets of the real numbers (as will always be the case in this class), this collection can be plotted in the plane. We already saw many examples. One can do this procedure in reverse: a graph defines a function. More precisely, a curve C in the plane is the graph of a function exactly when it intersects each vertical line at most once. The domain of the function will be the x-values for which the vertical line intersects exactly once. The function is given by sending x 0 to the y-coordinate of the intersection of the vertical line x = x 0 with C.

134 Functions and graphs

135 Functions and graphs (It is more correct to say this is/is not the graph of a function.)

136 Increasing and decreasing

137 Increasing and decreasing The function f is increasing on the interval [a, b]. To express this in a formula, note that for a x < x b, one has f (x) < f (x ).

138 Increasing and decreasing The function f is increasing on the interval [a, b]. To express this in a formula, note that for a x < x b, one has f (x) < f (x ). The function is also increasing on [c, d], and decreasing on [b, c].

139 Even and odd

140 Even and odd A function f is said to be even if f (x) = f ( x), and said to be odd if f (x) = f ( x).

141 Even and odd A function f is said to be even if f (x) = f ( x), and said to be odd if f (x) = f ( x). f (x) = x n is even if n is even, and odd if n is odd.

142 Even and odd A function f is said to be even if f (x) = f ( x), and said to be odd if f (x) = f ( x). f (x) = x n is even if n is even, and odd if n is odd. Sums of even functions are even; sums of odd functions are odd.

143 Even and odd A function f is said to be even if f (x) = f ( x), and said to be odd if f (x) = f ( x). f (x) = x n is even if n is even, and odd if n is odd. Sums of even functions are even; sums of odd functions are odd. A product of two even functions, or two odd functions, is even. A product of an even and an odd function is odd.

144 Even and odd A function f is said to be even if f (x) = f ( x), and said to be odd if f (x) = f ( x). f (x) = x n is even if n is even, and odd if n is odd. Sums of even functions are even; sums of odd functions are odd. A product of two even functions, or two odd functions, is even. A product of an even and an odd function is odd. This notion seems (and maybe is) a bit silly, but turns out to be often helpful in simplifying and sanity checking computations.