Calculus (Math 1A) Lecture 1 Vivek Shende August 23, 2017 Hello - - PowerPoint PPT Presentation
Calculus (Math 1A) Lecture 1 Vivek Shende August 23, 2017 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
Vivek Shende August 23, 2017
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
Enrolling in the class/sections:
enrollment@math.berkeley.edu
The book
James Stewart, Single Variable Calculus: Early Transcendentals for UC Berkeley, 8th edition
Prerequisites
Trigonometry, coordinate geometry, plus a satisfactory grade in one
diagnostic test, or 32.
Compute (x + 4)(x + 3).
Compute (x + 4)(x + 3). Compute
1 x+1 + 1 x−1.
Compute (x + 4)(x + 3). Compute
1 x+1 + 1 x−1.
Sketch y = x2 + 4x + 4.
Compute (x + 4)(x + 3). Compute
1 x+1 + 1 x−1.
Sketch y = x2 + 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.
Can you compute
√ a2 − x2
Can you compute
√ a2 − x2 If so, you should take a more advanced class.
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).
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 www.berkeleytime.com.
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.
http://math.berkeley.edu/~vivek/1A.html 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.
Topic Objects What you do
Topic Objects What you do Arithmetic Numbers Add, subtract, multiply, divide
Topic Objects What you do Arithmetic Numbers Add, subtract, multiply, divide Algebra Indeterminates Solve equations
Topic Objects What you do Arithmetic Numbers Add, subtract, multiply, divide Algebra Indeterminates Solve equations Geometry Shapes Draw lines, circles, ...
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
To emphasize: in calculus, the basic objects of study are functions,
To emphasize: in calculus, the basic objects of study are functions, which you may not be accustomed to thinking about.
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,
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,
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.
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.
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
At each time t, the earth is at some distance d(t) from the sun.
At each time t, the earth is at some distance d(t) from the sun.
At each time t, the earth is at some distance d(t) from the sun. The picture gets more interesting when you look closer:
At each time t, there is some number of living humans l(t).
At each time t, there is some number of living humans l(t).
At each time t, a stock has some price p(t).
At each time t, a stock has some price p(t).
A sample of carbon contains a certain amount of the unstable isotope carbon 14, which decays:
A sample of carbon contains a certain amount of the unstable isotope carbon 14, which decays:
A sample of carbon contains a certain amount of the unstable isotope carbon 14, which decays, hence takes different values c(t)
At each point of space there is a temperature, T(x, y, z); you could measure it with a thermometer.
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.
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.
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.
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
in a well.
A function from {A+, A, A−, B+, B, B−, C+, C, C−, D, F} to percents:
A function from {A+, A, A−, B+, B, B−, C+, C, C−, D, F} to percents:
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.
f (x) = 0
f (x) = 0
f (x) = x
f (x) = x
f (x) = 2x
f (x) = 2x
f (x) = x/2
f (x) = x/2
f (x) = x2
f (x) = x2
f (x) = x2 + 4x + 4
f (x) = x2 + 4x + 4
f (x) = 1/x
f (x) = 1/x
f (x) = (x + 3)/(x + 4)
f (x) = (x + 3)/(x + 4)
In your book, you will find:
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.
It is somewhat better to say less:
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.
The difference between these arises when considering the question
The difference between these arises when considering the question
following two ‘rules’, which can be applied to positive integers:
The difference between these arises when considering the question
following two ‘rules’, which can be applied to positive integers: Rule 1.
The difference between these arises when considering the question
following two ‘rules’, which can be applied to positive integers: Rule 1. Divide by 9 and consider the remainder.
The difference between these arises when considering the question
following two ‘rules’, which can be applied to positive integers: Rule 1. Divide by 9 and consider the remainder. E.g., 421 = 46 × 9 + 7,
The difference between these arises when considering the question
following two ‘rules’, which can be applied to positive integers: Rule 1. Divide by 9 and consider the remainder. E.g., 421 = 46 × 9 + 7, so we get 7.
The difference between these arises when considering the question
following two ‘rules’, which can be applied to positive integers: Rule 1. Divide by 9 and consider the remainder. E.g., 421 = 46 × 9 + 7, so we get 7. Rule 2.
The difference between these arises when considering the question
following two ‘rules’, which can be applied to positive integers: Rule 1. Divide by 9 and consider the remainder. E.g., 421 = 46 × 9 + 7, so we get 7. Rule 2. Add the digits together. Repeat until the result has fewer than one digit.
The difference between these arises when considering the question
following two ‘rules’, which can be applied to positive integers: Rule 1. Divide by 9 and consider the remainder. E.g., 421 = 46 × 9 + 7, so we get 7. Rule 2. Add the digits together. Repeat until the result has fewer than one digit. E.g., 421 → 4 + 2 + 1 = 7.
The difference between these arises when considering the question
following two ‘rules’, which can be applied to positive integers: Rule 1. Divide by 9 and consider the remainder. E.g., 421 = 46 × 9 + 7, so we get 7. Rule 2. Add the digits together. Repeat until the result has fewer than one digit. E.g., 421 → 4 + 2 + 1 = 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.
A function f assigns to each element x in a set D exactly one element, called f (x), in a set E.
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
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
In particular, to precisely specify a function, one is strictly speaking supposed to say what the domain D and codomain E are.
In particular, to precisely specify a function, one is strictly speaking supposed to say what the domain D and codomain E are.
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.
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 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
What is the domain of the function √x Strictly speaking, this question is somewhat ambiguous.
What is the domain of the function √x Strictly speaking, this question is somewhat ambiguous. Indeed, to give √x as a function,
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.
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”.
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, ∞).
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.
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, ∞),
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.
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.
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,
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.
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
In particular, to precisely specify a function, one is strictly speaking supposed to say what the domain D and codomain E are.
What is the codomain is √x?
What is the codomain is √x? Is it (−∞, ∞)?
What is the codomain is √x? Is it (−∞, ∞)? Or [0, ∞)?
What is the codomain is √x? Is it (−∞, ∞)? Or [0, ∞)? Or maybe (−1001, −23) ∪ (−5.5249682, ∞)?
What is the codomain is √x? Is it (−∞, ∞)? Or [0, ∞)? Or maybe (−1001, −23) ∪ (−5.5249682, ∞)? 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.
What is the codomain is √x? Is it (−∞, ∞)? Or [0, ∞)? Or maybe (−1001, −23) ∪ (−5.5249682, ∞)? 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.
The range of a function is the set of values it takes.
The range of a function is the set of values it takes. The range of f (x) = 0 is
The range of a function is the set of values it takes. The range of f (x) = 0 is {0}.
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 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 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 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 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) = x2 is
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) = x2 is [0, ∞).
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) = x2 is [0, ∞). The range of f (x) = x2 + 2 is
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) = x2 is [0, ∞). The range of f (x) = x2 + 2 is [2, ∞).
The range of a function is the set of values it takes.
The range of a function is the set of values it takes. The range of f (x) = √x is
The range of a function is the set of values it takes. The range of f (x) = √x is [0, ∞).
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.
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
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, ∞).
Given a function f , its graph is the collection of pairs (x, f (x)).
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.
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.
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.
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.
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.
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 x0 to the y-coordinate of the intersection of the vertical line x = x0 with C.
(It is more correct to say “this is/is not the graph of a function”.)
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 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].
A function f is said to be even if f (x) = f (−x), and said to be
A function f is said to be even if f (x) = f (−x), and said to be
f (x) = xn is even if n is even, and odd if n is odd.
A function f is said to be even if f (x) = f (−x), and said to be
f (x) = xn 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 function f is said to be even if f (x) = f (−x), and said to be
f (x) = xn 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.
A function f is said to be even if f (x) = f (−x), and said to be
f (x) = xn 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