1 Counting mappings
... Example 2. Let a planar rooted tree T be placed with root up on the plane and so that the sons of every vertex are placed from left to right in their order. Then the walk around this tree starting from the edge rs and leaving the tree on the left hand all the time corresponds to a sequence of 1-s an ...
... Example 2. Let a planar rooted tree T be placed with root up on the plane and so that the sons of every vertex are placed from left to right in their order. Then the walk around this tree starting from the edge rs and leaving the tree on the left hand all the time corresponds to a sequence of 1-s an ...
Chapter 3 Functions
... Because f (1) 1 and there is no real number which has this value, 1 . For what values is f not defined? In fact f is not defined for any of the negative real numbers because ...
... Because f (1) 1 and there is no real number which has this value, 1 . For what values is f not defined? In fact f is not defined for any of the negative real numbers because ...
Sample Questions for Exam 1 (Limits – Sections 2.1 to 2.5) 1. Sketch
... Investigate the limit of f as x approaches 1. Does the limit exist? If so, justify your answer. ...
... Investigate the limit of f as x approaches 1. Does the limit exist? If so, justify your answer. ...
PreAP Pre Calculus
... • What are logarithmic functions, how are they graphed and how do they represent real world situations? • What is the relationship between exponential and logarithmic functions and how is this knowledge used to solve a logarithmic or exponential equation? • What is a system of equations/inequalitie ...
... • What are logarithmic functions, how are they graphed and how do they represent real world situations? • What is the relationship between exponential and logarithmic functions and how is this knowledge used to solve a logarithmic or exponential equation? • What is a system of equations/inequalitie ...
ramanujan
... There is a famous story about Ramanujan. When he was ill, Hardy arrived at his residence to visit him. Being generally awkward, Hardy did not know what to say and told Ramanujan that the number of the taxicab he had come in was 1729, a number that seemed to be uninteresting. Ramanujan replied, on th ...
... There is a famous story about Ramanujan. When he was ill, Hardy arrived at his residence to visit him. Being generally awkward, Hardy did not know what to say and told Ramanujan that the number of the taxicab he had come in was 1729, a number that seemed to be uninteresting. Ramanujan replied, on th ...
Functions - UCSD Mathematics
... Since one line notation is a simple, brief way to specify functions, we’ll use it frequently. If the domain is not a set of numbers, the notation is poor because we must first pause and order the domain. There are other ways to write functions which overcome this problem. For example, we could write ...
... Since one line notation is a simple, brief way to specify functions, we’ll use it frequently. If the domain is not a set of numbers, the notation is poor because we must first pause and order the domain. There are other ways to write functions which overcome this problem. For example, we could write ...
Consensus Map Grade Level
... the maximum/minimum number of turning points, and state the maximum/minimum number of xintercepts. Graph a polynomial function given its zeros. Graph a polynomial function written in factored form. State the multiplicity of each zero, and describe how the graph behaves at each zero (bounce, cross). ...
... the maximum/minimum number of turning points, and state the maximum/minimum number of xintercepts. Graph a polynomial function given its zeros. Graph a polynomial function written in factored form. State the multiplicity of each zero, and describe how the graph behaves at each zero (bounce, cross). ...
LECTURE 4. RATIONAL AND IRRATIONAL NUMBERS: ORDER
... maximality condition will automatically imply the “no gap condition”. We will allow non-maximal cuts for other reasons to be seen below. We will provisionally denote the set of all cuts by R (identifying “equal” cuts, as usual). Example 4 shows how Q can be “embedded” in R (i.e., each rational numbe ...
... maximality condition will automatically imply the “no gap condition”. We will allow non-maximal cuts for other reasons to be seen below. We will provisionally denote the set of all cuts by R (identifying “equal” cuts, as usual). Example 4 shows how Q can be “embedded” in R (i.e., each rational numbe ...
A Note on Formalizing Undefined Terms in Real Analysis
... Lemma 2.6 0 = limn→∞ xn . Finally, we would like to spend a few additional words on teaching goals. So far, we have recognized two teaching goals in adopting proof assistants and types in mathematical education. The first one is the emphasis on mathematical rigour, so as to augment both the comprehe ...
... Lemma 2.6 0 = limn→∞ xn . Finally, we would like to spend a few additional words on teaching goals. So far, we have recognized two teaching goals in adopting proof assistants and types in mathematical education. The first one is the emphasis on mathematical rigour, so as to augment both the comprehe ...
Graphing a Trigonometric Function
... None of the six basic trigonometric functions graphed is a one to one. These functions do not have inverses. However, in each case the domain can be restricted to produce a new function that does not have an inverse as in the next example. ...
... None of the six basic trigonometric functions graphed is a one to one. These functions do not have inverses. However, in each case the domain can be restricted to produce a new function that does not have an inverse as in the next example. ...
I. INTRODUCTION. ELEMENTS OF MATHEMATICAL LOGIC AND
... Theorem 1.4 (Supremum Theorem). If M ⊂ R is nonempty bounded from above then there exists the unique s = sup M . Sketch of the proof. sup M = − inf(−M ), where −M = {x ∈ R; −x ∈ M }. Definition 1.5. Let M ⊂ R. We call a ∈ R maximum of M and write a = max M if a ∈ M and (∀x ∈ M ) x ≤ a. Minimum of ...
... Theorem 1.4 (Supremum Theorem). If M ⊂ R is nonempty bounded from above then there exists the unique s = sup M . Sketch of the proof. sup M = − inf(−M ), where −M = {x ∈ R; −x ∈ M }. Definition 1.5. Let M ⊂ R. We call a ∈ R maximum of M and write a = max M if a ∈ M and (∀x ∈ M ) x ≤ a. Minimum of ...
