show1
... • A function of a variable x is a rule f that assigns to each value of x a unique number f(x) (read ”f of x”), called the value of the function at x. • x is called the independent variable. • The set of values that the independent variable is allowed to assume is called the domain of the function. A ...
... • A function of a variable x is a rule f that assigns to each value of x a unique number f(x) (read ”f of x”), called the value of the function at x. • x is called the independent variable. • The set of values that the independent variable is allowed to assume is called the domain of the function. A ...
Math 116 - Final Exam Review Sheet
... Below is an outline of the topics we’ve covered this quarter, along with suggestions of problems to practice or review from the text (some are old homework problems). The Final Exam will be cumulative, so you should study everything. However, the questions on the test will not require you to use a s ...
... Below is an outline of the topics we’ve covered this quarter, along with suggestions of problems to practice or review from the text (some are old homework problems). The Final Exam will be cumulative, so you should study everything. However, the questions on the test will not require you to use a s ...
Math 142A Homework Assignment 3 Selected Solutions 2.2.4 Show
... Math 142A Homework Assignment 3 Selected Solutions 2.2.4 Show that the set of irrational numbers fails to be closed. Solution Consider the sequence { πn }. Observe that nπ is an irrational number for every index n; for, if r = πn were rational, then n r would be a product of a natural number and a r ...
... Math 142A Homework Assignment 3 Selected Solutions 2.2.4 Show that the set of irrational numbers fails to be closed. Solution Consider the sequence { πn }. Observe that nπ is an irrational number for every index n; for, if r = πn were rational, then n r would be a product of a natural number and a r ...
POWER SUM IDENTITIES WITH GENERALIZED STIRLING
... Note that when x = −1, (2.14) turns into (1.1). Remark 2.6. Identity (2.14) for positive integers α = r can also be found in the treasure chest [7]. It is listed there (as number 1.126 on p.16) in the form µ ¶ µ ¶ j n µ ¶ r X X X n r k xj k j n j n ...
... Note that when x = −1, (2.14) turns into (1.1). Remark 2.6. Identity (2.14) for positive integers α = r can also be found in the treasure chest [7]. It is listed there (as number 1.126 on p.16) in the form µ ¶ µ ¶ j n µ ¶ r X X X n r k xj k j n j n ...
(pdf)
... been studied in many different forms for centuries. The harmonic series, ζ(1), was proven to be divergent as far back as the 14th century [1]. In the 18th century, the Swiss mathematician Leonhard Euler found a closed form expression for the sum of the reciprocals of the squared integers i.e. ζ(2). ...
... been studied in many different forms for centuries. The harmonic series, ζ(1), was proven to be divergent as far back as the 14th century [1]. In the 18th century, the Swiss mathematician Leonhard Euler found a closed form expression for the sum of the reciprocals of the squared integers i.e. ζ(2). ...
Fundamental Theorem of Calculus, Riemann Sums, Substitution
... Riemann Sum Let [a,b] = closed interval in the domain of function f Partition [a,b] into n subdivisions: { [x0,x1], [x1,x2], [x2,x3], … , [xn-1,xn]} where a = x0 < x1 < … < xn-1 < xn = b The Riemann sum of function f over interval [a,b] is: ...
... Riemann Sum Let [a,b] = closed interval in the domain of function f Partition [a,b] into n subdivisions: { [x0,x1], [x1,x2], [x2,x3], … , [xn-1,xn]} where a = x0 < x1 < … < xn-1 < xn = b The Riemann sum of function f over interval [a,b] is: ...
1.5 Function Notation
... and extracting even roots of negative numbers. The following example illustrates these concepts. Example 1.5.3. Find the domain2 of the following functions. 1. f (x) = ...
... and extracting even roots of negative numbers. The following example illustrates these concepts. Example 1.5.3. Find the domain2 of the following functions. 1. f (x) = ...
Harmonic and Fibonacci Sequences
... Sometimes it is easier to recognize a harmonic sequence if you create common NUMERATORS for your numbers. For example, consider the sequence: 6, 3, 2, …. There is not a common difference so it is not ________________, and there is not a common ratio, so it is not _________________.* However, the com ...
... Sometimes it is easier to recognize a harmonic sequence if you create common NUMERATORS for your numbers. For example, consider the sequence: 6, 3, 2, …. There is not a common difference so it is not ________________, and there is not a common ratio, so it is not _________________.* However, the com ...
Chain Rule
... As x increases to x + x there will be corresponding increases of u and y in u and y respectively. or means multiplication. Since these finite quantities it is possible to write down: dy dy ...
... As x increases to x + x there will be corresponding increases of u and y in u and y respectively. or means multiplication. Since these finite quantities it is possible to write down: dy dy ...
Find the next term - Dozenal Society of Great Britain
... pattern could go on 1,2,3,4,5,6,... or (Fibonacci) 1,2,3,5,8,11,19... (No, 11 is not a mistake; it’s still what we usually call thirteen - but here’s it is “a dozen and one”, 11). So three terms, for the sequence 1,2,3.. are not enough - maybe four would fix the pattern. What sparked this article of ...
... pattern could go on 1,2,3,4,5,6,... or (Fibonacci) 1,2,3,5,8,11,19... (No, 11 is not a mistake; it’s still what we usually call thirteen - but here’s it is “a dozen and one”, 11). So three terms, for the sequence 1,2,3.. are not enough - maybe four would fix the pattern. What sparked this article of ...
Lecture 4
... operation. Definition 2: Start with the concept of “two apples”, and remove all aspects of a single apple, e.g. redness, taste, etc.. You’ll be left with the number “2”. This definition is a bit problematic. Definition 3: 2 = The class of all sets of size 2 (this is indeed a very large class) ...
... operation. Definition 2: Start with the concept of “two apples”, and remove all aspects of a single apple, e.g. redness, taste, etc.. You’ll be left with the number “2”. This definition is a bit problematic. Definition 3: 2 = The class of all sets of size 2 (this is indeed a very large class) ...
Sequences - Finding a rule
... definite order, where the terms are obtained by some rule. A finite sequence ends after a certain number of terms. An infinite sequence is one that continues indefinitely. ...
... definite order, where the terms are obtained by some rule. A finite sequence ends after a certain number of terms. An infinite sequence is one that continues indefinitely. ...
4 + 3 + (2) + 1
... every single mobile number. (OK I’ll cook supper!) Approximate solution: Say a mobile number has 10 digits (actually 11) and each digit can be 0 – 9 (10 possibilitities). So the solution is 10! How long will this take? Calculatore! (fact 10) ...
... every single mobile number. (OK I’ll cook supper!) Approximate solution: Say a mobile number has 10 digits (actually 11) and each digit can be 0 – 9 (10 possibilitities). So the solution is 10! How long will this take? Calculatore! (fact 10) ...
PDF
... † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. ...
... † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. ...