This document provides step-by-step instructions for multiplying 365 by 112. It begins by stating the problem and includes guidelines for keeping work neat. It then lists the steps as multiplying by the ones place, placing a zero as a placeholder, multiplying by the tens place, and placing two zeroes as placeholders before arriving at the final solution.
6-2 Multiplying Decimals by Whole NumbersRudy Alfonso
The document describes the step-by-step process for multiplying decimals by whole numbers. It shows examples of 0.12 x 5 and 9.74 x 8. The steps include: using guidelines to keep work neat, multiplying the ones and regrouping if necessary, counting the number of decimal places in the factors, and including the proper number of decimal places in the final answer.
Division can be solved using repeated addition or repeated subtraction. With repeated addition, you think of division as sharing equally into groups. With repeated subtraction, you think of division as repeatedly subtracting the divisor from the dividend until you reach zero or a number less than the divisor. The number of subtractions is the quotient.
6. multiplying 2 to 4-digit numbers by multiplies of 10Annie Villamer
This document discusses multiplying numbers by multiples of 10. It provides examples of multiplying 2-4 digit numbers by multiples of 10. The key points are:
- To multiply by a multiple of 10, multiply the non-zero digits first and then add the appropriate number of zeros to the end based on the multiple of 10.
- The number of zeros in the product will equal the number of zeros in the multiplying factor.
- Examples are provided of word problems involving multiplying amounts by multiples of 10 to find totals, along with step-by-step workings. Practice problems and assignments are also included.
This document provides examples of order of operations problems to solve, with the numbers written out step-by-step on separate lines to illustrate the proper order. It begins with an anticipatory set and examples showing different operations like addition, subtraction, multiplication, division, exponents and fractions. The document aims to teach order of operations through worked examples broken down into individual steps.
The document discusses order of operations, which is the order in which mathematical operations should be performed in a problem containing multiple operations. The standard order is: 1) operations within parentheses, 2) exponents, 3) multiplication and division from left to right, 4) addition and subtraction from left to right. It provides examples of solving problems using the proper order of operations and asks readers to try solving sample problems on their own.
Multiplying 3 Digit Numbers by 1 Digit Numbers Using The Grid MethodChris James
The document describes the grid method for multiplying a 3-digit number by a 1-digit number. It explains that the 3-digit number is split into hundreds, tens, and ones and placed in the top of a grid. The 1-digit number is placed on the side of the grid. Each place value of the 3-digit number is then multiplied by the 1-digit number and those products are written in the grid. The totals are then added to find the final product.
This document discusses the importance of order and provides questions and activities about order of operations in math. It asks questions about how order relates to everyday life and provides examples of order in nature and society. Students are asked to come up with their own examples of order from their lives. The document then discusses the order of operations, with parentheses first, exponents second, multiplication/division third, and addition/subtraction fourth. Students work through practice problems applying the order of operations and discuss why the order matters and why we need agreed-upon conventions.
The document explains the BODMAS rule, which establishes the order of operations in math problems. It states that BODMAS is an acronym that represents the order: Brackets, Orders (exponents), Division, Multiplication, Addition, Subtraction. An example problem is worked through step-by-step to demonstrate how applying the BODMAS rule determines the correct order to evaluate the terms. Additional practice problems are provided and it concludes that the BODMAS rule is helpful for solving various math homework problems.
6-2 Multiplying Decimals by Whole NumbersRudy Alfonso
The document describes the step-by-step process for multiplying decimals by whole numbers. It shows examples of 0.12 x 5 and 9.74 x 8. The steps include: using guidelines to keep work neat, multiplying the ones and regrouping if necessary, counting the number of decimal places in the factors, and including the proper number of decimal places in the final answer.
Division can be solved using repeated addition or repeated subtraction. With repeated addition, you think of division as sharing equally into groups. With repeated subtraction, you think of division as repeatedly subtracting the divisor from the dividend until you reach zero or a number less than the divisor. The number of subtractions is the quotient.
6. multiplying 2 to 4-digit numbers by multiplies of 10Annie Villamer
This document discusses multiplying numbers by multiples of 10. It provides examples of multiplying 2-4 digit numbers by multiples of 10. The key points are:
- To multiply by a multiple of 10, multiply the non-zero digits first and then add the appropriate number of zeros to the end based on the multiple of 10.
- The number of zeros in the product will equal the number of zeros in the multiplying factor.
- Examples are provided of word problems involving multiplying amounts by multiples of 10 to find totals, along with step-by-step workings. Practice problems and assignments are also included.
This document provides examples of order of operations problems to solve, with the numbers written out step-by-step on separate lines to illustrate the proper order. It begins with an anticipatory set and examples showing different operations like addition, subtraction, multiplication, division, exponents and fractions. The document aims to teach order of operations through worked examples broken down into individual steps.
The document discusses order of operations, which is the order in which mathematical operations should be performed in a problem containing multiple operations. The standard order is: 1) operations within parentheses, 2) exponents, 3) multiplication and division from left to right, 4) addition and subtraction from left to right. It provides examples of solving problems using the proper order of operations and asks readers to try solving sample problems on their own.
Multiplying 3 Digit Numbers by 1 Digit Numbers Using The Grid MethodChris James
The document describes the grid method for multiplying a 3-digit number by a 1-digit number. It explains that the 3-digit number is split into hundreds, tens, and ones and placed in the top of a grid. The 1-digit number is placed on the side of the grid. Each place value of the 3-digit number is then multiplied by the 1-digit number and those products are written in the grid. The totals are then added to find the final product.
This document discusses the importance of order and provides questions and activities about order of operations in math. It asks questions about how order relates to everyday life and provides examples of order in nature and society. Students are asked to come up with their own examples of order from their lives. The document then discusses the order of operations, with parentheses first, exponents second, multiplication/division third, and addition/subtraction fourth. Students work through practice problems applying the order of operations and discuss why the order matters and why we need agreed-upon conventions.
The document explains the BODMAS rule, which establishes the order of operations in math problems. It states that BODMAS is an acronym that represents the order: Brackets, Orders (exponents), Division, Multiplication, Addition, Subtraction. An example problem is worked through step-by-step to demonstrate how applying the BODMAS rule determines the correct order to evaluate the terms. Additional practice problems are provided and it concludes that the BODMAS rule is helpful for solving various math homework problems.
5. multiplying 2 to 4 digit numbers by 1-to 2-digitAnnie Villamer
1) The document provides instructions and examples for multiplying 2-4 digit numbers by 1-2 digit numbers without regrouping. It includes step-by-step worked examples and exercises.
2) Students are asked to solve word problems that require multiplying numbers with 2-4 digits by 1-2 digits to find totals, lengths, or amounts spent.
3) The assignment asks students to solve 5 word problems that require multiplying given amounts, lengths, or prices to determine totals, lengths, or amounts spent.
This document provides notes and practice problems about the order of operations. It explains that parentheses, exponents, multiplication, division, addition, and subtraction have a specific order in which they must be evaluated when multiple operations are present in an expression. Students are given examples of each operation and a mnemonic to remember the order of operations as PEMDAS, which stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. Guided practice problems are included for students to evaluate expressions using the proper order of operations.
The document discusses the order of operations, which is a rule for evaluating mathematical expressions containing multiple operations. It explains that the order of operations is: 1) Evaluate expressions inside grouping symbols (parentheses, brackets, braces) from innermost to outermost. 2) Evaluate exponents. 3) Perform multiplication and division from left to right. 4) Perform addition and subtraction from left to right. Mnemonics like PEMDAS are provided to help remember the order. Examples of applying the order of operations to evaluate expressions are provided.
This document discusses different shapes including two-dimensional shapes like squares, rectangles, and triangles, and three-dimensional shapes like cubes, cylinders, and spheres. It provides examples of how these shapes appear in real life objects like clocks, soup cans, and computers. The document also describes how shapes can be combined by addition or taken apart, and how they can be rotated, flipped or slid. It suggests that understanding shapes is important for math, reading, and writing skills.
The document discusses the order of operations rules for evaluating arithmetic expressions. It explains that the order is: 1) operations within grouping symbols, 2) exponents, 3) multiplication and division from left to right, 4) addition and subtraction from left to right. Following the order of operations is important so that everyone calculates expressions the same way and achieves consistent results. The document provides examples of evaluating expressions with different operations like exponents, grouping symbols, and fractions.
The document explains the order of operations (PEMDAS) for evaluating mathematical expressions. PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. It should be followed from left to right. Parentheses contain expressions that are evaluated first. Exponents represent multiplication and are done before multiplication and division from left to right. The same order applies to multiplication/division and addition/subtraction from left to right. An example equation is worked through step-by-step to demonstrate PEMDAS.
Social networking connects individuals through relationships and common interests. Facebook allows users to create profiles and communicate with friends through messages and chat. Twitter enables users to send and read short text-based posts called tweets. Blogs are websites or parts of websites where individuals regularly post entries like commentary or descriptions that are displayed in reverse chronological order. Heello also allows users to send posts and share pictures and videos. Social networking knowledge continues to expand rapidly as the technology develops and provides useful ways for people to connect in their everyday lives.
A 3 m 1 estrategias para la efectividad de la educación a distanciaIsaias Velazquez
El documento describe estrategias para mejorar la efectividad de la educación a distancia, incluyendo la búsqueda y manejo de información en línea, como el uso de directorios electrónicos y correo electrónico. También discute técnicas de estudio como el resumen y cuadro sinóptico, los cuales ayudan a organizar la información, sintetizar ideas principales, y ahorrar tiempo en el estudio y repaso de temas.
The document discusses three articles related to using the internet in education. Article 1 focuses on using teacher podcasts to help students proofread and revise blog posts. Article 2 details how blogging can promote higher order thinking and provides a framework called HOT blogging. Article 3 emphasizes the importance of teaching students how to effectively search for information online using the acronym SEARCH.
The document summarizes the clothing choices for characters portrayed as bullies and their victim in a story. The bullies are dressed in blue sports tops and shorts or a blue top and black shorts to convey strength, power, and a bold personality. Their clothing includes Nike brands. The victim, Sam, is dressed more formally in a school blazer, white shirt, black trousers, and glasses to portray a smart but outsider image.
This document provides a list of different camera shots and angles including close ups, long shots, two shots, low and high angle shots, extreme close ups, over the shoulder shots, very long shots, and medium close ups with some having connotations of nature, isolation, friendship, and stress.
This document provides guidance for students creating a film opening as an assessment task. It outlines 9 steps for the process, including taking stock of the requirements, setting up documentation of the process, building skills in areas like camera work and editing, investigating what makes effective openings, brainstorming ideas, planning shots and logistics, filming, editing, and evaluating the final product. Students are advised to allow enough time, keep evidence of their work, experiment, get feedback, and focus on simplicity while representing their film.
El método Montessori se basa en el respeto al desarrollo natural del niño y en darle libertad para explorar y aprender por sí mismo a través de materiales didácticos. La Dra. Montessori observó que los niños son capaces de absorber conocimientos espontáneamente como "esponjas" y desarrolló actividades prácticas para apoyar su aprendizaje autodirigido en lectura, escritura y matemáticas antes de los 6 años. El método enfatiza la autonomía, independencia y autodisciplina del ni
The document provides step-by-step instructions for multiplying decimals by whole numbers. It demonstrates multiplying 0.12 x 5, 9.74 x 8, and 219 x 7.2. For each problem, it lists the steps of multiplying the numbers, placing placeholders, adding products, and including the correct number of decimal places in the final answer.
The document shows the step-by-step work for multiplying 28 x 12. It breaks 12 into 10 and 2. It then multiplies 28 by 10 to get 280, and 28 by 2 to get 56. Finally, it adds 280 + 56 to get the total of 336.
The document provides step-by-step instructions for multiplying multi-digit numbers. It shows worked examples of 28 x 12 and 354 x 23. For each problem, it breaks the multiplication into steps: 1) multiply the ones, 2) add a placeholder zero, 3) multiply the tens, 4) add the partial products. The goal is to demonstrate the standard algorithm for multiplying multi-digit numbers in a clear and organized manner.
4-7 Multiplying by One-Digit and Two-Digit NumbersRudy Alfonso
The document shows the step-by-step work for multiplying 28 x 12. It breaks 12 into 10 and 2. It then multiplies 28 by 10 to get 280, and 28 by 2 to get 56. Finally, it adds 280 + 56 to get the total of 336.
4-7 Multiplying by One-Digit and Two-Digit NumbersRudy Alfonso
The document shows the step-by-step work for multiplying 28 x 12. It breaks 12 into 10 and 2. It then multiplies 28 by 10 to get 280, and 28 by 2 to get 56. Finally, it adds 280 + 56 to get the total of 336.
The document provides step-by-step instructions for multiplying decimals. It shows the multiplication of 2.3 x 1.5 worked out over multiple steps: 1) multiply the ones, 2) add a placeholder zero, 3) multiply the tens, 4) add the products, and 5) determine the number of decimal places in the answer based on the number in the factors. The final answer is 3.45.
The document provides step-by-step instructions for multiplying decimals. It shows the multiplication of 2.3 x 1.5 worked out over multiple steps: 1) multiply the ones, 2) add a placeholder zero, 3) multiply the tens, 4) add the products, and 5) determine the number of decimal places in the answer based on the number in the factors. The final answer is 3.45.
5. multiplying 2 to 4 digit numbers by 1-to 2-digitAnnie Villamer
1) The document provides instructions and examples for multiplying 2-4 digit numbers by 1-2 digit numbers without regrouping. It includes step-by-step worked examples and exercises.
2) Students are asked to solve word problems that require multiplying numbers with 2-4 digits by 1-2 digits to find totals, lengths, or amounts spent.
3) The assignment asks students to solve 5 word problems that require multiplying given amounts, lengths, or prices to determine totals, lengths, or amounts spent.
This document provides notes and practice problems about the order of operations. It explains that parentheses, exponents, multiplication, division, addition, and subtraction have a specific order in which they must be evaluated when multiple operations are present in an expression. Students are given examples of each operation and a mnemonic to remember the order of operations as PEMDAS, which stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. Guided practice problems are included for students to evaluate expressions using the proper order of operations.
The document discusses the order of operations, which is a rule for evaluating mathematical expressions containing multiple operations. It explains that the order of operations is: 1) Evaluate expressions inside grouping symbols (parentheses, brackets, braces) from innermost to outermost. 2) Evaluate exponents. 3) Perform multiplication and division from left to right. 4) Perform addition and subtraction from left to right. Mnemonics like PEMDAS are provided to help remember the order. Examples of applying the order of operations to evaluate expressions are provided.
This document discusses different shapes including two-dimensional shapes like squares, rectangles, and triangles, and three-dimensional shapes like cubes, cylinders, and spheres. It provides examples of how these shapes appear in real life objects like clocks, soup cans, and computers. The document also describes how shapes can be combined by addition or taken apart, and how they can be rotated, flipped or slid. It suggests that understanding shapes is important for math, reading, and writing skills.
The document discusses the order of operations rules for evaluating arithmetic expressions. It explains that the order is: 1) operations within grouping symbols, 2) exponents, 3) multiplication and division from left to right, 4) addition and subtraction from left to right. Following the order of operations is important so that everyone calculates expressions the same way and achieves consistent results. The document provides examples of evaluating expressions with different operations like exponents, grouping symbols, and fractions.
The document explains the order of operations (PEMDAS) for evaluating mathematical expressions. PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. It should be followed from left to right. Parentheses contain expressions that are evaluated first. Exponents represent multiplication and are done before multiplication and division from left to right. The same order applies to multiplication/division and addition/subtraction from left to right. An example equation is worked through step-by-step to demonstrate PEMDAS.
Social networking connects individuals through relationships and common interests. Facebook allows users to create profiles and communicate with friends through messages and chat. Twitter enables users to send and read short text-based posts called tweets. Blogs are websites or parts of websites where individuals regularly post entries like commentary or descriptions that are displayed in reverse chronological order. Heello also allows users to send posts and share pictures and videos. Social networking knowledge continues to expand rapidly as the technology develops and provides useful ways for people to connect in their everyday lives.
A 3 m 1 estrategias para la efectividad de la educación a distanciaIsaias Velazquez
El documento describe estrategias para mejorar la efectividad de la educación a distancia, incluyendo la búsqueda y manejo de información en línea, como el uso de directorios electrónicos y correo electrónico. También discute técnicas de estudio como el resumen y cuadro sinóptico, los cuales ayudan a organizar la información, sintetizar ideas principales, y ahorrar tiempo en el estudio y repaso de temas.
The document discusses three articles related to using the internet in education. Article 1 focuses on using teacher podcasts to help students proofread and revise blog posts. Article 2 details how blogging can promote higher order thinking and provides a framework called HOT blogging. Article 3 emphasizes the importance of teaching students how to effectively search for information online using the acronym SEARCH.
The document summarizes the clothing choices for characters portrayed as bullies and their victim in a story. The bullies are dressed in blue sports tops and shorts or a blue top and black shorts to convey strength, power, and a bold personality. Their clothing includes Nike brands. The victim, Sam, is dressed more formally in a school blazer, white shirt, black trousers, and glasses to portray a smart but outsider image.
This document provides a list of different camera shots and angles including close ups, long shots, two shots, low and high angle shots, extreme close ups, over the shoulder shots, very long shots, and medium close ups with some having connotations of nature, isolation, friendship, and stress.
This document provides guidance for students creating a film opening as an assessment task. It outlines 9 steps for the process, including taking stock of the requirements, setting up documentation of the process, building skills in areas like camera work and editing, investigating what makes effective openings, brainstorming ideas, planning shots and logistics, filming, editing, and evaluating the final product. Students are advised to allow enough time, keep evidence of their work, experiment, get feedback, and focus on simplicity while representing their film.
El método Montessori se basa en el respeto al desarrollo natural del niño y en darle libertad para explorar y aprender por sí mismo a través de materiales didácticos. La Dra. Montessori observó que los niños son capaces de absorber conocimientos espontáneamente como "esponjas" y desarrolló actividades prácticas para apoyar su aprendizaje autodirigido en lectura, escritura y matemáticas antes de los 6 años. El método enfatiza la autonomía, independencia y autodisciplina del ni
The document provides step-by-step instructions for multiplying decimals by whole numbers. It demonstrates multiplying 0.12 x 5, 9.74 x 8, and 219 x 7.2. For each problem, it lists the steps of multiplying the numbers, placing placeholders, adding products, and including the correct number of decimal places in the final answer.
The document shows the step-by-step work for multiplying 28 x 12. It breaks 12 into 10 and 2. It then multiplies 28 by 10 to get 280, and 28 by 2 to get 56. Finally, it adds 280 + 56 to get the total of 336.
The document provides step-by-step instructions for multiplying multi-digit numbers. It shows worked examples of 28 x 12 and 354 x 23. For each problem, it breaks the multiplication into steps: 1) multiply the ones, 2) add a placeholder zero, 3) multiply the tens, 4) add the partial products. The goal is to demonstrate the standard algorithm for multiplying multi-digit numbers in a clear and organized manner.
4-7 Multiplying by One-Digit and Two-Digit NumbersRudy Alfonso
The document shows the step-by-step work for multiplying 28 x 12. It breaks 12 into 10 and 2. It then multiplies 28 by 10 to get 280, and 28 by 2 to get 56. Finally, it adds 280 + 56 to get the total of 336.
4-7 Multiplying by One-Digit and Two-Digit NumbersRudy Alfonso
The document shows the step-by-step work for multiplying 28 x 12. It breaks 12 into 10 and 2. It then multiplies 28 by 10 to get 280, and 28 by 2 to get 56. Finally, it adds 280 + 56 to get the total of 336.
The document provides step-by-step instructions for multiplying decimals. It shows the multiplication of 2.3 x 1.5 worked out over multiple steps: 1) multiply the ones, 2) add a placeholder zero, 3) multiply the tens, 4) add the products, and 5) determine the number of decimal places in the answer based on the number in the factors. The final answer is 3.45.
The document provides step-by-step instructions for multiplying decimals. It shows the multiplication of 2.3 x 1.5 worked out over multiple steps: 1) multiply the ones, 2) add a placeholder zero, 3) multiply the tens, 4) add the products, and 5) determine the number of decimal places in the answer based on the number in the factors. The final answer is 3.45.
The document is a lecture on inverse trigonometric functions from a Calculus I class at New York University. It defines inverse trigonometric functions as the inverses of restricted trigonometric functions, gives their domains and ranges, and discusses their derivatives. The document also provides examples of evaluating inverse trigonometric functions.
Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
This document summarizes a calculus lecture on linear approximations. It provides examples of using the tangent line to approximate the sine function at different points. Specifically, it estimates sin(61°) by taking linear approximations about 0 and about 60°. The linear approximation about 0 is x, giving a value of 1.06465. The linear approximation about 60° uses the fact that the sine is √3/2 and the derivative is √3/2 at π/3, giving a better approximation than using 0.
This document contains lecture notes on related rates from a Calculus I class at New York University. It begins with announcements about assignments and no class on a holiday. It then outlines the objectives of learning to use derivatives to understand rates of change and model word problems. Examples are provided, including an oil slick problem worked out in detail. Strategies for solving related rates problems are discussed. Further examples on people walking and electrical resistors are presented.
Lesson 14: Derivatives of Logarithmic and Exponential FunctionsMel Anthony Pepito
The document discusses conventions for defining exponential functions with exponents other than positive integers, such as negative exponents, fractional exponents, and exponents of zero. It defines exponential functions with these exponent types in a way that maintains important properties like ax+y = ax * ay. The goal is to extend the definition of exponential functions beyond positive integer exponents in a principled way.
This document contains lecture notes on exponential growth and decay from a Calculus I class at New York University. It begins with announcements about an upcoming review session, office hours, and midterm exam. It then outlines the topics to be covered, including the differential equation y=ky, modeling population growth, radioactive decay including carbon-14 dating, Newton's law of cooling, and continuously compounded interest. Examples are provided of solving various differential equations representing exponential growth or decay. The document explains that many real-world situations exhibit exponential behavior due to proportional growth rates.
This document is from a Calculus I class lecture on L'Hopital's rule given at New York University. It begins with announcements and objectives for the lecture. It then provides examples of limits that are indeterminate forms to motivate L'Hopital's rule. The document explains the rule and when it can be used to evaluate limits. It also introduces the mathematician L'Hopital and applies the rule to examples introduced earlier.
The document provides an overview of curve sketching in calculus including objectives, rationale, theorems for determining monotonicity and concavity, and a checklist for graphing functions. It then gives examples of graphing cubic and quartic functions step-by-step, demonstrating how to analyze critical points, inflection points, and asymptotic behavior to create the curve. The examples illustrate applying differentiation rules to determine monotonicity from the derivative sign chart and concavity from the second derivative test.
This document is a section from a Calculus I course at New York University covering maximum and minimum values. It begins with announcements about exams and assignments. The objectives are to understand the Extreme Value Theorem and Fermat's Theorem, and use the Closed Interval Method to find extreme values. The document then covers the definitions of extreme points/values and the statements of the Extreme Value Theorem and Fermat's Theorem. It provides examples to illustrate the necessity of the hypotheses in the theorems. The focus is on using calculus concepts like continuity and differentiability to determine maximum and minimum values of functions on closed intervals.
This document is the notes from a Calculus I class at New York University covering Section 4.2 on the Mean Value Theorem. The notes include objectives, an outline, explanations of Rolle's Theorem and the Mean Value Theorem, examples of using the theorems, and a food for thought question. The key points are that Rolle's Theorem states that if a function is continuous on an interval and differentiable inside the interval, and the function values at the endpoints are equal, then there exists a point in the interior where the derivative is 0. The Mean Value Theorem similarly states that if a function is continuous on an interval and differentiable inside, there exists a point where the average rate of change equals the instantaneous rate of
This document discusses the definite integral and its properties. It begins by stating the objectives of computing definite integrals using Riemann sums and limits, estimating integrals using approximations like the midpoint rule, and reasoning about integrals using their properties. The outline then reviews the integral as a limit of Riemann sums and how to estimate integrals. It also discusses properties of the integral and comparison properties. Finally, it restates the theorem that if a function is continuous, the limit of Riemann sums is the same regardless of the choice of sample points.
The document summarizes the steps to solve optimization problems using calculus. It begins with an example of finding the rectangle with maximum area given a fixed perimeter. It works through the solution, identifying the objective function, variables, constraints, and using calculus techniques like taking the derivative to find critical points. The document then outlines Polya's 4-step method for problem solving and provides guidance on setting up optimization problems by understanding the problem, introducing notation, drawing diagrams, and eliminating variables using given constraints. It emphasizes using the Closed Interval Method, evaluating the function at endpoints and critical points to determine maximums and minimums.
The document is about calculating areas and distances using calculus. It discusses approximating areas of curved regions by dividing them into rectangles and letting the number of rectangles approach infinity. It provides examples of calculating areas of basic shapes like rectangles, parallelograms, and triangles. It then discusses Archimedes' work approximating the area under a parabola by inscribing sequences of triangles. The objectives are to compute areas using limits of approximating rectangles and to compute distances as limits of approximating time intervals.
This document is from a Calculus I class at New York University and covers antiderivatives. It begins with announcements about an upcoming quiz. The objectives are to find antiderivatives of simple functions, remember that a function whose derivative is zero must be constant, and solve rectilinear motion problems. It then outlines finding antiderivatives through tabulation, graphically, and with rectilinear motion examples. The document provides examples of finding antiderivatives of power functions by using the power rule in reverse.
- The document is about evaluating definite integrals and contains sections on: evaluating definite integrals using the evaluation theorem, writing antiderivatives as indefinite integrals, interpreting definite integrals as net change over an interval, and examples of computing definite integrals.
- It discusses properties of definite integrals such as additivity and comparison properties, and provides examples of definite integrals that can be evaluated using known area formulas or by direct computation of antiderivatives.
Here are the key points about g given f:
- g represents the area under the curve of f over successive intervals of the x-axis
- As x increases over an interval, g will increase if f is positive over that interval and decrease if f is negative
- The concavity (convexity or concavity) of g will match the concavity of f over each interval
In summary, the area function g, as defined by the integral of f, will have properties that correspond directly to the sign and concavity of f over successive intervals of integration.
This document provides information about a Calculus I course taught by Professor Matthew Leingang at the Courant Institute of Mathematical Sciences at NYU. The course will include weekly lectures, recitations, homework assignments, quizzes, a midterm exam, and a final exam. Grades will be determined based on exam, homework, and quiz scores. The required textbook is available in hardcover, looseleaf, or online formats through the campus bookstore or WebAssign. Students are encouraged to contact the professor or TAs with any questions.
The document is a section from a Calculus I course at NYU from June 22, 2010. It discusses using the substitution method to evaluate indefinite integrals. Specifically, it provides an example of using the substitution u=x^2+3 to evaluate the integral of (x^2+3)^3 4x dx. The solution transforms the integral into an integral of u^3 du and evaluates it to be 1/4 u^4 + C.
This document outlines information about a Calculus I course taught by Professor Matthew Leingang at the Courant Institute of Mathematical Sciences. It provides details on the course staff, contact information for the professor, an overview of assessments including homework, quizzes, a midterm and final exam. Grading breakdown is also included, as well as information on purchasing the required textbook and accessing the course on Blackboard. The document aims to provide students with essential logistical information to succeed in the Calculus I course.
This document contains lecture notes from a Calculus I class at New York University on September 14, 2010. The notes cover announcements, guidelines for written homework, a rubric for grading homework, examples of good and bad homework, and objectives for the concept of limits. The bulk of the document discusses the heuristic definition of a limit using an error-tolerance game approach, provides examples to illustrate the game, and outlines the path to a precise definition of a limit.
5. STEP ZERO:
Use guidelines
Find 365 x 112 STEP ONE:
Multiply by
to keep work
the ones. Regroup
365
neat.
if necessary.
x 1 12
6. STEP ZERO:
Use guidelines
Find 365 x 112 STEP ONE:
Multiply by
to keep work
the ones. Regroup
365
neat.
if necessary.
x 1 12
7. STEP ZERO: Find 365 x 112 STEP ONE:
1
Use guidelines
Multiply by
to keep work
the ones. Regroup
365
neat.
if necessary.
x 1 12
0
8. STEP ZERO: Find 365 x 112 STEP ONE:
1
Use guidelines
Multiply by
to keep work
the ones. Regroup
365
neat.
if necessary.
x 1 12
0
9. STEP ZERO: Find 365 x 112 STEP ONE:
11
Use guidelines
Multiply by
to keep work
the ones. Regroup
365
neat.
if necessary.
x 1 12
30
10. STEP ZERO: Find 365 x 112 STEP ONE:
11
Use guidelines
Multiply by
to keep work
the ones. Regroup
365
neat.
if necessary.
x 1 12
30
11. STEP ZERO: Find 365 x 112 STEP ONE:
11
Use guidelines
Multiply by
to keep work
the ones. Regroup
365
neat.
if necessary.
x 1 12
73 0
12. STEP ZERO: Find 365 x 112 STEP ONE:
11
Use guidelines
Multiply by
to keep work
the ones. Regroup
365
neat.
if necessary.
x 1 12 STEP TWO:
73 0
Place a zero
in the ones place.
(PLACEHOLDER)
Scratch out
previous work.
13. STEP ZERO:
Use guidelines
Find 365 x 112 STEP ONE:
Multiply by
to keep work
the ones. Regroup
365
neat.
if necessary.
x 1 12 STEP TWO:
73 0
Place a zero
in the ones place.
(PLACEHOLDER)
0 Scratch out
previous work.
14. STEP ZERO:
Use guidelines
Find 365 x 112 STEP ONE:
Multiply by
to keep work
the ones. Regroup
365
neat.
if necessary.
STEP THREE:
x 1 12
Multiply by the
tens. Regroup STEP TWO:
if necessary.
73 0
Place a zero
in the ones place.
(PLACEHOLDER)
0 Scratch out
previous work.
15. STEP ZERO:
Use guidelines
Find 365 x 112 STEP ONE:
Multiply by
to keep work
the ones. Regroup
365
neat.
if necessary.
STEP THREE:
x 1 12
Multiply by the
tens. Regroup STEP TWO:
if necessary.
73 0
Place a zero
in the ones place.
(PLACEHOLDER)
0 Scratch out
previous work.
16. STEP ZERO:
Use guidelines
Find 365 x 112 STEP ONE:
Multiply by
to keep work
the ones. Regroup
365
neat.
if necessary.
STEP THREE:
x 1 12
Multiply by the
tens. Regroup STEP TWO:
if necessary.
73 0
Place a zero
in the ones place.
(PLACEHOLDER)
50 Scratch out
previous work.
17. STEP ZERO:
Use guidelines
Find 365 x 112 STEP ONE:
Multiply by
to keep work
the ones. Regroup
365
neat.
if necessary.
STEP THREE:
x 1 12
Multiply by the
tens. Regroup STEP TWO:
if necessary.
73 0
Place a zero
in the ones place.
(PLACEHOLDER)
50 Scratch out
previous work.
18. STEP ZERO:
Use guidelines
Find 365 x 112 STEP ONE:
Multiply by
to keep work
the ones. Regroup
365
neat.
if necessary.
STEP THREE:
x 1 12
Multiply by the
tens. Regroup STEP TWO:
if necessary.
73 0
Place a zero
in the ones place.
(PLACEHOLDER)
6 50 Scratch out
previous work.
19. STEP ZERO:
Use guidelines
Find 365 x 112 STEP ONE:
Multiply by
to keep work
the ones. Regroup
365
neat.
if necessary.
STEP THREE:
x 1 12
Multiply by the
tens. Regroup STEP TWO:
if necessary.
73 0
Place a zero
in the ones place.
(PLACEHOLDER)
6 50 Scratch out
previous work.
20. STEP ZERO:
Use guidelines
Find 365 x 112 STEP ONE:
Multiply by
to keep work
the ones. Regroup
365
neat.
if necessary.
STEP THREE:
x 1 12
Multiply by the
tens. Regroup STEP TWO:
if necessary.
73 0
Place a zero
in the ones place.
(PLACEHOLDER)
36 50 Scratch out
previous work.
21. STEP ZERO:
Use guidelines
Find 365 x 112 STEP ONE:
Multiply by
to keep work
the ones. Regroup
365
neat.
if necessary.
STEP THREE:
x 1 12
Multiply by the
tens. Regroup STEP TWO:
if necessary.
73 0
Place a zero
in the ones place.
(PLACEHOLDER)
36 50 Scratch out
previous work.
STEP FOUR:
Place two zeroes
in the ones/tens place.
(PLACEHOLDERS)
Scratch out
previous work.
22. STEP ZERO:
Use guidelines
Find 365 x 112 STEP ONE:
Multiply by
to keep work
the ones. Regroup
365
neat.
if necessary.
STEP THREE:
x 1 12
Multiply by the
tens. Regroup STEP TWO:
if necessary.
73 0
Place a zero
in the ones place.
(PLACEHOLDER)
36 50 Scratch out
previous work.
00 STEP FOUR:
Place two zeroes
in the ones/tens place.
(PLACEHOLDERS)
Scratch out
previous work.
23. STEP ZERO:
Use guidelines
Find 365 x 112 STEP ONE:
Multiply by
to keep work
the ones. Regroup
365
neat.
if necessary.
STEP THREE:
x 1 12
Multiply by the
tens. Regroup STEP TWO:
if necessary.
73 0
Place a zero
in the ones place.
STEP FIVE: (PLACEHOLDER)
Multiply by the
hundreds. Regroup 36 50 Scratch out
previous work.
if necessary.
00 STEP FOUR:
Place two zeroes
in the ones/tens place.
(PLACEHOLDERS)
Scratch out
previous work.
24. STEP ZERO:
Use guidelines
Find 365 x 112 STEP ONE:
Multiply by
to keep work
the ones. Regroup
365
neat.
if necessary.
STEP THREE:
x 1 12
Multiply by the
tens. Regroup STEP TWO:
if necessary.
73 0
Place a zero
in the ones place.
STEP FIVE: (PLACEHOLDER)
Multiply by the
hundreds. Regroup 36 50 Scratch out
previous work.
if necessary.
00 STEP FOUR:
Place two zeroes
in the ones/tens place.
(PLACEHOLDERS)
Scratch out
previous work.
25. STEP ZERO:
Use guidelines
Find 365 x 112 STEP ONE:
Multiply by
to keep work
the ones. Regroup
365
neat.
if necessary.
STEP THREE:
x 1 12
Multiply by the
tens. Regroup STEP TWO:
if necessary.
73 0
Place a zero
in the ones place.
STEP FIVE: (PLACEHOLDER)
Multiply by the
hundreds. Regroup 36 50 Scratch out
previous work.
if necessary.
500 STEP FOUR:
Place two zeroes
in the ones/tens place.
(PLACEHOLDERS)
Scratch out
previous work.
26. STEP ZERO:
Use guidelines
Find 365 x 112 STEP ONE:
Multiply by
to keep work
the ones. Regroup
365
neat.
if necessary.
STEP THREE:
x 1 12
Multiply by the
tens. Regroup STEP TWO:
if necessary.
73 0
Place a zero
in the ones place.
STEP FIVE: (PLACEHOLDER)
Multiply by the
hundreds. Regroup 36 50 Scratch out
previous work.
if necessary.
500 STEP FOUR:
Place two zeroes
in the ones/tens place.
(PLACEHOLDERS)
Scratch out
previous work.
27. STEP ZERO:
Use guidelines
Find 365 x 112 STEP ONE:
Multiply by
to keep work
the ones. Regroup
365
neat.
if necessary.
STEP THREE:
x 1 12
Multiply by the
tens. Regroup STEP TWO:
if necessary.
73 0
Place a zero
in the ones place.
STEP FIVE: (PLACEHOLDER)
Multiply by the
hundreds. Regroup 36 50 Scratch out
previous work.
if necessary.
6 500 STEP FOUR:
Place two zeroes
in the ones/tens place.
(PLACEHOLDERS)
Scratch out
previous work.
28. STEP ZERO:
Use guidelines
Find 365 x 112 STEP ONE:
Multiply by
to keep work
the ones. Regroup
365
neat.
if necessary.
STEP THREE:
x 1 12
Multiply by the
tens. Regroup STEP TWO:
if necessary.
73 0
Place a zero
in the ones place.
STEP FIVE: (PLACEHOLDER)
Multiply by the
hundreds. Regroup 36 50 Scratch out
previous work.
if necessary.
6 500 STEP FOUR:
Place two zeroes
in the ones/tens place.
(PLACEHOLDERS)
Scratch out
previous work.
29. STEP ZERO:
Use guidelines
Find 365 x 112 STEP ONE:
Multiply by
to keep work
the ones. Regroup
365
neat.
if necessary.
STEP THREE:
x 1 12
Multiply by the
tens. Regroup STEP TWO:
if necessary.
73 0
Place a zero
in the ones place.
STEP FIVE: (PLACEHOLDER)
Multiply by the
hundreds. Regroup 36 50 Scratch out
previous work.
if necessary.
36 500 STEP FOUR:
Place two zeroes
in the ones/tens place.
(PLACEHOLDERS)
Scratch out
previous work.
30. STEP ZERO:
Use guidelines
Find 365 x 112 STEP ONE:
Multiply by
to keep work
the ones. Regroup
365
neat.
if necessary.
STEP THREE:
x 1 12
Multiply by the
tens. Regroup STEP TWO:
if necessary.
73 0
Place a zero
in the ones place.
STEP FIVE: (PLACEHOLDER)
Multiply by the
hundreds. Regroup 36 50 Scratch out
previous work.
if necessary.
36 500 STEP FOUR:
Place two zeroes
STEP SIX:
in the ones/tens place.
Add the products.
(PLACEHOLDERS)
Regroup if necessary. Scratch out
previous work.
31. STEP ZERO:
Use guidelines
Find 365 x 112 STEP ONE:
Multiply by
to keep work
the ones. Regroup
365
neat.
if necessary.
STEP THREE:
x 1 12
Multiply by the
tens. Regroup STEP TWO:
if necessary.
73 0
Place a zero
in the ones place.
STEP FIVE: (PLACEHOLDER)
Multiply by the
hundreds. Regroup 36 50 Scratch out
previous work.
if necessary.
STEP SIX:
+ 36 500 STEP FOUR:
Place two zeroes
in the ones/tens place.
Add the products.
(PLACEHOLDERS)
Regroup if necessary. Scratch out
previous work.
32. STEP ZERO:
Use guidelines
Find 365 x 112 STEP ONE:
Multiply by
to keep work
the ones. Regroup
365
neat.
if necessary.
STEP THREE:
x 1 12
Multiply by the
tens. Regroup STEP TWO:
if necessary.
1 73 0
Place a zero
in the ones place.
STEP FIVE: (PLACEHOLDER)
Multiply by the
hundreds. Regroup 1 36 50 Scratch out
previous work.
if necessary.
+ 36 500 STEP FOUR:
4 0880
Place two zeroes
STEP SIX:
in the ones/tens place.
Add the products.
(PLACEHOLDERS)
Regroup if necessary. Scratch out
previous work.
34. STEP ZERO:
Find 287 x 104
Use guidelines
287
to keep work
neat.
x 104
35. STEP ZERO:
Find 287 x 104
Use guidelines
287
to keep work
neat.
x 104
36. STEP ZERO:
Find 287 x 104 STEP ONE:
Multiply by
Use guidelines the ones. Regroup
287
to keep work if necessary.
neat.
x 104
37. STEP ZERO:
Find 287 x 104 STEP ONE:
Multiply by
Use guidelines the ones. Regroup
287
to keep work if necessary.
neat.
x 104
38. Find 287 x 104 STEP ONE:
2
STEP ZERO: Multiply by
Use guidelines the ones. Regroup
287
to keep work if necessary.
neat.
x 104
8
39. Find 287 x 104 STEP ONE:
2
STEP ZERO: Multiply by
Use guidelines the ones. Regroup
287
to keep work if necessary.
neat.
x 104
8
40. Find 287 x 104 STEP ONE:
32
STEP ZERO: Multiply by
Use guidelines the ones. Regroup
287
to keep work if necessary.
neat.
x 104
48
41. Find 287 x 104 STEP ONE:
32
STEP ZERO: Multiply by
Use guidelines the ones. Regroup
287
to keep work if necessary.
neat.
x 104
48
42. Find 287 x 104 STEP ONE:
32
STEP ZERO: Multiply by
Use guidelines the ones. Regroup
287
to keep work if necessary.
neat.
x 104
11 4 8
43. Find 287 x 104 STEP ONE:
32
STEP ZERO: Multiply by
Use guidelines the ones. Regroup
287
to keep work if necessary.
neat.
x 104 STEP TWO:
11 4 8 Show a row of
PLACEHOLDERS.
Scratch out
previous work.
44. STEP ZERO:
Find 287 x 104 STEP ONE:
Multiply by
Use guidelines the ones. Regroup
287
to keep work if necessary.
neat.
x 104 STEP TWO:
11 4 8 Show a row of
PLACEHOLDERS.
00 00
Scratch out
previous work.
45. STEP ZERO:
Find 287 x 104 STEP ONE:
Multiply by
Use guidelines the ones. Regroup
287
to keep work if necessary.
neat.
x 104 STEP TWO:
11 4 8 Show a row of
PLACEHOLDERS.
00 00
Scratch out
previous work.
STEP THREE:
Place two zeroes
in the ones/tens place.
(PLACEHOLDERS)
Scratch out
previous work.
46. STEP ZERO:
Find 287 x 104 STEP ONE:
Multiply by
Use guidelines the ones. Regroup
287
to keep work if necessary.
neat.
x 104 STEP TWO:
11 4 8 Show a row of
PLACEHOLDERS.
00 00
Scratch out
previous work.
00 STEP THREE:
Place two zeroes
in the ones/tens place.
(PLACEHOLDERS)
Scratch out
previous work.
47. STEP ZERO:
Find 287 x 104 STEP ONE:
Multiply by
Use guidelines the ones. Regroup
287
to keep work if necessary.
neat.
x 104 STEP TWO:
11 4 8
STEP FOUR:
Show a row of
Multiply by the
PLACEHOLDERS.
hundreds. Regroup
00 00
Scratch out
if necessary. previous work.
00 STEP THREE:
Place two zeroes
in the ones/tens place.
(PLACEHOLDERS)
Scratch out
previous work.
48. STEP ZERO:
Find 287 x 104 STEP ONE:
Multiply by
Use guidelines the ones. Regroup
287
to keep work if necessary.
neat.
x 104 STEP TWO:
11 4 8
STEP FOUR:
Show a row of
Multiply by the
PLACEHOLDERS.
hundreds. Regroup
00 00
Scratch out
if necessary. previous work.
00 STEP THREE:
Place two zeroes
in the ones/tens place.
(PLACEHOLDERS)
Scratch out
previous work.
49. STEP ZERO:
Find 287 x 104 STEP ONE:
Multiply by
Use guidelines the ones. Regroup
287
to keep work if necessary.
neat.
x 104 STEP TWO:
11 4 8
STEP FOUR:
Show a row of
Multiply by the
PLACEHOLDERS.
hundreds. Regroup
00 00
Scratch out
if necessary. previous work.
700 STEP THREE:
Place two zeroes
in the ones/tens place.
(PLACEHOLDERS)
Scratch out
previous work.
50. STEP ZERO:
Find 287 x 104 STEP ONE:
Multiply by
Use guidelines the ones. Regroup
287
to keep work if necessary.
neat.
x 104 STEP TWO:
11 4 8
STEP FOUR:
Show a row of
Multiply by the
PLACEHOLDERS.
hundreds. Regroup
00 00
Scratch out
if necessary. previous work.
700 STEP THREE:
Place two zeroes
in the ones/tens place.
(PLACEHOLDERS)
Scratch out
previous work.
51. STEP ZERO:
Find 287 x 104 STEP ONE:
Multiply by
Use guidelines the ones. Regroup
287
to keep work if necessary.
neat.
x 104 STEP TWO:
11 4 8
STEP FOUR:
Show a row of
Multiply by the
PLACEHOLDERS.
hundreds. Regroup
00 00
Scratch out
if necessary. previous work.
8 700 STEP THREE:
Place two zeroes
in the ones/tens place.
(PLACEHOLDERS)
Scratch out
previous work.
52. STEP ZERO:
Find 287 x 104 STEP ONE:
Multiply by
Use guidelines the ones. Regroup
287
to keep work if necessary.
neat.
x 104 STEP TWO:
11 4 8
STEP FOUR:
Show a row of
Multiply by the
PLACEHOLDERS.
hundreds. Regroup
00 00
Scratch out
if necessary. previous work.
8 700 STEP THREE:
Place two zeroes
in the ones/tens place.
(PLACEHOLDERS)
Scratch out
previous work.
53. STEP ZERO:
Find 287 x 104 STEP ONE:
Multiply by
Use guidelines the ones. Regroup
287
to keep work if necessary.
neat.
x 104 STEP TWO:
11 4 8
STEP FOUR:
Show a row of
Multiply by the
PLACEHOLDERS.
hundreds. Regroup
00 00
Scratch out
if necessary. previous work.
28 700 STEP THREE:
Place two zeroes
in the ones/tens place.
(PLACEHOLDERS)
Scratch out
previous work.
54. STEP ZERO:
Find 287 x 104 STEP ONE:
Multiply by
Use guidelines the ones. Regroup
287
to keep work if necessary.
neat.
x 104 STEP TWO:
11 4 8
STEP FOUR:
Show a row of
Multiply by the
PLACEHOLDERS.
hundreds. Regroup
00 00
Scratch out
if necessary. previous work.
STEP FIVE:
28 700 STEP THREE:
Place two zeroes
Add the products.
in the ones/tens place.
Regroup if necessary.
(PLACEHOLDERS)
Scratch out
previous work.
55. STEP ZERO:
Find 287 x 104 STEP ONE:
Multiply by
Use guidelines the ones. Regroup
287
to keep work if necessary.
neat.
x 104 STEP TWO:
11 4 8
STEP FOUR:
Show a row of
Multiply by the
PLACEHOLDERS.
hundreds. Regroup
00 00
Scratch out
if necessary. previous work.
STEP FIVE:
Add the products.
+ 28 700 STEP THREE:
Place two zeroes
in the ones/tens place.
Regroup if necessary.
(PLACEHOLDERS)
Scratch out
previous work.
56. STEP ZERO:
Find 287 x 104 STEP ONE:
Multiply by
Use guidelines the ones. Regroup
287
to keep work if necessary.
neat.
x 104 STEP TWO:
11 4 8
STEP FOUR:
Show a row of
Multiply by the
PLACEHOLDERS.
hundreds. Regroup
00 00
Scratch out
if necessary. previous work.
STEP FIVE: + 28 700 STEP THREE:
2 9848
Place two zeroes
Add the products.
in the ones/tens place.
Regroup if necessary.
(PLACEHOLDERS)
Scratch out
previous work.