Sample3
0 likes17 views
This document contains a 3 question math quiz for Math 231 in Spring 2015. The first question asks to define the Taylor series of a function f(x) at a point c. The second asks to state the precise definition of the binomial theorem. The third asks to find the McLaurin series of the function f(x) = x2/(1 + x3) and use it to evaluate f(14) at 0.
1 of 1
Download to read offline
Recommended
Final exam mariluz 1
Final exam mariluz 1Maestra Luz Torres
油
1. This document is a final exam for a bilingual high school differential calculus course. It consists of 10 multiple choice and free response questions testing concepts like limits, derivatives, optimization, and graph sketching.
2. The exam is worth a total of 100 points and will be graded based on various components like exams, homework, quizzes and class activities. Partial credit is given for showing work even if the final answer is incorrect.
3. Questions cover topics like finding the domain and range of functions, limits, derivatives using definitions and rules, implicit differentiation, optimization, related rates, and graph sketching through identifying key features.graphs plotting in MATLAB
graphs plotting in MATLABApurva Patil
油
This document provides code examples for creating different types of plots in MATLAB, including line plots, bar plots, polar plots, and pie charts. It shows how to generate a sine wave plot and customize it by adding titles, labels, legends and adjusting axes. It also demonstrates how to create subplot layouts to show multiple graphs together and explores functions like bar, stairs, compass and pie for generating other common plot types from data.Limits2
Limits2maryjaneo2003
油
The document defines three functions f(x) with different values of 2, 0, and does not exist. It also provides an end behavior model of a polynomial function 2x^3+x^2+x+1 as 2x^3 and identifies several function types including jump, removable, infinite, and piecewise. It defines a function (x)=x^2 with a condition of x=3.2.2 Polynomial Function Notes
2.2 Polynomial Function Noteslgemgnani
油
The document discusses polynomial functions, including how to graph common polynomials, find zeros of polynomials, and write polynomials given their roots. It provides examples of matching polynomial equations to their graphs, finding the real zeros of polynomials by factoring, and writing polynomials when given the roots. The document also covers how to use a graphing calculator to find the zeros of polynomials.125 5.4
125 5.4Jeneva Clark
油
The document defines a definite integral as the integral of a function over a bounded interval from a to b, written as f(x)dx from a to b. This represents the area under the curve of the function f(x) between the bounds a and b. Several examples are provided of calculating definite integrals to find the area under curves over given intervals using the Fundamental Theorem of Calculus. It is noted that definite integrals cannot result in negative area values.7.2 abs value function
7.2 abs value functionMichael Cellini
油
This document provides notes for a math chapter on absolute value and reciprocal functions. It covers graphing and expressing absolute value functions as piecewise functions. Key points include: absolute value is defined as the distance from zero on the number line; absolute value functions are continuous and have the same x-intercept as the linear function inside the absolute value; taking the absolute value of a quadratic function preserves the vertex but changes the range. Care must be taken with the domains of absolute value functions depending on if the coefficient of the x-term inside is positive or negative.Radical functions
Radical functionsTarun Gehlot
油
1) A radical function is of the form y = f(x) = ax + b, where changing a and b affects the graph.
2) Graphing shows that if a > 0 the graph increases, if a < 0 the graph decreases, larger a makes the graph steeper, and closer to 0 makes the graph flatter.
3) The value of b is the y-intercept, and the domain is all x 0 while the range is all y above or below b depending on if the graph increases or decreases.Lesson 1: Functions and their representations (slides)
Lesson 1: Functions and their representations (slides)Matthew Leingang
油
The function is the fundamental unit in calculus. There are many ways to describe functions: with words, pictures, symbols, or numbers.
Difrentiation
Difrentiationlecturer
油
This document discusses differentiation and defines the derivative. It begins by formally defining the derivative as a limit and then provides formulas to find the derivatives of simple functions like constants, linear functions, and power functions. It also covers numerical derivatives, implicit differentiation, and higher-order derivatives. Examples are provided to illustrate each concept.Exercise set 3.5
Exercise set 3.5math265
油
1. Graph and analyze the critical points, extrema, inflection points, intervals of increasing/decreasing, and intervals of concave up/down for 10 functions.
2. Review homework on finding derivatives using the definition of the difference quotient and evaluating limits. Find the derivatives of 6 functions.
3. Use implicit differentiation to find the derivative of one function defined implicitly and to find points with tangent lines of slope 1 for another implicit function.
4. Find the second derivatives of two functions.Math1000 section2.1
Math1000 section2.1StuartJones92
油
This document contains lecture notes from a Math 1000 class taught by Stuart Jones. It covers several topics related to functions, including definitions of functions, evaluating functions, piecewise functions, and the difference quotient. Key points covered include the definition of a function as a mapping from a domain to a range, evaluating functions at different inputs, how piecewise functions use different formulas in different intervals, and the formula for the difference quotient.Functions
FunctionsMalikahmad105
油
The document discusses functions and relations. It defines functions, relations, and domain and range. It provides examples of expressing relations in set notation, tabular form, equations, graphs, and mappings. It also discusses evaluating, adding, multiplying, dividing, and composing functions. Graphs of various functions like absolute value, piecewise, greatest integer, and least integer functions are also explained.X2 T04 03 cuve sketching - addition, subtraction, multiplication and division
X2 T04 03 cuve sketching - addition, subtraction, multiplication and divisionNigel Simmons
油
The document outlines methods for graphing functions that are combinations of other functions using addition, subtraction, multiplication, and division. It provides steps for graphing each type of combination function, which include separately graphing the individual functions and then using properties of the combination to determine points and features of the combined graph. An example is worked through for each type of combination function.Matlab differential
Matlab differentialpramodkumar1804
油
The document discusses various methods for computing derivatives and solving differential equations in MATLAB. Some key points:
- MATLAB's diff command is used to compute symbolic derivatives of functions. It can find 1st, 2nd and higher order derivatives.
- Common rules of differentiation like product rule, quotient rule, chain rule are demonstrated. Derivatives of exponential, logarithmic and trigonometric functions are provided.
- Finding critical points/stationary points of a function involves taking the derivative and setting it equal to zero.
- The dsolve command can be used to find symbolic solutions to differential equations, including specifying initial/boundary conditions.Matlab ploting
Matlab plotingAmeen San
油
This document provides an overview of plotting functions in MATLAB. It discusses how to generate basic and 3D plots, customize plots using options like color, style and labels, and control the plot appearance using functions such as axis, title, legend. Examples are given to illustrate plotting a simple function, holding multiple plots, using subplots, and generating surface plots. The document also covers plotting in 3D using functions like surf, plot3 and manipulating axes properties.Polynomial Function and Synthetic Division
Polynomial Function and Synthetic DivisionAleczQ1414
油
This file is about Polynomial Function and Synthetic Division. A project passed to Mrs. Marissa De Ocampo. Submitted by Group 6 of Grade 10-Galilei of Caloocan National Science and Technology High School '15-'16Exponential functions
Exponential functionscarljeffmorris
油
The document discusses exponential functions of the form f(x) = ax, where a is the base. It defines exponential functions and provides examples of evaluating them. The key aspects of exponential graphs are that they increase rapidly as x increases and have a horizontal asymptote of y = 0 if a > 1 or y = 0 if 0 < a < 1. Examples are given of sketching graphs of exponential functions and stating their domains and ranges. The graph of the natural exponential function f(x) = ex is also discussed.Graphs of polynomial functions
Graphs of polynomial functionsCarlos Erepol
油
The document summarizes key characteristics of polynomial functions:
1) Polynomial functions produce smooth, continuous curves on their domains which are the set of real numbers.
2) The graph's x-intercepts, turning points, and absolute/relative maxima and minima are defined.
3) As the degree of a polynomial increases, so do the possible number of x-intercepts and turning points, up to the degree value. The leading coefficient and degree determine whether the graph rises or falls.10. functions
10. functionsUniversity Of Johannesburg
油
The document discusses functions and relations. It defines functions as special relations where each element of the domain is mapped to only one element in the range. It provides examples of determining whether a relation represented by ordered pairs or a graph is a function. It also discusses determining the domain and range of functions from ordered pairs or graphs. Finally, it introduces mapping and functional notation used to represent functions.Relations and functions 
Relations and functions Nabeel Simair
油
The document discusses relations and functions. A relation is defined as a set of ordered pairs, with a domain (set of x-values) and range (set of y-values). A function is a special type of relation where each x-value is assigned to exactly one y-value. Examples of relations and functions are provided to illustrate the differences. Function notation is introduced, where f(x) represents the function f with variable x. The domain of a function is defined as the set of input values that do not result in illegal operations like division by zero or taking the square root of a negative number. Methods for finding the domain of a function are presented.Graphing rational functions
Graphing rational functionsJessica Garcia
油
1. The rational function f(x) = 1/x^2 has a vertical asymptote at x = 2 because the denominator is 0 at that point.
2. As x values approach 2 from either side, y values approach positive or negative infinity, respectively.
3. The graph gets closer and closer to the vertical line x = 2 but never touches it. This line is called a vertical asymptote.
4. The graph also has a horizontal asymptote at y = 0, as y values approach 0 as x values increase or decrease indefinitely.Exercise #11 notes
Exercise #11 notesKelly Scallion
油
This document discusses graphing absolute value functions. It provides examples of graphing various absolute value functions, including f(x)=|x|^2, f(x)=2|x|-1, f(x)=|x|^2-1, f(x)=3|x|^2-4, f(x)=|cos(x)|, f(x)=2|x|-3-1, and shows how to write a piecewise function definition for f(x)=|x|^2. The graphs are V-shaped and symmetric about the y-axis, with vertices at the points where the absolute value terms are equal to zero.Matlab dc circuit analysis
Matlab dc circuit analysisAmeen San
油
This document discusses solving sets of linear equations and analyzing DC circuits. It explains that to solve a set of linear equations in matrix form A*x=y, the matrix A and augmented matrix [A y] must have equal rank. It then discusses calculating the condition number of A, which should be close to 1 for an accurate solution. The document provides an example circuit in matrix form Z*I=V and shows using Matlab commands to calculate the rank of Z and [Z V], the condition number of Z, and solve for the current I.Notes - Graphs of Polynomials
Notes - Graphs of PolynomialsLori Rapp
油
The document discusses graphing and interpreting various polynomial functions using a graphing calculator. It examines the graphs of y=x-1, y=x^2-4, and y=x^3+x^2-12 to note their similarities and differences in terms of degree, number of turns, and x-intercepts. It then explores how changing the coefficients of a cubic function impacts its graph. Finally, it interprets key features of the graph of y=x^3-5x^2+x-5 such as its degree, number of turns, x-intercept, maximum and minimum points.2.7 Curve Sketching Day 2
2.7 Curve Sketching Day 2gagnonator
油
This document discusses curve sketching for the function f(x) = 3x^3 - 18x^2 - 72x + 192. It asks the student to sketch the first and second derivatives of f(x) on the same axes, explains how the second derivative can help determine critical points, and asks why the lesson is titled "The Second Derivative Test". It also provides practice problems applying the second derivative test.1 13s-f
1 13s-fKamarat Kumanukit
油
This document contains the instructions for a Calculus I final exam consisting of two parts. Part I has 10 multiple choice questions worth 5 points each involving derivatives, anti-derivatives, limits, and finding absolute maxima and minima. Part II has 6 longer form problems worth between 1-10 points each involving implicit differentiation, linearization, graphing functions, finding absolute extrema, and a word problem about the speed of a shadow. Students are instructed to show their work and simplify answers where possible to earn full credit.Lesson 20: Derivatives and the Shapes of Curves (slides)
Lesson 20: Derivatives and the Shapes of Curves (slides)Matthew Leingang
油
This document contains lecture notes on derivatives and the shapes of curves from a Calculus I class taught by Professor Matthew Leingang at New York University. The notes cover using derivatives to determine the intervals where a function is increasing or decreasing, classifying critical points as maxima or minima, using the second derivative to determine concavity, and applying the first and second derivative tests. Examples are provided to illustrate finding intervals of monotonicity for various functions.Sample3
Sample3Nima Rasekh
油
This document contains a 4 question math quiz with questions that involve: 1) determining the area between two graphs on an interval, 2) writing integrals to represent volumes obtained by rotating an area around axes, 3) finding the volume of a solid shape with a triangular cross-section, and 4) finding the distance an object travels given its initial velocity and acceleration functions over a time period.Sample2
Sample2Nima Rasekh
油
This document contains a math worksheet with 7 problems involving graphing and analyzing functions. The problems include sketching graphs of functions with given properties, finding examples of functions with vertical tangents or local maxima, graphing polynomial functions, determining intervals where functions are positive or negative, and using templates to analyze additional functions by finding domains, intercepts, asymptotes, critical points, and concavity.Sample1
Sample1Nima Rasekh
油
The document summarizes key topics covered in a Math 220 class, including a review of Pythagorean theorem and Thales' theorem. It then introduces the six basic trigonometric functions of sine, cosine, tangent, cotangent, secant, and cosecant. Various angle relationships for these trig functions are also defined, along with recommended values to memorize for specific angles. Finally, the document defines the trig functions in terms of right triangles with non-unit hypotenuses and provides a graph of the trig functions.More Related Content
What's hot (19)
Difrentiation
Difrentiationlecturer
油
This document discusses differentiation and defines the derivative. It begins by formally defining the derivative as a limit and then provides formulas to find the derivatives of simple functions like constants, linear functions, and power functions. It also covers numerical derivatives, implicit differentiation, and higher-order derivatives. Examples are provided to illustrate each concept.Exercise set 3.5
Exercise set 3.5math265
油
1. Graph and analyze the critical points, extrema, inflection points, intervals of increasing/decreasing, and intervals of concave up/down for 10 functions.
2. Review homework on finding derivatives using the definition of the difference quotient and evaluating limits. Find the derivatives of 6 functions.
3. Use implicit differentiation to find the derivative of one function defined implicitly and to find points with tangent lines of slope 1 for another implicit function.
4. Find the second derivatives of two functions.Math1000 section2.1
Math1000 section2.1StuartJones92
油
This document contains lecture notes from a Math 1000 class taught by Stuart Jones. It covers several topics related to functions, including definitions of functions, evaluating functions, piecewise functions, and the difference quotient. Key points covered include the definition of a function as a mapping from a domain to a range, evaluating functions at different inputs, how piecewise functions use different formulas in different intervals, and the formula for the difference quotient.Functions
FunctionsMalikahmad105
油
The document discusses functions and relations. It defines functions, relations, and domain and range. It provides examples of expressing relations in set notation, tabular form, equations, graphs, and mappings. It also discusses evaluating, adding, multiplying, dividing, and composing functions. Graphs of various functions like absolute value, piecewise, greatest integer, and least integer functions are also explained.X2 T04 03 cuve sketching - addition, subtraction, multiplication and division
X2 T04 03 cuve sketching - addition, subtraction, multiplication and divisionNigel Simmons
油
The document outlines methods for graphing functions that are combinations of other functions using addition, subtraction, multiplication, and division. It provides steps for graphing each type of combination function, which include separately graphing the individual functions and then using properties of the combination to determine points and features of the combined graph. An example is worked through for each type of combination function.Matlab differential
Matlab differentialpramodkumar1804
油
The document discusses various methods for computing derivatives and solving differential equations in MATLAB. Some key points:
- MATLAB's diff command is used to compute symbolic derivatives of functions. It can find 1st, 2nd and higher order derivatives.
- Common rules of differentiation like product rule, quotient rule, chain rule are demonstrated. Derivatives of exponential, logarithmic and trigonometric functions are provided.
- Finding critical points/stationary points of a function involves taking the derivative and setting it equal to zero.
- The dsolve command can be used to find symbolic solutions to differential equations, including specifying initial/boundary conditions.Matlab ploting
Matlab plotingAmeen San
油
This document provides an overview of plotting functions in MATLAB. It discusses how to generate basic and 3D plots, customize plots using options like color, style and labels, and control the plot appearance using functions such as axis, title, legend. Examples are given to illustrate plotting a simple function, holding multiple plots, using subplots, and generating surface plots. The document also covers plotting in 3D using functions like surf, plot3 and manipulating axes properties.Polynomial Function and Synthetic Division
Polynomial Function and Synthetic DivisionAleczQ1414
油
This file is about Polynomial Function and Synthetic Division. A project passed to Mrs. Marissa De Ocampo. Submitted by Group 6 of Grade 10-Galilei of Caloocan National Science and Technology High School '15-'16Exponential functions
Exponential functionscarljeffmorris
油
The document discusses exponential functions of the form f(x) = ax, where a is the base. It defines exponential functions and provides examples of evaluating them. The key aspects of exponential graphs are that they increase rapidly as x increases and have a horizontal asymptote of y = 0 if a > 1 or y = 0 if 0 < a < 1. Examples are given of sketching graphs of exponential functions and stating their domains and ranges. The graph of the natural exponential function f(x) = ex is also discussed.Graphs of polynomial functions
Graphs of polynomial functionsCarlos Erepol
油
The document summarizes key characteristics of polynomial functions:
1) Polynomial functions produce smooth, continuous curves on their domains which are the set of real numbers.
2) The graph's x-intercepts, turning points, and absolute/relative maxima and minima are defined.
3) As the degree of a polynomial increases, so do the possible number of x-intercepts and turning points, up to the degree value. The leading coefficient and degree determine whether the graph rises or falls.10. functions
10. functionsUniversity Of Johannesburg
油
The document discusses functions and relations. It defines functions as special relations where each element of the domain is mapped to only one element in the range. It provides examples of determining whether a relation represented by ordered pairs or a graph is a function. It also discusses determining the domain and range of functions from ordered pairs or graphs. Finally, it introduces mapping and functional notation used to represent functions.Relations and functions
Relations and functions Nabeel Simair
油
The document discusses relations and functions. A relation is defined as a set of ordered pairs, with a domain (set of x-values) and range (set of y-values). A function is a special type of relation where each x-value is assigned to exactly one y-value. Examples of relations and functions are provided to illustrate the differences. Function notation is introduced, where f(x) represents the function f with variable x. The domain of a function is defined as the set of input values that do not result in illegal operations like division by zero or taking the square root of a negative number. Methods for finding the domain of a function are presented.Graphing rational functions
Graphing rational functionsJessica Garcia
油
1. The rational function f(x) = 1/x^2 has a vertical asymptote at x = 2 because the denominator is 0 at that point.
2. As x values approach 2 from either side, y values approach positive or negative infinity, respectively.
3. The graph gets closer and closer to the vertical line x = 2 but never touches it. This line is called a vertical asymptote.
4. The graph also has a horizontal asymptote at y = 0, as y values approach 0 as x values increase or decrease indefinitely.Exercise #11 notes
Exercise #11 notesKelly Scallion
油
This document discusses graphing absolute value functions. It provides examples of graphing various absolute value functions, including f(x)=|x|^2, f(x)=2|x|-1, f(x)=|x|^2-1, f(x)=3|x|^2-4, f(x)=|cos(x)|, f(x)=2|x|-3-1, and shows how to write a piecewise function definition for f(x)=|x|^2. The graphs are V-shaped and symmetric about the y-axis, with vertices at the points where the absolute value terms are equal to zero.Matlab dc circuit analysis
Matlab dc circuit analysisAmeen San
油
This document discusses solving sets of linear equations and analyzing DC circuits. It explains that to solve a set of linear equations in matrix form A*x=y, the matrix A and augmented matrix [A y] must have equal rank. It then discusses calculating the condition number of A, which should be close to 1 for an accurate solution. The document provides an example circuit in matrix form Z*I=V and shows using Matlab commands to calculate the rank of Z and [Z V], the condition number of Z, and solve for the current I.Notes - Graphs of Polynomials
Notes - Graphs of PolynomialsLori Rapp
油
The document discusses graphing and interpreting various polynomial functions using a graphing calculator. It examines the graphs of y=x-1, y=x^2-4, and y=x^3+x^2-12 to note their similarities and differences in terms of degree, number of turns, and x-intercepts. It then explores how changing the coefficients of a cubic function impacts its graph. Finally, it interprets key features of the graph of y=x^3-5x^2+x-5 such as its degree, number of turns, x-intercept, maximum and minimum points.2.7 Curve Sketching Day 2
2.7 Curve Sketching Day 2gagnonator
油
This document discusses curve sketching for the function f(x) = 3x^3 - 18x^2 - 72x + 192. It asks the student to sketch the first and second derivatives of f(x) on the same axes, explains how the second derivative can help determine critical points, and asks why the lesson is titled "The Second Derivative Test". It also provides practice problems applying the second derivative test.1 13s-f
1 13s-fKamarat Kumanukit
油
This document contains the instructions for a Calculus I final exam consisting of two parts. Part I has 10 multiple choice questions worth 5 points each involving derivatives, anti-derivatives, limits, and finding absolute maxima and minima. Part II has 6 longer form problems worth between 1-10 points each involving implicit differentiation, linearization, graphing functions, finding absolute extrema, and a word problem about the speed of a shadow. Students are instructed to show their work and simplify answers where possible to earn full credit.Lesson 20: Derivatives and the Shapes of Curves (slides)
Lesson 20: Derivatives and the Shapes of Curves (slides)Matthew Leingang
油
This document contains lecture notes on derivatives and the shapes of curves from a Calculus I class taught by Professor Matthew Leingang at New York University. The notes cover using derivatives to determine the intervals where a function is increasing or decreasing, classifying critical points as maxima or minima, using the second derivative to determine concavity, and applying the first and second derivative tests. Examples are provided to illustrate finding intervals of monotonicity for various functions.More from Nima Rasekh (6)
Sample3
Sample3Nima Rasekh
油
This document contains a 4 question math quiz with questions that involve: 1) determining the area between two graphs on an interval, 2) writing integrals to represent volumes obtained by rotating an area around axes, 3) finding the volume of a solid shape with a triangular cross-section, and 4) finding the distance an object travels given its initial velocity and acceleration functions over a time period.Sample2
Sample2Nima Rasekh
油
This document contains a math worksheet with 7 problems involving graphing and analyzing functions. The problems include sketching graphs of functions with given properties, finding examples of functions with vertical tangents or local maxima, graphing polynomial functions, determining intervals where functions are positive or negative, and using templates to analyze additional functions by finding domains, intercepts, asymptotes, critical points, and concavity.Sample1
Sample1Nima Rasekh
油
The document summarizes key topics covered in a Math 220 class, including a review of Pythagorean theorem and Thales' theorem. It then introduces the six basic trigonometric functions of sine, cosine, tangent, cotangent, secant, and cosecant. Various angle relationships for these trig functions are also defined, along with recommended values to memorize for specific angles. Finally, the document defines the trig functions in terms of right triangles with non-unit hypotenuses and provides a graph of the trig functions.Sample2
Sample2Nima Rasekh
油
This document contains a math worksheet with 20 practice problems on integration using the substitution rule. It begins with an explanation of the substitution rule and its relationship to the chain rule. Students are asked to use substitutions to evaluate definite and indefinite integrals involving trigonometric, exponential, logarithmic and algebraic functions. The difficulty of the problems increases throughout the worksheet.Sample1
Sample1Nima Rasekh
油
1) The document discusses a scenario where a person repeatedly turns a lamp on and off while forgetting and remembering parts of their homework. Each time the interval to remember something new is halved.
2) It asks if the lamp would be on or off at 12am, and what that would say about the scenario.
3) It introduces sequences and convergence, asking questions to explore properties of different sequences and how summing sequences relates to convergence.CV, Nima Rasekh
CV, Nima RasekhNima Rasekh
油
Nima Rasekh is a PhD candidate in mathematics at the University of Illinois at Urbana-Champaign. He has extensive teaching experience as a discussion leader and instructor for calculus courses. His research is in homotopy theory and higher category theory, and he has presented his work at several conferences. He has also taken on leadership roles organizing seminars and workshops for graduate students.Sample3
- 1. Math 231 Quiz 7 Spring 2015 Name: Grade: 1. De鍖ne the Taylor series of the function f(x) at the point c. 2. State the binomial theorem (make sure your de鍖nition is precise). 3. Find the McLaurin series of f(x) = x2 1 + x3 and use it to 鍖nd f(14) (0).