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This document defines the function f(x) = 3√(x)/(x^2-1) and provides 8 exercises to analyze properties of the function: (1) find the domain, (2) compute limits as x approaches positive/negative infinity, (3) compute derivatives, (4) find critical numbers and points of inflection, (5) find asymptotes, and (6) draw the graph of the function.
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Hw9
This document defines the function f(x) = 3√(x)/√(|x| + 1) and provides exercises to analyze properties of the function including its domain, limits, continuity, derivatives, critical numbers, points of inflection, asymptotes, and sketching its graph. The exercises require computing limits, derivatives, critical numbers, points of inflection, and asymptotes then using these to draw the graph of the defined function.
Day 7 examples u1w14
This document provides examples of sketching the graphs of reciprocal functions by dividing 1 by various linear functions of x and finding their domains and ranges. It includes 4 examples of sketching the graphs of reciprocal functions by dividing 1 by expressions of the form ax + b, finding any horizontal or vertical asymptotes, and stating their domains and ranges.
Pre Calculus notes 4 1 Exponential Functions
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This document discusses how to sketch graphs resulting from transformations of basic parent functions, including horizontal and vertical stretches as well as translations. It provides examples of stretching and translating graphs of f(x) = x^2 and generalizes the effects of stretches and translations on basic parent functions. The document concludes with instructions on combining multiple transformations and an example problem determining the x-intercepts resulting from a composite transformation.
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This document provides examples for graphing reciprocals of functions. It explains that for reciprocals, smaller numbers on the x-axis become larger on the y-axis, and vice versa. An example graphs the reciprocal of f(x)=1/x^2. It notes that reciprocals can be written in the form f(x)=k/x+h, where k and h are determined from the original function. Further examples graph the reciprocals of f(x)=2/x+4 and f(x)=sin(x).
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Hw9
This document defines the function f(x) = 3√(x)/√(|x| + 1) and provides exercises to analyze properties of the function including its domain, limits, continuity, derivatives, critical numbers, points of inflection, asymptotes, and sketching its graph. The exercises require computing limits, derivatives, critical numbers, points of inflection, and asymptotes then using these to draw the graph of the defined function.
Day 7 examples u1w14
This document provides examples of sketching the graphs of reciprocal functions by dividing 1 by various linear functions of x and finding their domains and ranges. It includes 4 examples of sketching the graphs of reciprocal functions by dividing 1 by expressions of the form ax + b, finding any horizontal or vertical asymptotes, and stating their domains and ranges.
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This document discusses how to sketch graphs resulting from transformations of basic parent functions, including horizontal and vertical stretches as well as translations. It provides examples of stretching and translating graphs of f(x) = x^2 and generalizes the effects of stretches and translations on basic parent functions. The document concludes with instructions on combining multiple transformations and an example problem determining the x-intercepts resulting from a composite transformation.
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This document provides examples for graphing reciprocals of functions. It explains that for reciprocals, smaller numbers on the x-axis become larger on the y-axis, and vice versa. An example graphs the reciprocal of f(x)=1/x^2. It notes that reciprocals can be written in the form f(x)=k/x+h, where k and h are determined from the original function. Further examples graph the reciprocals of f(x)=2/x+4 and f(x)=sin(x).
X2 T04 06 curve sketching - roots of functions
The document discusses how to sketch the graph of y = f(x). It notes that f(x) must be greater than or equal to 0 within its domain. The graph passes the x-axis where f(x) = 0. Critical points occur where the derivative f'(x) is 0. The graph has the same general shape as y = f(x) but is reflected above or below the line y = x depending on if f(x) is greater than or less than 1.
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This document contains solutions to problems involving spherical and Cartesian coordinate systems.
It begins by presenting the solution to problem 2(a), which defines a function z=f(x,y) for both coordinate systems. Next, it solves problem 2(b) by deriving and plotting the equation r=f(θ) for both systems. Finally, it solves problem 2(c) and 2(d) through similar methods, presenting the equations and Maple plotting commands with figures for each part. Maple code is included to generate 3D plots visualizing the surfaces defined by the equations.
ลิมิต
1. The document discusses 4 problems involving limits, derivatives, and integrals. It provides the questions, solutions, and point values for each part.
2. The problems cover a range of calculus topics - finding limits, determining derivatives, solving differential equations, and evaluating definite integrals.
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Exercise #9 notes
1) Reflections of functions f(x) over the x-axis or y-axis produce new equations by changing the sign of the y-values or x-values respectively.
2) The inverse of a function f(x) is denoted f^-1(x) and is found by switching the x and y values when graphed, effectively reflecting over the line y=x.
3) For example, the inverse of f(x)=2x+5 is found by graphing f^-1(x) which treats x as the output and y as the input.
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The document discusses how to sketch the graph of functions of the form y = f(x). It notes that f(x) must be greater than or equal to 0 within its domain. The graph of y = f(x) can be drawn and critical points occur where f(x) = 0 or f'(x) = 0. The shape of the graph is determined by whether f(x) is greater than or less than 1.
جلسه اول الگوریتم و فلوچارت
The document discusses the importance of learning algorithms and flowcharts. It provides some examples of simple algorithms and explains that algorithms are step-by-step methods for solving problems. Flowcharts are used to visually represent algorithms using standard notation to ensure clarity and precision.
Exercise #11 notes
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.
جلسه اول الگوریتم و فلوچارت
توضیحات مربوط به این پاورپوینت را در آدرس زیر ببینید:
http://jahangirics.ir/%d8%ac%d9%84%d8%b3%d9%87-%d8%a7%d9%88%d9%84-%d8%a7%d9%84%da%af%d9%88%d8%b1%db%8c%d8%aa%d9%85-%d9%88-%d9%81%d9%84%d9%88%da%86%d8%a7%d8%b1%d8%aa/
ویدیوی مربوط به این پاورپوینت در یوتیوب
https://www.youtube.com/watch?v=Ucg1TUbnzp8&t=2002s
Lesson 1 Feb 10 2010
This document provides an introduction to quadratic functions. It defines the standard form of a quadratic function as f(x) = ax^2 + bx + c, and shows how to graph simple quadratic functions like f(x) = x^2 by creating a table of x and y-values. It also introduces key concepts for quadratic functions like domain, range, vertex, axis of symmetry, and maximum/minimum values.
Integration worksheet.
This document provides a tutorial on topics related to calculus including:
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2) Evaluating definite integrals of functions including trigonometric, exponential, and rational functions
3) Finding areas bounded by curves, axes, and lines by evaluating definite integrals
4) Sketching graphs of functions and finding relevant information like minimum/maximum points
5) Finding equations of tangents and normals to curves at given points
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The document discusses how to determine if a function is increasing or decreasing based on the sign of its derivative, and provides an example of finding the intervals where a function is increasing or decreasing. It also discusses how to find and classify critical points of a function by taking the derivative, setting it equal to 0, and using the first derivative test to determine if the critical points are local maxima or minima.
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This document contains a 14 question mathematics exam with questions covering a range of topics including:
- Matrix algebra and inverses
- Integration using substitution
- Graphing functions
- Probability
- Arithmetic and geometric progressions
- Complex numbers
- Linear transformations
- Logical statements
The exam consists of two sections - Section A contains 8 short answer questions, and Section B contains 6 multi-part questions where students must show working for partial or full marks. Overall the exam covers a wide breadth of mathematical concepts and techniques.
Mαθηματικά Γ Λυκείου προσανατολισμού
Επιμέλεια: Νίκος Ράπτης από το Αγρίνιο αποκλειστικά για το lisari.blogspot.gr
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The document discusses how to sketch the graph of a reciprocal function y = 1/f(x). Key points include:
1) Vertical asymptotes occur where f(x) = 0.
2) Horizontal asymptotes occur where f(x) approaches infinity.
3) The graph of the reciprocal function is decreasing where the original function is increasing, and vice versa.
4) Stationary points of the original function correspond to stationary points of the reciprocal function.
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Hw6 extra for calculus
This document contains exercises to prove properties of the function f(x) = 1/(x^2+1). The first exercise shows that f(x) = -x^2/(x^2+1)(f(x)+1) for all real numbers x. The second exercise shows that 0 < f(x)+1 ≤ 1 for all real x. The third exercise uses the epsilon-delta definition of limits to show that the limit of f(x) as x approaches 0 equals 1.
Hw6sol
1) The document provides solutions to homework problems from a complex analysis course. It solves problems involving properties of polynomials, including showing two polynomials are equal if they have the same roots, and investigating properties of roots.
2) It also analyzes singularities of complex functions, determining whether singularities are removable, poles, or essential singularities. Functions include rational, trigonometric, exponential and combined functions.
3) The solutions demonstrate techniques for analyzing complex functions at isolated singular points using principles like series expansions and the Casorati-Weierstrass theorem.
Hw5sol
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Hw4sol
This document contains solutions to homework problems from a complex analysis course. It solves problems involving logarithms, exponentials, trigonometric and hyperbolic functions of complex numbers. Key steps and solutions are shown for problems involving contour integrals of complex functions along curves in the complex plane. The length of one such curve is computed to be 8a. Several contour integrals are evaluated, with one found to be equal to 1 - i. Bounds on contour integrals are also determined using theorems.
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ลิมิต
1. The document discusses 4 problems involving limits, derivatives, and integrals. It provides the questions, solutions, and point values for each part.
2. The problems cover a range of calculus topics - finding limits, determining derivatives, solving differential equations, and evaluating definite integrals.
3. Overall, the document presents 4 multi-part problems that require applying core calculus concepts, with the goal of summarizing essential information at a high level in 3 sentences or less.
Exercise #9 notes
1) Reflections of functions f(x) over the x-axis or y-axis produce new equations by changing the sign of the y-values or x-values respectively.
2) The inverse of a function f(x) is denoted f^-1(x) and is found by switching the x and y values when graphed, effectively reflecting over the line y=x.
3) For example, the inverse of f(x)=2x+5 is found by graphing f^-1(x) which treats x as the output and y as the input.
X2 t07 06 roots of functions (2012)
The document discusses how to sketch the graph of functions of the form y = f(x). It notes that f(x) must be greater than or equal to 0 within its domain. The graph of y = f(x) can be drawn and critical points occur where f(x) = 0 or f'(x) = 0. The shape of the graph is determined by whether f(x) is greater than or less than 1.
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The document discusses the importance of learning algorithms and flowcharts. It provides some examples of simple algorithms and explains that algorithms are step-by-step methods for solving problems. Flowcharts are used to visually represent algorithms using standard notation to ensure clarity and precision.
Exercise #11 notes
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.
جلسه اول الگوریتم و فلوچارت
توضیحات مربوط به این پاورپوینت را در آدرس زیر ببینید:
http://jahangirics.ir/%d8%ac%d9%84%d8%b3%d9%87-%d8%a7%d9%88%d9%84-%d8%a7%d9%84%da%af%d9%88%d8%b1%db%8c%d8%aa%d9%85-%d9%88-%d9%81%d9%84%d9%88%da%86%d8%a7%d8%b1%d8%aa/
ویدیوی مربوط به این پاورپوینت در یوتیوب
https://www.youtube.com/watch?v=Ucg1TUbnzp8&t=2002s
Lesson 1 Feb 10 2010
This document provides an introduction to quadratic functions. It defines the standard form of a quadratic function as f(x) = ax^2 + bx + c, and shows how to graph simple quadratic functions like f(x) = x^2 by creating a table of x and y-values. It also introduces key concepts for quadratic functions like domain, range, vertex, axis of symmetry, and maximum/minimum values.
Integration worksheet.
This document provides a tutorial on topics related to calculus including:
1) Differentiating various functions and finding points where the gradient is zero
2) Evaluating definite integrals of functions including trigonometric, exponential, and rational functions
3) Finding areas bounded by curves, axes, and lines by evaluating definite integrals
4) Sketching graphs of functions and finding relevant information like minimum/maximum points
5) Finding equations of tangents and normals to curves at given points
Day 1a examples
The document discusses how to determine if a function is increasing or decreasing based on the sign of its derivative, and provides an example of finding the intervals where a function is increasing or decreasing. It also discusses how to find and classify critical points of a function by taking the derivative, setting it equal to 0, and using the first derivative test to determine if the critical points are local maxima or minima.
Quadratic Function by Taylor & Asia
Quadratic functions are functions that can be described by an equation in the form f(x)=ax^2+bx+c. Examples of quadratic functions include f(x) = -2x^2 + x - 1 and f(x) = x^2 + 3x + 2. Quadratic equations were used to describe the orbits of planets around the sun and allow observation of planetary motion. Structural engineers also use quadratic equations to design tall skyscrapers.
Probability5
This document discusses probability and related concepts like sample space, events, and dependent and independent events. It provides examples of calculating probabilities by listing outcomes and constructing tree diagrams for combined events. It also discusses the probability of complementary events and the probability of unions and intersections of events. Examples include calculating the probability of obtaining certain numbers when rolling dice or outcomes when drawing balls from bags. The document emphasizes that two events are independent if the outcome of one does not affect the other, while dependent events influence each other's probabilities.
Rational functions precalc
The document discusses graphing rational functions by hand. It explains that to graph rational functions, one should find all intercepts by setting the numerator and denominator equal to zero, find and sketch all asymptotes using dashed lines, and plot points near intercepts and asymptotes. The document provides examples of sketching various rational functions step-by-step and asks the reader to sketch additional examples on their own.
Lesson 5: Functions and surfaces
A function of two variables is defined similar to a function of one variable. It has a domain (in the plane) and a range. The graph of such a function is a surface in space and we try to sketch some.
Additional math 1999
This document contains a 14 question mathematics exam with questions covering a range of topics including:
- Matrix algebra and inverses
- Integration using substitution
- Graphing functions
- Probability
- Arithmetic and geometric progressions
- Complex numbers
- Linear transformations
- Logical statements
The exam consists of two sections - Section A contains 8 short answer questions, and Section B contains 6 multi-part questions where students must show working for partial or full marks. Overall the exam covers a wide breadth of mathematical concepts and techniques.
Mαθηματικά Γ Λυκείου προσανατολισμού
Επιμέλεια: Νίκος Ράπτης από το Αγρίνιο αποκλειστικά για το lisari.blogspot.gr
X2 T04 04 curve sketching - reciprocal functions
The document discusses how to sketch the graph of a reciprocal function y = 1/f(x). Key points include:
1) Vertical asymptotes occur where f(x) = 0.
2) Horizontal asymptotes occur where f(x) approaches infinity.
3) The graph of the reciprocal function is decreasing where the original function is increasing, and vice versa.
4) Stationary points of the original function correspond to stationary points of the reciprocal function.
March 19 Quadratic Test Review
The function is a quadratic function in the form f(x) = ax2 + bx + c, with a = 2, b = -4, and c = -1. It has a vertex of (1, -3), an axis of symmetry at x = 1, a domain of all real numbers, and a range of [-3,∞). The y-intercept is -1 and the x-intercepts (roots) are 1 ± 0.5√6.
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Hw6 extra for calculus
This document contains exercises to prove properties of the function f(x) = 1/(x^2+1). The first exercise shows that f(x) = -x^2/(x^2+1)(f(x)+1) for all real numbers x. The second exercise shows that 0 < f(x)+1 ≤ 1 for all real x. The third exercise uses the epsilon-delta definition of limits to show that the limit of f(x) as x approaches 0 equals 1.
Hw6sol
1) The document provides solutions to homework problems from a complex analysis course. It solves problems involving properties of polynomials, including showing two polynomials are equal if they have the same roots, and investigating properties of roots.
2) It also analyzes singularities of complex functions, determining whether singularities are removable, poles, or essential singularities. Functions include rational, trigonometric, exponential and combined functions.
3) The solutions demonstrate techniques for analyzing complex functions at isolated singular points using principles like series expansions and the Casorati-Weierstrass theorem.
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This document provides solutions to problems from a complex analysis homework assignment. The solutions involve applying theorems related to contour integrals, Cauchy's integral formula, and derivatives to evaluate definite integrals over curves and find derivatives of functions. Key steps include using the Cauchy-Goursat theorem when functions are analytic over regions, and applying Cauchy's integral formula and its generalization to derivatives to evaluate integrals that involve singularities inside curves.
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This document contains solutions to homework problems from a complex analysis course. It solves problems involving logarithms, exponentials, trigonometric and hyperbolic functions of complex numbers. Key steps and solutions are shown for problems involving contour integrals of complex functions along curves in the complex plane. The length of one such curve is computed to be 8a. Several contour integrals are evaluated, with one found to be equal to 1 - i. Bounds on contour integrals are also determined using theorems.
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