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Esta es la presentación en el pizarrón electrónico de la tercera semana de mi curso de Cálculo diferencial
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Dif calc10week1
Verify the following trigonometric identities:
1. sin^2(θ) + cos^2(θ) = 1
2. tan^2(θ) + 1 = sec^2(θ)
3. cot^2(θ) + 1 = csc^2(θ)
Show the steps for transforming the left-hand side into the right-hand side for each identity.
Tengo 10 minutos
Esta es la primer versión de mi keynote Tengo 10 minutos.
Es una plática, estilo Darren Kuropatwa, ya que compartiré mis experiencias con el uso de la tecnología en mis clases.
Tomé bastante del prof. Kuropatwa, como se puede ver ;-)
Update:
Agh. Puse 31 de agosto de 2012, cuando debió haber dicho 31 de julio de 2012 :-(
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Verify the following trigonometric identities:
1. sin^2(θ) + cos^2(θ) = 1
2. tan^2(θ) + 1 = sec^2(θ)
3. cot^2(θ) + 1 = csc^2(θ)
Show the steps for transforming the left-hand side into the right-hand side for each identity.
Tengo 10 minutos
Esta es la primer versión de mi keynote Tengo 10 minutos.
Es una plática, estilo Darren Kuropatwa, ya que compartiré mis experiencias con el uso de la tecnología en mis clases.
Tomé bastante del prof. Kuropatwa, como se puede ver ;-)
Update:
Agh. Puse 31 de agosto de 2012, cuando debió haber dicho 31 de julio de 2012 :-(
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Update:
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Chapter 1 (functions).
The document discusses various topics in functions including:
- Types of relations such as one-to-one, one-to-many, and many-to-one.
- Ordered pairs, domains, codomains, and defining functions.
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Exploration 8 – Shifting and Stretching Rational Functions .docx
Exploration 8 – Shifting and Stretching Rational Functions
1. Sketch the graph of each function.
3( )f x
x
3
( ) 1
2
f x
x
Domain: Range: Domain: Range:
vertical horizontal vertical horizontal
asymptote: asymptote: asymptote: asymptote:
x-intercept: y-intercept: x-intercept: y-intercept:
How do you find the domain and vertical asymptote of a rational function?
How did you find the range and horizontal asymptote of THIS rational function?
How do you find the x-intercept of a function?
How do you find the y-intercept of a function?
Graphing
3
( ) 1
2
f x
x
is relatively easy.
Re-write the function rule as a single fraction by
subtracting the 1. Then find each of the following
for the newly written function.
Domain: Range: x-intercept: y-intercept:
vertical horizontal
asymptote: asymptote:
How do you find the equation of the horizontal asymptote for THIS type of function?
WebAssign Problem:
Graph the function,
2 4
( )
1
x
f x
x
, by shifting and stretching the function, 1( )f x
x
.
The horizontal shift is ______________________ because ________________________________.
The vertical shift is ______________________ because ___________________________________.
To find the stretch, you must re-write the function,
2 4
( )
1
x
f x
x
, in 1( )f x
x
form, by setting the
two rules equal and solving for c. Then sketch the graph below.
For the group submission:
Graph the function,
2 2
( )
1
x
f x
x
, by shifting and stretching the function, 1( )f x
x
.
Horizontal Shift:
Vertical Shift:
Stretch:
vertical horizontal x-intercept: y-intercept:
asymptote: asymptote:
Domain: Range:
Group Submission for Investigation #8
Write group member names legibly here:
Graph the function,
2 2
( )
1
x
f x
x
, by shifting and stretching the function, 1( )f x
x
.
Horizontal Shift:
Vertical Shift:
Stretch:
vertical horizontal x-intercept: y-intercept:
asymptote: asymptote:
Domain: Range:
...
Exponential and logarithmic functions
This document provides an overview of exponential and logarithmic functions. It defines one-to-one functions and inverse functions. It explains how to find the inverse of a one-to-one function and shows that the inverse of f(x) is f-1(x). Properties of exponential functions like f(x)=ax and logarithmic functions like f(x)=logax are described. The product, quotient, and power rules for logarithms are outlined along with examples. Finally, it discusses how to solve exponential and logarithmic equations using properties of these functions.
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This document discusses graphing rational functions. It defines key concepts like domain, range, intercepts, zeros, and asymptotes. An example rational function f(x)=x-2/(x+2) is used to demonstrate how to find these values and graph the function. The domain is all real numbers except -2, the x-intercept is 2, and the y-intercept is -1. The vertical asymptote is at x=-2. Horizontal asymptotes occur when the degree of the numerator is less than, equal to, or greater than the degree of the denominator.
Radical functions
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.
Lecture 1
The document defines and provides examples of various types of functions, including:
- Polynomial functions including constant, linear, and general polynomial functions.
- Rational functions defined as the ratio of two polynomial functions.
- Trigonometric functions including sine, cosine, and their inverses.
- Other common functions like absolute value, square root, exponential, logarithmic, floor, and ceiling functions.
It also defines properties of functions like being one-to-one, even, or odd and provides examples of each.
Lesson 1
A function is a set of ordered pairs where each x-value is paired with exactly one y-value. The domain is the set of all possible x-values, and the range is the resulting y-values. To find the rate of change, use the difference quotient which is similar to the slope formula. The vertical line test determines if a graph represents a function - if any vertical line intersects the graph more than once, it is not a function. Domain restrictions specify allowed input values.
Lesson 1
A function is a set of ordered pairs where each x-value is paired with exactly one y-value. The domain is the set of all possible x-values, and the range is the resulting y-values. The rate of change of a function can be found using the difference quotient or slope formula. A graph is a function if it passes the vertical line test, meaning no vertical line intersects the graph at more than one point. Domain restrictions limit the possible input values of a function. Horizontal and vertical asymptotes provide information about the behavior of a rational function as x approaches positive or negative infinity.
Exponential_Functions.ppt
This document discusses exponential functions. It begins by stating the objectives of understanding properties of exponents, simplifying exponential expressions, solving exponential equations, and sketching graphs of exponential functions. It then defines exponential functions and discusses their general form. Several properties of exponents are listed and examples of simplifying expressions using these properties are shown. The document also demonstrates how to solve various exponential equations. Finally, it discusses graphing and properties of exponential functions, including the special number e and the exponential function with base e.
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R lecture co4_math 21-1
This document discusses concepts related to calculus including functions, limits, continuity, and derivatives. The objective is for students to be able to evaluate limits and determine derivatives of algebraic functions. It defines functions and function notation. It discusses limits, continuity, and the definition of the derivative. It provides examples of evaluating limits using theorems and the squeeze principle. It also defines types of discontinuities and conditions for continuity.
Rational functions
Rational functions are functions of the form f(x)=polynomial/polynomial. There are six key aspects to analyze in rational functions: y-intercept, x-intercepts, vertical asymptotes, horizontal/slant asymptotes, and the graph. Vertical asymptotes occur when the denominator is 0, x-intercepts when the numerator is 0, and horizontal/slant asymptotes depend on the relative degrees of the numerator and denominator polynomials.
Vertical asymptotes to rational functions
This document discusses how to graph rational functions by identifying key characteristics from the function expression. These include:
1) The y-intercept by setting x=0.
2) X-intercepts by setting the numerator equal to 0.
3) Vertical asymptotes by setting the denominator equal to 0.
4) Horizontal or slant asymptotes using rules based on the degrees of the numerator and denominator polynomials.
5) The graph by considering the intercepts, asymptotes, and a "sign property" that determines whether the graph is above or below the x-axis between intercepts/asymptotes. Examples are worked through step-by-step.
Exercise #19
The document discusses exponential functions of the form f(x) = a^x where a is a constant. It provides examples of exponential functions with a = 2 and a = 1/2. It describes the domain, range, and common point of exponential functions. The document also discusses transformations of exponential functions by adding or subtracting constants, and provides examples of sketching and describing transformed exponential functions. Finally, it lists common exponential expressions.
0101: Graphing Quadratic Functions
This document provides instruction on graphing quadratic functions. It covers:
1) Graphing quadratic functions by determining the axis of symmetry from the factored form, finding points where the function equals zero, and using those points to locate the vertex.
2) Finding the axis of symmetry and vertex of a quadratic function given its standard form.
3) Graphing additional points on a quadratic function by moving left and right of the vertex based on the value of a in the standard form.
Edsc 304 lesson 1
A function is a set of ordered pairs where each x-value is paired with exactly one y-value. The rate of change of a function is found by taking the difference quotient, which is similar to the slope formula. The domain of a function is the set of all possible x-values, while the range is the resulting y-values. A graph is a function if it passes the vertical line test, meaning no vertical line intersects the graph at more than one point.
11 X1 T02 06 relations and functions (2010)
The document discusses relations and functions. A relation is any set of ordered pairs, while a function assigns each input exactly one output. It also discusses domains and ranges of functions. The domain is the set of all possible inputs, and is found by determining values that would make the function undefined. Examples show how to determine domains based on fractions, roots, and inequality restrictions.
11X1 T02 06 relations & functions (2011)
The document discusses relations and functions. A relation is any set of ordered pairs, while a function assigns each input exactly one output. It also discusses domains and ranges of functions. The domain is the set of all possible inputs, and is found by determining values that would make the function undefined. Examples show how to determine domains based on fractions, roots, and inequality restrictions.
11 x1 t02 06 relations & functions (2013)
The document discusses relations and functions. A relation is any set of ordered pairs, while a function assigns each input value to at most one output value. The domain of a function is the set of all possible input values, while looking for values the input could not be to determine the domain. Examples are provided of determining domains for various functions, such as ensuring denominators are not equal to zero and that roots are not taken of negative numbers.
11 x1 t02 06 relations & functions (2012)
The document discusses relations and functions. A relation is any set of ordered pairs, while a function assigns each input exactly one output. It also discusses domains and ranges of functions. The domain is the set of all possible inputs, and is found by determining values that would make the function undefined. Examples show how to determine domains based on fractions, roots, and inequality restrictions.
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Difcalc10week3
- 1. Exponential and Logarithmic Functions. Graphs of Exponential Functions. a>1 0<a<1 y y x x Equation: Equation: y-intercept: y-intercept: Domain: Domain: Range: Range: Asymptote: Asymptote:
- 2. Exponential and Logarithmic Functions. Graphs of Logarithmic Functions. a>1 0<a<1 y y x x Equation: Equation: y-intercept: y-intercept: Domain: Domain: Range: Range: Asymptote: Asymptote:
- 3. Examples. Sketch the graph of the following functions: y y x x y y x x
- 4. Exercises. Sketch the graph of the following functions:
- 5. Homework. Posted on Blackboard (Assignments). Collaborative Activity 1. Posted on Blackboard (Course Documents). You will work in teams up to four people. Due date: September 3.
- 6. Piecewise functions. Opener. Sketch the graph of the functions on the same XY-plane.
- 7. Piecewise function is a function that is defined piecewise, that is, f(x) is given by different expressions on various intervals.
- 8. Examples. Sketch the graph of the following functions:
- 11. Exercises.
- 12. Homework. Posted on Blackboard (Assignments).
- 13. Absolute Value Functions. Opener. 1. Find the following: 2. Sketch the graph of the following function:
- 14. Official definition of the Absolute Value: The Absolute Value Function is given by:
- 15. Examples. Sketch the graph of: