Day 8 examples
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This document defines and provides examples of trigonometric functions such as sine, cosine, and tangent. It discusses their periodic and asymptotic properties including period, amplitude, domain, range, and zero crossings. Graphs of sine from 0 to 2π, cosine from -2 to 2, and tangent from -2 to 2 are shown along with their characteristics and equations for locating zeroes and asymptotes.
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![Graphs of sin , cos , tan
also called trig functions
sin(
)
sin(
)
sin(
)
Note: as we change , different values of sin result
Let
=x
Resulting values are Dependent variable, ie. y
So we get,
f( ) = sin
Let's graph f( ) = sin over [0, 2 ]](https://image.slidesharecdn.com/day8examples-131017163359-phpapp02/85/Day-8-examples-1-320.jpg)
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This document discusses graphs of trigonometric functions such as sine, cosine, and tangent. It provides definitions of key terms like periodic function, period, amplitude, and sinusoidal axis. It also examines the domain, range, zeros and other characteristics of graphs of f(x)=sin, f(x)=cos, and f(x)=tan over specific intervals. The document notes that as the input x changes, different output values of sine result, and defines the dependent variable y. It explores where the graphs of these trig functions are undefined, such as when cosine equals 0.
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The document provides an overview of functions and limits. It defines a function as a relation where each input is paired with exactly one output. Functions can be represented graphically and include types like linear, quadratic, polynomial, and trigonometric. Limits describe the behavior of a function as inputs approach a value. For single-variable functions, limits can be evaluated at finite and infinite points. For multi-variable functions, limits may not exist if the function approaches different values along different paths to the point. Continuity requires a function's limit to exist and agree with its actual value at a point.
The Application of Derivatives
This document discusses several topics related to calculus including:
1) Derivatives of position, velocity, and acceleration and how they relate to each other.
2) An example problem calculating velocity from a position function.
3) The Mean Value Theorem and how to apply it to find critical points of a function.
4) How the first and second derivatives of a function relate to critical points, maxima, minima, and points of inflection or concavity.
5) Related rates problems and how to set them up using derivatives and relationships between variables.
Analytic function 1
- The document discusses complex analytic functions and their derivatives.
- A complex function is analytic if it has a complex derivative everywhere in some open region of the complex plane. This allows drawing richer conclusions than for real differentiable functions.
- The derivative of a complex function f at z0 is defined as the limit of (f(z)-f(z0))/(z-z0) as z approaches z0. If this limit exists, f is said to be differentiable or analytic at z0.
Principle of Definite Integra - Integral Calculus - by Arun Umrao
Definite integral notes. Best for quick preparation. Easy to understand and colored graphics. Step by step description. Suitable for CBSE board and State Board students in Class XI & XII
Definite Integral
The document discusses definite integration and its properties. It defines definite integration as the integration of a continuous function between two fixed limits. Some key properties discussed include:
1) A definite integral is independent of the variable of integration and depends only on the limits.
2) Flipping the limits of integration changes the sign of the integral.
3) Splitting the limits of integration allows breaking the integral into multiple parts summed together.
Dokumen.tips mathematics ii-institute-of-aeronautical-engineering-pptpdfadvan...
This document outlines the contents of a Mathematics II course, including five units: vector calculus, Fourier series and Fourier transforms, interpolation and curve fitting, solutions to algebraic/transcendental equations and linear systems of equations, and numerical integration and solutions to differential equations. It lists three textbooks and four references used in the course. It then provides examples and explanations of key concepts from the first two units, including vector differential operators, gradient, divergence, curl, and Fourier series representations of functions.
Functions of several variables.pdf
This document provides an overview of functions of several variables. It discusses notation for functions with multiple independent variables, domains of such functions, and graphs of functions with two or more variables. Specifically, it gives examples of finding the domain of functions defined by equations, sketching the graph of a function as a surface in 3D space using traces in coordinate planes and parallel planes, and creating a contour map using level curves representing different values of the dependent variable.
Trigonometry
The document summarizes trigonometric functions and their relationships. It discusses the unit circle, radians vs degrees, computing trig ratios using the unit circle, trig identities, trig functions and their properties including amplitude, frequency, phase and period. It provides examples of trig functions and applications to modeling periodic behavior like sound waves and tides. Practice problems at the end involve converting between radians and degrees and expressing trig ratios as radicals.
Derivación e integración de funcione variables
This document provides an introduction and overview of calculus topics for functions of several real variables, including:
1) Limits and continuity of functions in R3 space.
2) Derivatives of functions of several variables, including partial derivatives and gradients.
3) Total differential, divergence, and rotor.
4) Tangent planes and normal lines to surfaces.
The document defines scalar and vector functions, and discusses concepts like limits, continuity, derivatives, and geometrical interpretations in multi-dimensional spaces. It also outlines further calculus topics to be covered like integrals of functions with parameters, line and surface integrals.
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More from jchartiersjsd
Combs perms review pascals
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive function. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms for those who already suffer from conditions like anxiety and depression.
Day 8 examples u7w14
This document contains 5 problems involving finding terms in expansions of polynomials. The problems involve finding specific terms that contain a given power of x in expansions of polynomials such as (3x^4 - 1)^9, (-x^3 + 2)^6, (x + 1)^3x, (x + 1)^x, and determining the value of m if one term in the expansion of (2x - m)^7 is -262500x^2y^5.
Day 7 examples u7w14
The document discusses binomial expansion, which is the process of multiplying out terms with two variables according to their power using the binomial theorem. It provides examples of expanding binomial expressions like (x + y)2, (x + y)3, and (x + y)4. It also notes that the sum of the exponents in each term equals the overall power, and the number of terms is always one more than the power. Finally, it provides the binomial theorem for expanding any binomial expression and finding a particular term.
Day 4 examples u7w14
This document discusses how to calculate arrangements when some items must be together or apart. It explains that when items need to be together, they should be counted as a single item to reduce the total items being arranged. Then the total number of arrangements is calculated by finding the total possible arrangements and subtracting the arrangements that do not satisfy the constraints of certain items being together or apart. Examples provided include arranging people in a row when some must or cannot sit together and arranging books on a shelf keeping books of each subject together.
Day 3 examples u7w14
Permutations refer to arrangements of objects in a definite order. Some key points:
- Permutations are represented by "nPn" where n is the total number of objects and r is the number being arranged.
- Permutations are used to calculate possibilities like license plates, phone numbers, and locker combinations.
- Restrictions like starting/ending conditions or requiring alternating arrangements reduce the number of possible permutations.
- Objects that are identical only count once toward the total number of permutations rather than being distinguishable.
Day 2 examples u7w14
Factorial notation represents the product of all positive integers less than or equal to the given number. For example, 5! = 5 x 4 x 3 x 2 x 1 and 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1. The document also provides examples of simplifying factorials without a calculator by using properties such as 5! + 4! = 6 x 4! and (k + 1)! + k! = (k + 2)k!.
Day 1 examples u7w14
This document provides examples and explanations of the fundamental counting principle and addition counting principle to solve combinatorics problems. It gives 8 examples of using the fundamental counting principle to count the number of possible outcomes of independent events. These include counting the number of volleyball shoe combinations, outfits that can be created from different clothing items, ways to select committees from groups of people, and 3-digit numbers with no repeating digits. It also provides 5 examples of using the addition counting principle to count outcomes when events are dependent, such as selecting a president and vice president of opposite sexes from a group.
Day 7 examples u6w14
This document discusses graphing composite functions. It provides examples of determining the composite functions f(g(x)) and g(f(x)) for various functions f(x) and g(x), sketching the graphs of the composite functions, and stating their domains. It also gives examples of determining possible functions f(x) and g(x) that satisfy given composite functions.
Day 5 examples u6w14
1. The document discusses composite functions, which involve combining two functions f(x) and g(x) where the output of one is used as the input of the other. It provides examples of evaluating composite functions using tables and graphs.
2. Key steps for evaluating composite functions are: 1) Substitute the inner function into the outer function and 2) Simplify the expression. Order matters as f(g(x)) and g(f(x)) may have different values.
3. Examples are worked through to find composite functions given basic functions like f(x) = x + 1 and g(x) = 2x as well as more complex rational functions.
Day 4 examples u6w14
The graph is a linear function with a domain of all real numbers and a range of real numbers greater than or equal to 3. The graph is a line with a y-intercept of 3 that increases at a rate of 1 as x increases.
Day 3 examples u6w14
Rational functions are functions of the form f(x) = p(x)/q(x) where p(x) and q(x) are polynomials. For example, comparing rational functions like 2x/(x^2 - 4) and (x-1)/(x+1). Horizontal asymptotes of rational functions occur when the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator.
Day 2 examples u6w14
This document discusses combining functions by graphing. When two functions f(x) and g(x) are combined, their graphs are overlayed on the same coordinate plane. The result is a new combined function where the output is determined by applying both functions f(x) and g(x) to the same input x.
Day 1 examples u6w14
This document discusses how to find the sum, difference, product, and quotient of functions. The sum of functions is found by adding the y-coordinates of each function. The difference is found by subtracting the y-coordinates. The product is represented as h(x) = f(x)g(x) and the quotient is represented as h(x) = f(x)/g(x). Examples are provided for adding and subtracting functions.
Mental math test outline
The document outlines a mental math test covering polynomials. It includes short answer questions testing long division, synthetic division, the remainder theorem, and finding the degree, leading coefficient, and y-intercept of polynomials. The test also covers matching graphs to polynomial equations and word problems involving fully factoring polynomials and two graphs. Multiple choice questions will require explaining solutions, while long answer questions involve fully factoring polynomials and word problems.
Day 8 examples u5w14
The document contains two polynomial word problems. The first asks to write a function V(x) to express the volume of a box with dimensions x, x+2, x+10 in terms of x, and find possible x values if the volume is 96 cm^3. The second problem describes a block of ice that is initially 3 ft by 4 ft by 5 ft, and asks to write a function to model reducing each dimension by the same amount to reach a volume of 24 ft^3, and determine how much to remove from each dimension.
Day 7 examples u5w14
The document provides 3 polynomial word problems: 1) finding the equation for a polynomial given its graph f(x) = -(x - 2)2(x + 1), 2) determining the polynomial P(x) when divided by (x - 3) with a quotient of 2x^2 + x - 6 and remainder of 4, and 3) finding the value of a if (x - 2) is a factor of ax^3 + 4x^2 + x - 2. It also gives a 4th problem of determining the value of k when 2x^3 + kx^2 - 3x + 2 is divided by x - 2 with a remainder of 4.
Day 5 examples u5w14
Polynomial functions are described by their degree and have certain characteristics. The graph of a polynomial is smooth and continuous without sharp corners. Odd degree polynomials rise on the left and fall on the right, while even degree polynomials rise on both sides. The number of x-intercepts and local maxima/minima are limited by the degree. Polynomials can be matched based on their degree, leading coefficient, even/odd nature, and number of x-intercepts and local extrema. The x-intercepts of a polynomial correspond to the roots of the equation, and a repeated root indicates a zero of higher multiplicity which affects the graph.
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Day 8 examples
- 1. Graphs of sin , cos , tan also called trig functions sin( ) sin( ) sin( ) Note: as we change , different values of sin result Let =x Resulting values are Dependent variable, ie. y So we get, f( ) = sin Let's graph f( ) = sin over [0, 2 ]
- 3. Definitions Periodic Function = function whose graph has a pattern Period = the length of one full pattern. ex. in sine graph its 2 Amplitude = vertical deviation of graph from middle (sinusoidal axis). Sinusoidial Axis = horizontal axis above and below which the graph fluctuates. It defines the amplitude. Characteristics of sine function Domain: In general Rangle: In general Zeroes: In general Graph f( ) = cos over [-2 , 2 ] Domain: In general Rangle: In general Zeroes: In general + k ,k I
- 4. Graph f( ) = tan over [-2 ,2 ] Locating the zeroes: f( ) = 0 ie. tan = 0 sin = 0 when sin = 0, occurs when = 0, , 2 , - , -2 What happens when cos = 0 in tan = sin = sin = - an invisible line where the graph or function is undefined when cos = 0, occurs when Domain: In general Range: In general Equations for asymptotes: x = = , 3 , - , -3