Caculus II - 37
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The document discusses tangent, normal, and binormal vectors for curves. It defines these vectors for a space curve at a point and explains that they form an orthogonal TNB frame. It then gives examples of finding these vectors and the normal and osculating planes for a helix curve at a specific point.
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The dot product of two vectors a and b is defined as a·b and is calculated by multiplying their corresponding components and adding them. It can be used to calculate the projection of one vector onto another and the cosine of the angle between the vectors. The cross product of two vectors a and b is defined as axb and produces a third vector that is perpendicular to both a and b. It has various properties including being anticommutative and having a magnitude equal to the area of the parallelogram determined by the two vectors.
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1) Methods for solving quadratic equations such as factoring and the quadratic formula.
2) Properties and equations of parabolas, circles, ellipses, and hyperbolas including their standard forms and key components.
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3. Methods for finding the foot of a perpendicular from a point to a line or plane, reflecting lines and planes, and determining relationships between lines and planes are summarized.
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This document discusses numerical integration using the trapezoidal rule. It begins by introducing the concept of numerical integration as a way to evaluate integrals numerically. It then describes the trapezoidal rule, explaining that it approximates the integral of a function between intervals by calculating the area of trapezoids under the function curve. The rule takes the average of the function values at the interval endpoints to estimate the area of each trapezoid. An example calculates the integral of e^-x^2 from 0 to 1 using the trapezoidal rule with n=10 subintervals. It finds the result to be approximately 0.7617562, demonstrating how to apply the rule.
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The trapezoidal rule is used to approximate the area under a curve by dividing it into trapezoids. It takes the average of the function values at the beginning and end of each sub-interval. The area is calculated as the sum of the areas of each trapezoid multiplied by the width of the sub-interval. An example calculates the area under y=1+x^3 from 0 to 1 using n=4 sub-intervals, giving an approximate result of 1.26953125. The document also provides an example of using the trapezoidal rule with n=8 sub-intervals to estimate the area under the curve of the function y=x from 0 to 3.
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1) A sequence is an ordered list of numbers written as {an}. A sequence converges to a limit L if the terms an get arbitrarily close to L as n increases.
2) Common properties of convergent sequences include that the sum, difference, and scalar multiple of two convergent sequences is also convergent.
3) Useful theorems include if {an} converges to L and {bn} converges to M, then {an + bn} converges to L + M. If |an| ≤ M for all n and M is a constant, then the sequence converges.
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The document discusses 3D coordinate systems and vectors. It defines:
1) The distance formula in 3D space between two points (x1,y1,z1) and (x2,y2,z2) as the square root of (x2-x1)2 + (y2-y1)2 + (z2-z1)2.
2) Vectors as quantities that have both magnitude and direction. It shows examples of vector addition and subtraction using the triangle and parallelogram laws.
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This document discusses tests for determining if infinite series converge or diverge, including:
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The document discusses Taylor series and Maclaurin series. It provides the formulas for determining the coefficients of a Taylor series centered at a point a. It also discusses Taylor's inequality which relates the remainder of a Taylor series to the derivative of the function. Several examples are worked out of finding the Maclaurin series for common functions and proving they represent the functions for all values. Additional power series are derived from fundamental series like e^x, sin(x), and cos(x).
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The document discusses Taylor series and how they can be used to represent general functions as power series. It provides the formulas for determining the coefficients in the Taylor series expansion of a function f(x) about a point a. Examples are given of finding the Maclaurin series, which is the Taylor series expansion about x=0, for common functions and determining their radii of convergence. The key steps in proving that the Taylor series equals the original function within the radius of convergence are outlined using Taylor's inequality.
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The document discusses formulas for calculating the area and length of regions defined by polar curves. It gives the formula for finding the area inside a polar curve between rays at specific theta values. Similarly, it provides the formula for calculating the arc length of a polar curve between two theta values. It then works through examples of using these formulas to find the area of a four-leaved rose curve and the area and length of a cardioid curve.
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The document discusses using Taylor series approximations to evaluate expressions. It finds the maximum error of approximating e-0.5 as 1 - 0.5 + 0.5^2, determines the value of x for which this approximation is accurate to within 0.01, and establishes that an 8th degree Taylor polynomial is needed to approximate e^x in [-1,1] with an error less than 0.001.
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This document discusses convergence of infinite series. It defines partial sums of a series and states that if the sequence of partial sums converges, then the series converges to that limit. If the partial sums do not converge or their limit does not exist, then the series diverges. It provides examples of geometric series, which converge if the common ratio is less than 1 in absolute value, and harmonic series, which diverges. It also discusses using known convergent series to determine if new series formed from their sums or products converge.
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The Comparison Test allows determining if a series converges or diverges by comparing it to standard p-series or geometric series. If a series is less than or equal to a convergent p-series, or greater than or equal to a divergent p-series, then the series has the same behavior. Examples demonstrate applying the test to various series.
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Calculus II - 16
The document discusses calculus concepts related to parametric curves including calculating tangents, areas, arc length, and surfaces of revolution. It provides the formulas for finding the tangent of a parametric curve, area under a parametric curve, arc length of a parametric curve, and surface area obtained by rotating a parametric curve around an axis. Examples are given for each concept using the parametric cycloid curve.
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Calculus II - 11
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MT102 Лекц 5
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MT102 Лекц 6
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Set theory, sample space, events, concepts of randomness and uncertainty, basic principles of probability, axioms and properties of probability, conditional probability, independent events, Baye’s formula, Bernoulli trails, sequential experiments, discrete and continuous random variable, distribution and density functions, one and two dimensional random variables, marginal and joint distributions and density functions. Expectations, probability distribution families (binomial, poisson, hyper geometric, geometric distribution, normal, uniform and exponential), mean, variance, standard deviations, moments and moment generating functions, law of large numbers, limits theorems
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35 tangent and arc length in polar coordinates
The document discusses parametric representations of polar curves. It begins by explaining that a rectangular curve given by y=f(x) can be parametrized as x=t and y=f(t). It then explains that a polar curve given by r=f(θ) can be parametrized as x=f(θ)cos(θ) and y=f(θ)sin(θ). Examples are given of parametrizing the Archimedean spiral r=θ and the cardioid r=1+sin(θ). Formulas are derived for calculating the slope of the tangent line to a polar curve at a given point in terms of r and θ. Examples are worked out for finding the slope
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This document summarizes a lecture on fuzzy logic and neural networks. It introduces fuzzy sets and compares them to classical or crisp sets. Key concepts covered include fuzzy set representation using membership functions, common membership function types like triangular and trapezoidal, fuzzy set operations, and properties of fuzzy and crisp sets. Examples are provided to demonstrate calculating membership values and performing operations on fuzzy sets.
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In this slide we explain the application of cylindrical and Spherical coordinate system and their applications in double and Triple integrations. Calculus II - 10
The document discusses calculating the area of a surface obtained by rotating a curve around an axis. It provides the general formulas for finding the area which involves integrating the function squared plus the integral of its derivative. Several examples are worked out for rotating curves and arcs around the x-axis and y-axis. The document also gives an example problem of finding the force on a dam due to hydrostatic water pressure.
Chpt 2-sets v.3
This document provides definitions and notation for set theory concepts. It defines what a set is, ways to describe sets (explicitly by listing elements or implicitly using set builder notation), and basic set relationships like subset, proper subset, union, intersection, complement, power set, and Cartesian product. It also discusses Russell's paradox and defines important sets like the natural numbers. Key identities for set operations like idempotent, commutative, associative, distributive, De Morgan's laws, and complement laws are presented. Proofs of identities using logical equivalences and membership tables are demonstrated.
Truth, deduction, computation lecture g
This document provides information about lambda calculus and combinators. It includes definitions and examples of:
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- Defining basic operations like addition and multiplication
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Caculus II - 37
- 1. 13.3 The Normal and Binormal Vectors At a given point on a space curve ( ), the unit tangent vector is () ()= | ( )| Since ( )· ( )= we have ( )· ( )= so () () We define the principal unit normal vector as () ()= | ( )|
- 2. We define the principal unit normal vector as () ()= | ( )| We define the binormal vector as ()= () () it is also unit. ( ), ( ), ( ) are three unit vectors, perpendicular to each other. They form a TNB frame at point ( ) .
- 3. The plane determined by the normal and binormal vectors at point is called the normal plane at . The plane determined by the tangent and normal vectors at point is called the osculating plane at . The circle that lies in the osculating plane towards the direction of , has the same tangent at and has radius / ( ) is called the osculating circle at . This circle describes the behavior of the curve at : it shares the same tangent, normal, curvature and osculating plane.
- 4. Ex: Find the unit normal and binormal vectors and the normal and osculating plane of the helix ( ) = , , at the point ( , , ).
- 5. Ex: Find the unit normal and binormal vectors and the normal and osculating plane of the helix ( ) = , , at the point ( , , ). ()= , ,
- 6. Ex: Find the unit normal and binormal vectors and the normal and osculating plane of the helix ( ) = , , at the point ( , , ). ()= , , () ()= = , , | ( )|
- 7. Ex: Find the unit normal and binormal vectors and the normal and osculating plane of the helix ( ) = , , at the point ( , , ). ()= , , () ()= = , , | ( )| ()= , ,
- 8. Ex: Find the unit normal and binormal vectors and the normal and osculating plane of the helix ( ) = , , at the point ( , , ). ()= , , () ()= = , , | ( )| ()= , , () ()= = , , | ( )|
- 9. Ex: Find the unit normal and binormal vectors and the normal and osculating plane of the helix ( ) = , , at the point ( , , ). ()= , , () ()= = , , | ( )| ()= , , () ()= = , , | ( )| ()= () ()= , ,
- 10. () ()= = , , | ( )| () ()= = , , | ( )| ()= () ()= , ,
- 11. () ()= = , , | ( )| () ()= = , , | ( )| ()= () ()= , , At the point ( , , ), = . ( )= , , ( )= , , ( )= , ,
- 12. ( )= , , ( )= , , ( )= , ,
- 13. ( )= , , ( )= , , ( )= , , The normal vector of the normal plane is ( ).
- 14. ( )= , , ( )= , , ( )= , , The normal vector of the normal plane is ( ). The normal plane is ( )+ ( )+ ( )= or + =
- 15. ( )= , , ( )= , , ( )= , , The normal vector of the normal plane is ( ). The normal plane is ( )+ ( )+ ( )= or + = The normal vector of the osculating plane is ( ).
- 16. ( )= , , ( )= , , ( )= , , The normal vector of the normal plane is ( ). The normal plane is ( )+ ( )+ ( )= or + = The normal vector of the osculating plane is ( ). The osculating plane is ( ) ( )+ ( )= or =
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