HOW TO USE INTEGRALS - University of Hawaii Mathematics
... strips, one arrives at the concept of a Riemann sum (or some variation on this idea). A theorem is then proved stating that under reasonable conditions such a Riemann sum will converge to a limit as the width of the rectangles goes to zero. The integral is then defined to be the limit guaranteed to ...
... strips, one arrives at the concept of a Riemann sum (or some variation on this idea). A theorem is then proved stating that under reasonable conditions such a Riemann sum will converge to a limit as the width of the rectangles goes to zero. The integral is then defined to be the limit guaranteed to ...
Antiderivatives - John Abbott College
... Problem 3: Given f ′′ (x) = 15 x − 12 with conditions f ′ (4) = 1 , f (1) = 3 . Find f (x) Z Solution 3: f ′ (x) = (15x1/2 − 12) dx = 10x3/2 − 12x + C1 replace x = 4 and f ′ (4) = 1 to solve for C1 : 1 = 10(4)3/2 − 12(4) + C1 =⇒ C1 = −30 =⇒ f ′ (x) = 10x3/2 − 12x − 30 Z f (x) = (10x3/2 − 12x − 30) d ...
... Problem 3: Given f ′′ (x) = 15 x − 12 with conditions f ′ (4) = 1 , f (1) = 3 . Find f (x) Z Solution 3: f ′ (x) = (15x1/2 − 12) dx = 10x3/2 − 12x + C1 replace x = 4 and f ′ (4) = 1 to solve for C1 : 1 = 10(4)3/2 − 12(4) + C1 =⇒ C1 = −30 =⇒ f ′ (x) = 10x3/2 − 12x − 30 Z f (x) = (10x3/2 − 12x − 30) d ...
4.6 Finding Antiderivatives
... The value of a definite integral of cos( x2 ) could still be approximated as accurately as needed by using Riemann sums or one of the numerical techniques in Sections 4.9 and 8.7, but no matter how hard we try, we cannot find a concise formula for an antiderivative of cos( x2 ) in order to use the F ...
... The value of a definite integral of cos( x2 ) could still be approximated as accurately as needed by using Riemann sums or one of the numerical techniques in Sections 4.9 and 8.7, but no matter how hard we try, we cannot find a concise formula for an antiderivative of cos( x2 ) in order to use the F ...
Notes on space complexity of integration of computable real
... It is known [1] that real function g(x) = 0 f (t)dt is polynomial-time computable real function on interval [0, 1] iff FP = #P wherein f is a polynomial-time computable real function on interval [0, 1]. It means integration of polynomial-time computable real functions is as hard as string functions ...
... It is known [1] that real function g(x) = 0 f (t)dt is polynomial-time computable real function on interval [0, 1] iff FP = #P wherein f is a polynomial-time computable real function on interval [0, 1]. It means integration of polynomial-time computable real functions is as hard as string functions ...
Multidimensional Calculus. Lectures content. Week 10 22. Tests for
... (Alternating series test or Leibniz test) Let bk ≥ 0 for all k and let {bk } be non-increasing. The series (−1)k bk converges if and only if lim (bk ) = 0. k→∞ ...
... (Alternating series test or Leibniz test) Let bk ≥ 0 for all k and let {bk } be non-increasing. The series (−1)k bk converges if and only if lim (bk ) = 0. k→∞ ...
For this assignment, we must write three definitions of a term we
... infinitely small numbers. The word calculus origins from Latin word, calculus, which means small pebbles used for counting on a counting frame. Although in general calculus means methods of calculation, in mathematics calculus is often referred as the study of calculation of continuous change by wor ...
... infinitely small numbers. The word calculus origins from Latin word, calculus, which means small pebbles used for counting on a counting frame. Although in general calculus means methods of calculation, in mathematics calculus is often referred as the study of calculation of continuous change by wor ...
Using Mapping Diagrams to Understand Functions
... • There is a sensible way to visualize them using “mapping diagrams.” • Examples of important function features (like slope and intercepts) can be illustrated with mapping diagrams. • Activities for students engage understanding both function and linearity concepts. • Mapping diagrams use simple str ...
... • There is a sensible way to visualize them using “mapping diagrams.” • Examples of important function features (like slope and intercepts) can be illustrated with mapping diagrams. • Activities for students engage understanding both function and linearity concepts. • Mapping diagrams use simple str ...
educative commentary on jee 2014 advanced mathematics papers
... same as the JEE in two tiers which prevailed for a few years. It was hoped that now that the number of candidates appearing for the JEE (Advanced) is manageable enough to permit evaluation by humans, the classic practice of requiring the candidates to give justifications for their answers would be ...
... same as the JEE in two tiers which prevailed for a few years. It was hoped that now that the number of candidates appearing for the JEE (Advanced) is manageable enough to permit evaluation by humans, the classic practice of requiring the candidates to give justifications for their answers would be ...
Continuity of Local Time: An applied perspective
... discontinuities in large scale concentration, diffusion coefficients, and transmission rates. For these purposes one may ignore processes with drift, and focus on discontinuities in diffusion rates and/or specific rate coefficients. This builds on related work of the authors where the focus was on i ...
... discontinuities in large scale concentration, diffusion coefficients, and transmission rates. For these purposes one may ignore processes with drift, and focus on discontinuities in diffusion rates and/or specific rate coefficients. This builds on related work of the authors where the focus was on i ...
Math 20B. Lecture Examples. Section 10.3. Convergence of series
... X (0.6)n converge or diverge and why? n+1 ...
... X (0.6)n converge or diverge and why? n+1 ...
De nition and some Properties of Generalized Elementary Functions
... of a real variable, which is a most broader class of functions that includes all the elementary functions. It is not claimed to be an original research article, but rather a note that could serve the students to see a proper mathematical denition of the term Generalized Elementary Function of a Rea ...
... of a real variable, which is a most broader class of functions that includes all the elementary functions. It is not claimed to be an original research article, but rather a note that could serve the students to see a proper mathematical denition of the term Generalized Elementary Function of a Rea ...
AP Calculus AB Course Outline
... Students will rise to the occasion, given the opportunity. For most students, Calculus is the culmination of all those years studying mathematics in school. It becomes the “tie that binds.” The course is designed to be challenging, but not overbearing; with an emphasis using the multifaceted approac ...
... Students will rise to the occasion, given the opportunity. For most students, Calculus is the culmination of all those years studying mathematics in school. It becomes the “tie that binds.” The course is designed to be challenging, but not overbearing; with an emphasis using the multifaceted approac ...
lecture - Dartmouth Math Home
... be contained in f −1 (V ), say a ∈ f −1 (V ). Then f (a) ∈ V . Thus an arbitrary nbhd of y must contain an element of f (a), and y ∈ f (A). Thus f (A) ⊂ f (A). (b implies c): Suppose that B is a closed subset of Y . Then A = f −1 (B) is a subset of X, and by the assumption of (b) we obtain f (A) ⊂ ...
... be contained in f −1 (V ), say a ∈ f −1 (V ). Then f (a) ∈ V . Thus an arbitrary nbhd of y must contain an element of f (a), and y ∈ f (A). Thus f (A) ⊂ f (A). (b implies c): Suppose that B is a closed subset of Y . Then A = f −1 (B) is a subset of X, and by the assumption of (b) we obtain f (A) ⊂ ...
Calculus II
... Besides addition of vectors and multiplication by the scalar there two different operation which allows to multiply vectors. Definition 11.3.1 The dot product (or scalar product, or inner product) a·b is a · b = a1 b1 + a2 b2 + a3 b3 . Theorem 11.3.2 Properties of the dot product are: (i). a · a = k ...
... Besides addition of vectors and multiplication by the scalar there two different operation which allows to multiply vectors. Definition 11.3.1 The dot product (or scalar product, or inner product) a·b is a · b = a1 b1 + a2 b2 + a3 b3 . Theorem 11.3.2 Properties of the dot product are: (i). a · a = k ...
On Malliavin`s proof of Hörmander`s theorem
... every x ∈ M, the tangent space of the leaf passing through x is given by Ex . In view of this S result, Hörmander’s condition is not surprising. Indeed, if we define E(x,t) = k≥0 V̂k (x, t), then this gives us a subbundle of Rn+1 which is integrable by construction of the V̂k . Note that the dimens ...
... every x ∈ M, the tangent space of the leaf passing through x is given by Ex . In view of this S result, Hörmander’s condition is not surprising. Indeed, if we define E(x,t) = k≥0 V̂k (x, t), then this gives us a subbundle of Rn+1 which is integrable by construction of the V̂k . Note that the dimens ...
Solutions 1. - UC Davis Mathematics
... • (a) The sequence diverges since it oscillates between 1 and 3. For example, if = 1, there is no number L such that |an − L| < for all sufficiently large n, since then we would have both |1 − L| < 1 (or 0 < L < 2) and |3 − L| < 1 or (2 < L < 4), which is impossible. So there is no L that satisf ...
... • (a) The sequence diverges since it oscillates between 1 and 3. For example, if = 1, there is no number L such that |an − L| < for all sufficiently large n, since then we would have both |1 − L| < 1 (or 0 < L < 2) and |3 − L| < 1 or (2 < L < 4), which is impossible. So there is no L that satisf ...
1 Prerequisites: conditional expectation, stopping time
... 1. the Gt+ -conditional characteristic of the vector (Bu , Bz ), z, u > t is the limit of Gw -conditional characteristic function of the vector (Bu , Bz ), when w decreases to t, 2. this limit is equal to the Gt -conditional characteristic of the vector (Bu , Bz ), z, u > t, 3. thus for any integrab ...
... 1. the Gt+ -conditional characteristic of the vector (Bu , Bz ), z, u > t is the limit of Gw -conditional characteristic function of the vector (Bu , Bz ), when w decreases to t, 2. this limit is equal to the Gt -conditional characteristic of the vector (Bu , Bz ), z, u > t, 3. thus for any integrab ...
Analysis of Functions - Chariho Regional School District
... In algebra 1, students analyzed linear, exponential, quadratic, absolute value, step, and piece-defined functions using different representations, and they interpreted linear, exponential, and quadratic functions that arose in applications within a context. In algebra 2, students analyzed and interp ...
... In algebra 1, students analyzed linear, exponential, quadratic, absolute value, step, and piece-defined functions using different representations, and they interpreted linear, exponential, and quadratic functions that arose in applications within a context. In algebra 2, students analyzed and interp ...
Trig Course Outline - Northwest Arkansas Community College
... shift, and 5 key points. Students should also be required to accurately graph two full periods of the function on paper. Familiarity with the unit circle and the ability to memorize or quickly determine the values of the trigonometric functions for common angles without the aid of a calculator is vi ...
... shift, and 5 key points. Students should also be required to accurately graph two full periods of the function on paper. Familiarity with the unit circle and the ability to memorize or quickly determine the values of the trigonometric functions for common angles without the aid of a calculator is vi ...
REVIEW FOR FINAL EXAM April 08, 2014 • Final Exam Review Session:
... n=1 approach a unique number L as n increase, that is, if an can be made arbitrarily close to L by taking n sufficiently large, then we say that the sequence {an } converges to L, denoted by lim an = L. If the terms of the sequence do not approach a single number as n increases, n→∞ ...
... n=1 approach a unique number L as n increase, that is, if an can be made arbitrarily close to L by taking n sufficiently large, then we say that the sequence {an } converges to L, denoted by lim an = L. If the terms of the sequence do not approach a single number as n increases, n→∞ ...
Advanced Stochastic Calculus I Fall 2007 Prof. K. Ramanan Chris Almost
... 1.2.2 Theorem (Daniell-Kolmogorov). Let {Q t (·), t ∈ I} be a family of finite dimensional distributions that satisfies (i) Q t (A) is invariant under permutation of the elements of t; and (ii) For any t = (t 1 , . . . , t n ), if s = (t 1 , . . . , t n−1 ) and A ∈ B(Rn−1 ) then Q t (A × R) = Q s (A ...
... 1.2.2 Theorem (Daniell-Kolmogorov). Let {Q t (·), t ∈ I} be a family of finite dimensional distributions that satisfies (i) Q t (A) is invariant under permutation of the elements of t; and (ii) For any t = (t 1 , . . . , t n ), if s = (t 1 , . . . , t n−1 ) and A ∈ B(Rn−1 ) then Q t (A × R) = Q s (A ...
RATIONAL FUNCTIONS AND REAL SCHUBERT CALCULUS 1
... So p0 = 0 and we conclude from (16) that p0 = 0. This proves the theorem. In the limit, when in each group Aj all points collide to one point xj we recover from Theorem 10 the main result of [1]: all rational functions whose critical points are real are equivalent to real rational functions. In this ...
... So p0 = 0 and we conclude from (16) that p0 = 0. This proves the theorem. In the limit, when in each group Aj all points collide to one point xj we recover from Theorem 10 the main result of [1]: all rational functions whose critical points are real are equivalent to real rational functions. In this ...
PDF
... every continuous function has a well defined tangent - at least at “almost all” points. As the Weierstrass function shows that this is clearly not the case. The function is named after Karl Weierstrass who presented it in a lecture for the Berlin Academy in 1872 [?]. Alternative examples of continuo ...
... every continuous function has a well defined tangent - at least at “almost all” points. As the Weierstrass function shows that this is clearly not the case. The function is named after Karl Weierstrass who presented it in a lecture for the Berlin Academy in 1872 [?]. Alternative examples of continuo ...
Sets and functions
... X1 ∪ X2 = {x : x ∈ X1 or x ∈ X2 }. Thus X1 ⊆ (X1 ∪ X2 ) and X2 ⊆ (X1 ∪ X2 ). 2. The intersection of X1 and X2 is: X1 ∩ X2 = {x : x ∈ X1 and x ∈ X2 }. Thus (X1 ∩ X2 ) ⊆ X1 and (X1 ∩ X2 ) ⊆ X2 . 3. Given two sets X1 and X2 , the complement of X2 in X1 , written X1 − X2 , is the set {x ∈ X1 : x ∈ / X2 ...
... X1 ∪ X2 = {x : x ∈ X1 or x ∈ X2 }. Thus X1 ⊆ (X1 ∪ X2 ) and X2 ⊆ (X1 ∪ X2 ). 2. The intersection of X1 and X2 is: X1 ∩ X2 = {x : x ∈ X1 and x ∈ X2 }. Thus (X1 ∩ X2 ) ⊆ X1 and (X1 ∩ X2 ) ⊆ X2 . 3. Given two sets X1 and X2 , the complement of X2 in X1 , written X1 − X2 , is the set {x ∈ X1 : x ∈ / X2 ...
Lebesgue integration
In mathematics, the integral of a non-negative function can be regarded, in the simplest case, as the area between the graph of that function and the x-axis. The Lebesgue integral extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined.Mathematicians had long understood that for non-negative functions with a smooth enough graph—such as continuous functions on closed bounded intervals—the area under the curve could be defined as the integral, and computed using approximation techniques on the region by polygons. However, as the need to consider more irregular functions arose—e.g., as a result of the limiting processes of mathematical analysis and the mathematical theory of probability—it became clear that more careful approximation techniques were needed to define a suitable integral. Also, we might wish to integrate on spaces more general than the real line. The Lebesgue integral provides the right abstractions needed to do this important job.The Lebesgue integral plays an important role in the branch of mathematics called real analysis, and in many other mathematical sciences fields. It is named after Henri Lebesgue (1875–1941), who introduced the integral (Lebesgue 1904). It is also a pivotal part of the axiomatic theory of probability.The term Lebesgue integration can mean either the general theory of integration of a function with respect to a general measure, as introduced by Lebesgue—or the specific case of integration of a function defined on a sub-domain of the real line with respect to Lebesgue measure.