Here are the key points about g given f:
- g represents the area under the curve of f over successive intervals of the x-axis
- As x increases over an interval, g will increase if f is positive over that interval and decrease if f is negative
- The concavity (convexity or concavity) of g will match the concavity of f over each interval
In summary, the area function g, as defined by the integral of f, will have properties that correspond directly to the sign and concavity of f over successive intervals of integration.
This document discusses geodesic data processing on Riemannian manifolds. It defines geodesic distances as the shortest path between two points on the manifold according to the Riemannian metric. Methods are presented for computing geodesic distances and curves, including iterative schemes and fast marching. Applications discussed include shape recognition using geodesic statistics and geodesic meshing.
1. Geodesic sampling and meshing techniques can be used to generate adaptive triangulations and meshes on Riemannian manifolds based on a metric tensor.
2. Anisotropic metrics can be defined to generate meshes adapted to features like edges in images or curvature on surfaces. Triangles will be elongated along strong features to better approximate functions.
3. Farthest point sampling can be used to generate well-spaced point distributions over manifolds according to a metric, which can then be triangulated using geodesic Delaunay refinement.
Lesson 26: The Fundamental Theorem of Calculus (Section 041 handout)Matthew Leingang
This document contains lecture notes on the fundamental theorem of calculus from a Calculus I class. The notes discuss:
1) The first and second fundamental theorems of calculus, which relate differentiation and integration as inverse processes.
2) How to use the first fundamental theorem to differentiate functions defined by integrals.
3) Biographies of several mathematicians involved in the development of calculus, including Newton, Leibniz, Gregory and Barrow.
We study QPT (quasi-polynomial tractability) in the worst case setting of linear tensor product problems defined over Hilbert spaces. We prove QPT for algorithms that use only function values under three assumptions'
1. the minimal errors for the univariate case decay polynomially fast to zero,
2. the largest singular value for the univariate case is simple,
3. the eigenfunction corresponding to the largest singular value is a multiple of the function value at some point.
The first two assumptions are necessary for QPT. The third assumption is necessary for QPT for some Hilbert spaces.
Joint work with Erich Novak
Coincidence points for mappings under generalized contractionAlexander Decker
1. The document presents a theorem that establishes conditions for the existence of coincidence points between multi-valued and single-valued mappings.
2. It generalizes previous results by Feng and Liu (2004) and Liu et al. (2005) by relaxing the contraction conditions.
3. The main theorem proves that if mappings T and f satisfy generalized contraction conditions involving α and β functions, and the space is orbitally complete, then the mappings have a coincidence point.
ABC convergence under well- and mis-specified modelsChristian Robert
1. Approximate Bayesian computation (ABC) is a simulation-based method for performing Bayesian inference when the likelihood function is intractable or unavailable. ABC works by simulating data from the model, accepting simulations where the simulated and observed data are close according to some distance measure.
2. Advances in ABC include modifying the proposal distribution to increase efficiency, viewing it as a conditional density estimation problem to allow for larger tolerances, and including a tolerance parameter in the inferential framework.
3. Recent studies have analyzed the asymptotic properties of ABC, showing the posterior distributions and means can be consistent under certain conditions on the summary statistics and tolerance decreasing rates.
Coordinate sampler: A non-reversible Gibbs-like samplerChristian Robert
This document describes a new MCMC method called the Coordinate Sampler. It is a non-reversible Gibbs-like sampler based on a piecewise deterministic Markov process (PDMP). The Coordinate Sampler generalizes the Bouncy Particle Sampler by making the bounce direction partly random and orthogonal to the gradient. It is proven that under certain conditions, the PDMP induced by the Coordinate Sampler has a unique invariant distribution of the target distribution multiplied by a uniform auxiliary variable distribution. The Coordinate Sampler is also shown to exhibit geometric ergodicity, an important convergence property, under additional regularity conditions on the target distribution.
Slides: A glance at information-geometric signal processingFrank Nielsen
This document discusses information geometry and its applications in statistical signal processing. It introduces several key concepts:
1) Statistical signal processing models data with probability distributions like Gaussians and histograms. Information geometry provides a geometric framework for intuitive reasoning about these statistical models.
2) Exponential family mixture models generalize Gaussian and Rayleigh mixtures and are algorithmically useful in dually flat spaces.
3) Distances between statistical models, like Kullback-Leibler divergence and Bregman divergences, can be interpreted geometrically in terms of convex conjugates and Legendre transformations.
Patch Matching with Polynomial Exponential Families and Projective DivergencesFrank Nielsen
This document presents a method called Polynomial Exponential Family-Patch Matching (PEF-PM) to solve the patch matching problem. PEF-PM models patch colors using polynomial exponential families (PEFs), which are universal smooth positive densities. It estimates PEFs using a Score Matching Estimator and accelerates batch estimation using Summed Area Tables. Patch similarity is measured using a statistical projective divergence called the symmetrized γ-divergence. Experiments show PEF-PM handles noise robustly, symmetries, and outperforms baseline methods.
This document discusses Bayesian model comparison in cosmology using population Monte Carlo methods. It provides background on key questions in cosmology that can be addressed using cosmic microwave background data from experiments like WMAP and Planck. Population Monte Carlo and adaptive importance sampling methods are introduced to help approximate Bayesian evidence for different cosmological models given the immense computational challenges of working with this cosmological data.
The generation of Gaussian random fields over a physical domain is a challenging problem in computational mathematics, especially when the correlation length is short and the field is rough. The traditional approach is to make use of a truncated Karhunen-Loeve (KL) expansion, but the generation of even a single realisation of the field may then be effectively beyond reach (especially for 3-dimensional domains) if the need is to obtain an expected L2 error of say 5%, because of the potentially very slow convergence of the KL expansion. In this talk, based on joint work with Ivan Graham, Frances Kuo, Dirk Nuyens, and Rob Scheichl, a completely different approach is used, in which the field is initially generated at a regular grid on a 2- or 3-dimensional rectangle that contains the physical domain, and then possibly interpolated to obtain the field at other points. In that case there is no need for any truncation. Rather the main problem becomes the factorisation of a large dense matrix. For this we use circulant embedding and FFT ideas. Quasi-Monte Carlo integration is then used to evaluate the expected value of some functional of the finite-element solution of an elliptic PDE with a random field as input.
Multiple estimators for Monte Carlo approximationsChristian Robert
This document discusses multiple estimators that can be used to approximate integrals using Monte Carlo simulations. It begins by introducing concepts like multiple importance sampling, Rao-Blackwellisation, and delayed acceptance that allow combining multiple estimators to improve accuracy. It then discusses approaches like mixtures as proposals, global adaptation, and nonparametric maximum likelihood estimation (NPMLE) that frame Monte Carlo estimation as a statistical estimation problem. The document notes various advantages of the statistical formulation, like the ability to directly estimate simulation error from the Fisher information. Overall, the document presents an overview of different techniques for combining Monte Carlo simulations to obtain more accurate integral approximations.
This document summarizes multivariate extreme value theory and methods for analyzing the joint behavior of extremes from multiple variables. It discusses three main approaches:
1) Limit theorems for multivariate sample maxima, which characterize the limiting distribution of component-wise maxima.
2) Alternative formulations by Ledford-Tawn and Heffernan-Tawn that allow for more flexible dependence structures between variables.
3) Max-stable processes, which generalize univariate extreme value distributions to the multivariate case through the use of exponent measures.
Estimation of multivariate extreme value models poses challenges due to their nonregular behavior and potential for high dimensionality. Most methods transform to unit Fréchet margins before modeling dependence structure.
k-MLE: A fast algorithm for learning statistical mixture modelsFrank Nielsen
This document describes a fast algorithm called k-MLE for learning statistical mixture models. k-MLE is based on the connection between exponential family mixture models and Bregman divergences. It extends Lloyd's k-means clustering algorithm to optimize the complete log-likelihood of an exponential family mixture model using Bregman divergences. The algorithm iterates between assigning data points to clusters based on Bregman divergence, and updating the cluster parameters by taking the Bregman centroid of each cluster's assigned points. This provides a fast method for maximum likelihood estimation of exponential family mixture models.
The document summarizes Approximate Bayesian Computation (ABC). It discusses how ABC provides a way to approximate Bayesian inference when the likelihood function is intractable or too computationally expensive to evaluate directly. ABC works by simulating data under different parameter values and accepting simulations that are close to the observed data according to a distance measure and tolerance level. Key points discussed include:
- ABC provides an approximation to the posterior distribution by sampling from simulations that fall within a tolerance of the observed data.
- Summary statistics are often used to reduce the dimension of the data and improve the signal-to-noise ratio when applying the tolerance criterion.
- Random forests can help select informative summary statistics and provide semi-automated ABC
1) Likelihood-free Bayesian experimental design is discussed as an intractable likelihood optimization problem, where the goal is to find the optimal design d that minimizes expected loss without using the full posterior distribution.
2) Several Bayesian tools are proposed to make the design problem more Bayesian, including Bayesian non-parametrics, annealing algorithms, and placing a posterior on the design d.
3) Gaussian processes are a default modeling choice for complex unknown functions in these problems, but their accuracy is difficult to assess and they may incur a dimension curse.
1. The document proposes a method for making approximate Bayesian computation (ABC) inferences accurate by modeling the distribution of summary statistics calculated from simulated and observed data.
2. It involves constructing an auxiliary probability space (ρ-space) based on these summary values, and performing classification on ρ-space to determine whether simulated and observed data are from the same population.
3. Indirect inference is then used to link ρ-space back to the original parameter space, allowing the ABC approximation to match the true posterior distribution if the ABC tolerances and number of simulations are properly calibrated.
Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)Matthew Leingang
The First Fundamental Theorem of Calculus looks at the area function and its derivative. It so happens that the derivative of the area function is the original integrand.
This document summarizes a calculus lecture on the Mean Value Theorem. It begins with announcements about exams and assignments. It then outlines the topics to be covered: Rolle's Theorem and the Mean Value Theorem, applications of the MVT, and why the MVT is important. It provides heuristic motivations and mathematical statements of Rolle's Theorem and the Mean Value Theorem. It also includes a proof of the Mean Value Theorem using Rolle's Theorem.
- The document is a section from a calculus course at NYU that discusses using derivatives to determine the shapes of curves.
- It covers using the first derivative to determine if a function is increasing or decreasing over an interval using the Increasing/Decreasing Test. It also discusses using the second derivative to determine if a function is concave up or down over an interval using the Second Derivative Test.
- Examples are provided to demonstrate finding intervals of monotonicity for functions and classifying critical points as local maxima, minima or neither using the First Derivative Test.
This document provides information about a Calculus I course taught by Professor Matthew Leingang at the Courant Institute of Mathematical Sciences at NYU. The course will cover fundamentals of calculus including limits, derivatives, integrals, and optimization. It will meet twice a week for lectures and recitations. Assessment will include weekly homework, biweekly quizzes, a midterm exam, and a final exam. Grades will be calculated based on scores on these assessments. The required textbook can be purchased in hardcover, looseleaf, or online formats. Students are encouraged to contact the professor or TAs with any questions.
This document is a section from a Calculus I course at New York University covering maximum and minimum values. It begins with announcements about exams and assignments. The objectives are to understand the Extreme Value Theorem and Fermat's Theorem, and to use the Closed Interval Method to find extreme values. The document then covers the definitions of extreme points/values and the Extreme Value Theorem, which states that a continuous function on a closed interval attains maximum and minimum values. Examples are given to show the importance of the hypotheses in the theorem.
This document is from a Calculus I class at New York University and covers antiderivatives. It begins with announcements about an upcoming quiz. The objectives are to find antiderivatives of simple functions, remember that a function whose derivative is zero must be constant, and solve rectilinear motion problems. It then outlines finding antiderivatives through tabulation, graphically, and with rectilinear motion examples. Examples are provided of finding the antiderivative of power functions like x^3 through identifying the power rule relationship between a function and its derivative.
Lesson 17: Indeterminate Forms and L'Hôpital's RuleMatthew Leingang
This document appears to be a lecture on indeterminate forms and L'Hopital's rule from a Calculus I course at New York University. It includes:
1) An introduction to different types of indeterminate forms such as 0/0, infinity/infinity, 0*infinity, etc.
2) Examples of limits that are indeterminate forms and experiments calculating their values.
3) A discussion of dividing limits and exceptions where the limit may not exist even when the numerator approaches a finite number and denominator approaches zero.
4) An outline of the topics to be covered, including L'Hopital's rule, other indeterminate limits, and a summary.
This document provides an overview of integration by substitution. It begins with announcements about an upcoming review session, evaluations, and final exam. It then discusses the objectives and outline of the section, which are to transform integrals using substitutions, evaluate indefinite integrals using substitutions, and evaluate definite integrals using substitutions. An example is provided to illustrate how to use substitution to evaluate the indefinite integral of x/(x^2 + 1) by letting u = x^2 + 1. The solution uses a new notation of letting u = x^2 + 1 and du = 2x dx to rewrite the integral in terms of u.
The document is a lecture on inverse trigonometric functions from a Calculus I class at New York University. It defines inverse trig functions like arcsin, arccos, and arctan and discusses their domains, ranges, and relationships to the original trig functions. It also provides examples of evaluating inverse trig functions at specific values.
This document is from a Calculus I class at New York University. It provides an overview of functions, including the definition of a function, different ways functions can be represented (formulas, tables, graphs, verbal descriptions), properties of functions like monotonicity and symmetry, and examples of determining domains and ranges of functions. It aims to help students understand functions and their representations as a foundation for calculus.
This document contains notes from a Calculus I class at New York University. It discusses related rates problems, which involve taking derivatives of equations relating changing quantities to determine rates of change. The document provides examples of related rates problems involving an oil slick, two people walking towards and away from each other, and electrical resistors. It also outlines strategies for solving related rates problems, such as drawing diagrams, introducing notation, relating quantities with equations, and using the chain rule to solve for unknown rates.
Silberberg Chemistry Molecular Nature Of Matter And Change 4e Copy2jeksespina
There are three main types of radioactive decay: alpha, beta, and gamma. Alpha decay involves emitting an alpha particle (helium nucleus) which decreases the mass and atomic numbers by 4 and 2 respectively. Beta decay involves emitting an electron or positron, which does not change mass number but increases or decreases the atomic number by 1. Gamma decay involves emitting high-energy photons without changing the nucleus. Nuclear equations must balance the total numbers of protons and nucleons between reactants and products.
The document is about calculating areas and distances using calculus. It discusses calculating areas of rectangles, parallelograms, triangles and other polygons. For curved regions, it discusses Archimedes' method of approximating areas using inscribed polygons and letting the number of sides approach infinity. It also discusses calculating distances traveled as the limit of approximating distances over smaller time intervals. The objectives are to compute areas using approximating rectangles and distances using approximating time intervals.
- The document is from a Calculus I class at New York University and covers evaluating definite integrals.
- It discusses using the Evaluation Theorem to evaluate definite integrals, writing antiderivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval.
- Examples are provided of using the midpoint rule to estimate a definite integral, and properties of definite integrals like additivity and comparison properties are reviewed.
The document discusses the definite integral, including computing it using Riemann sums, estimating it using approximations like the midpoint rule, and reasoning about its properties. It outlines the topics to be covered, such as recalling previous concepts and comparing properties of integrals. Formulas are provided for calculating Riemann sums using different representative points within the intervals.
This document discusses using linear approximations to estimate functions. It provides an example estimating sin(61°) using linear approximations about a=0 and a=60°. When approximating about a=0, the estimate is 1.06465. When approximating about a=60°, the estimate is 0.87475, which is closer to the actual value of sin(61°) according to a calculator check. The document teaches that the tangent line provides the best linear approximation near a point, and its equation can be used to estimate function values.
The document provides an overview of curve sketching in calculus. It discusses objectives like sketching a function graph completely by identifying zeros, asymptotes, critical points, maxima/minima and inflection points. Examples are provided to demonstrate the increasing/decreasing test using derivatives and the concavity test to determine concave up/down regions. A step-by-step process is outlined to graph functions which involves analyzing monotonicity using the sign chart of the derivative and concavity using the second derivative sign chart. This is demonstrated on sketching the graph of a cubic function f(x)=2x^3-3x^2-12x.
The document summarizes the steps to solve optimization problems using calculus. It begins with an example of finding the rectangle with maximum area given a fixed perimeter. It works through the solution, identifying the objective function, variables, constraints, and using calculus techniques like taking the derivative to find critical points. The document then outlines Polya's 4-step method for problem solving and provides guidance on setting up optimization problems by understanding the problem, introducing notation, drawing diagrams, and eliminating variables using given constraints. It emphasizes using the Closed Interval Method, evaluating the function at endpoints and critical points to find extreme values over a domain.
The basic rules of differentiation, including the power rule, the sum rule, the constant multiple rule, and the rule for differentiating sine and cosine.
Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)Mel Anthony Pepito
The document provides an overview of Section 5.4 on the Fundamental Theorem of Calculus from a Calculus I course at New York University. It outlines topics to be covered, including recalling the Second Fundamental Theorem, stating the First Fundamental Theorem, and differentiating functions defined by integrals. Examples are provided to illustrate using the theorems to find derivatives and integrals.
Lesson 26: The Fundamental Theorem of Calculus (Section 041 slides)Mel Anthony Pepito
The document discusses the fundamental theorem of calculus. It begins by outlining the topics to be covered, including a review of the second fundamental theorem of calculus, an explanation of the first fundamental theorem of calculus, and examples of differentiating functions defined by integrals. It then provides more details on these topics, such as defining the integral as a limit, stating the second fundamental theorem, using integrals to represent concepts like distance traveled and mass, and working through an example of finding the derivative of a function defined by an integral.
This document contains notes from a calculus class. It provides the outline and key points about the Fundamental Theorem of Calculus. It discusses the first and second Fundamental Theorems of Calculus, including proofs and examples. It also provides brief biographies of several important mathematicians that contributed to the development of calculus, including the Fundamental Theorem of Calculus, such as Isaac Newton, Gottfried Leibniz, James Gregory, and Isaac Barrow.
This document contains lecture notes on the Fundamental Theorem of Calculus. It begins with announcements about the final exam date and current grade distribution. The outline then reviews the Evaluation Theorem and introduces the First and Second Fundamental Theorems of Calculus. It provides examples of how the integral can represent total change in concepts like distance, cost, and mass. Biographies are also included of mathematicians like Gregory, Barrow, Newton, and Leibniz who contributed to the development of calculus.
The document is a lecture note on the fundamental theorem of calculus from a Calculus I class at New York University. It provides announcements about upcoming exams and assignments. It then outlines the key topics to be covered, including the first fundamental theorem of calculus and how to differentiate functions defined by integrals. Examples are provided to illustrate using integrals to find the area under a curve and how this relates to the derivative of the area function.
The document is a section from a Calculus I course at NYU from June 22, 2010. It discusses using the substitution method to evaluate indefinite integrals. Specifically, it provides an example of using the substitution u=x^2+3 to evaluate the integral of (x^2+3)^3 4x dx. The solution transforms the integral into an integral of u^3 du and evaluates it to be 1/4 u^4 + C.
Lesson 27: Integration by Substitution (Section 041 slides)Matthew Leingang
The document contains notes from a Calculus I class at New York University on December 13, 2010. It discusses using the substitution method for indefinite and definite integrals. Examples are provided to demonstrate how to use substitutions to evaluate integrals involving trigonometric, exponential, and polynomial functions. The key steps are to make a substitution for the variable in terms of a new variable, determine the differential of the substitution, and substitute into the integral to transform it into an integral involving only the new variable.
Lesson 27: Integration by Substitution (Section 041 slides)Mel Anthony Pepito
The document provides notes from a Calculus I class at New York University on December 13, 2010. It includes announcements about upcoming review sessions and the final exam. It then outlines the objectives and topics to be covered, which is integration by substitution. Examples are provided to demonstrate how to use u-substitution to evaluate indefinite integrals. The key steps are to let u equal an expression involving x, find du in terms of dx, and then substitute into the integral.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
This document is a section from a Calculus I course at New York University that discusses basic differentiation rules. It covers objectives like differentiating constant, sum, and difference functions. It also covers derivatives of sine and cosine. Examples are provided, like finding the derivative of the squaring function x^2, which is 2x. Notation for derivatives is explained, including Leibniz notation. The concept of the second derivative is also introduced.
This document contains lecture notes on differentiation rules from a Calculus I class at New York University. It begins with objectives to understand basic differentiation rules like the derivative of a constant function, the Constant Multiple Rule, the Sum Rule, and derivatives of sine and cosine. It then provides examples of using the definition of the derivative to find the derivatives of squaring and cubing functions. It illustrates the functions and their derivatives on graphs and discusses properties like a function being increasing when its derivative is positive.
This document is a section from a Calculus I course at New York University that discusses basic differentiation rules. It provides objectives for understanding rules like the derivative of a constant function, the constant multiple rule, sum rule, and derivatives of sine and cosine functions. It then gives examples of finding the derivative of a squaring function using the definition, and introduces the concept of the second derivative.
This document is a section from a Calculus I course at New York University that discusses basic differentiation rules. It covers objectives like differentiating constant, sum, and difference functions. It also reviews derivatives of sine and cosine. Examples are provided, like finding the derivative of a squaring function x^2 using the definition of a derivative. The document outlines the topics and provides explanations and step-by-step solutions.
The document is a lecture note on derivatives and the shapes of curves. It discusses the mean value theorem and its applications to determining monotonicity and concavity of functions. Specifically, it covers:
- Using the first derivative test to find intervals where a function is increasing or decreasing by determining where the derivative is positive or negative
- Examples of applying this process to functions like x2 - 1 and x2/3(x + 2)
- Definitions of increasing, decreasing, and concavity
- How the second derivative test can determine concavity by examining the sign of the second derivative
Lesson 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
The document discusses the Fundamental Theorem of Calculus, which has two parts. The first part states that if a function f is continuous on an interval, then the derivative of the integral of f is equal to f. This is proven using Riemann sums. The second part relates the integral of a function f to the integral of its derivative F'. Examples are provided to illustrate how the area under a curve relates to these concepts.
Lesson 26: The Fundamental Theorem of Calculus (slides)Mel Anthony Pepito
The document discusses the Fundamental Theorem of Calculus, which has two parts. The first part states that if a function f is continuous, then the derivative of the integral of f is equal to f. This is proven using Riemann sums. The second part relates the integral of a function f to the anti-derivative F of f. Examples are provided to illustrate how to use the Fundamental Theorem to find derivatives and integrals.
Lesson 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
g(x) represents the area under the curve of f(t) between 0 and x.
.
x
What can you say about g? 2 4 6 8 10f
The First Fundamental Theorem of Calculus
Theorem (First Fundamental Theorem of Calculus)
Let f be a con nuous func on on [a, b]. Define the func on F on [a, b] by
∫ x
F(x) = f(t) dt
a
Then F is con nuous on [a, b] and differentiable on (a, b) and for all x in (a, b),
F′(x
Lesson 20: Derivatives and the Shape of Curves (Section 041 handout)Matthew Leingang
1) The document discusses using derivatives to determine the shapes of curves, including intervals of increasing/decreasing behavior, local extrema, and concavity.
2) Key tests covered are the increasing/decreasing test, first derivative test, concavity test, and second derivative test. These allow determining monotonicity, local extrema, and concavity from the signs of the first and second derivatives.
3) Examples demonstrate applying these tests to find the intervals of monotonicity, points of local extrema, and intervals of concavity for various functions.
No, this is not a function because the same input of 2 is mapped to two different outputs of 4 and 5. For a relation to be a function, each input must map to a unique output.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 17 / 33
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This document provides guidance on developing effective lesson plans for calculus instructors. It recommends starting by defining specific learning objectives and assessments. Examples should be chosen carefully to illustrate concepts and engage students at a variety of levels. The lesson plan should include an introductory problem, definitions, theorems, examples, and group work. Timing for each section should be estimated. After teaching, the lesson can be improved by analyzing what was effective and what needs adjustment for the next time. Advanced preparation is key to looking prepared and ensuring students learn.
Streamlining assessment, feedback, and archival with auto-multiple-choiceMatthew Leingang
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This document discusses electronic grading of paper assessments using PDF forms. Key points include:
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Integration by substitution is the chain rule in reverse.
NOTE: the final location is section specific. Section 1 (morning) is in SILV 703, Section 11 (afternoon) is in CANT 200
Lesson 27: Integration by Substitution (handout)Matthew Leingang
This document contains lecture notes on integration by substitution from a Calculus I class. It introduces the technique of substitution for both indefinite and definite integrals. For indefinite integrals, the substitution rule is presented, along with examples of using substitutions to evaluate integrals involving polynomials, trigonometric, exponential, and other functions. For definite integrals, the substitution rule is extended and examples are worked through both with and without first finding the indefinite integral. The document emphasizes that substitution often simplifies integrals and makes them easier to evaluate.
Lesson 26: The Fundamental Theorem of Calculus (handout)Matthew Leingang
1) The document discusses lecture notes on Section 5.4: The Fundamental Theorem of Calculus from a Calculus I course. 2) It covers stating and explaining the Fundamental Theorems of Calculus and using the first fundamental theorem to find derivatives of functions defined by integrals. 3) The lecture outlines the first fundamental theorem, which relates differentiation and integration, and gives examples of applying it.
This document contains notes from a calculus class lecture on evaluating definite integrals. It discusses using the evaluation theorem to evaluate definite integrals, writing derivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. The document also contains examples of evaluating definite integrals, properties of integrals, and an outline of the key topics covered.
This document contains lecture notes from a Calculus I class covering Section 5.3 on evaluating definite integrals. The notes discuss using the Evaluation Theorem to calculate definite integrals, writing derivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. Examples are provided to demonstrate evaluating definite integrals using the midpoint rule approximation. Properties of integrals such as additivity and the relationship between definite and indefinite integrals are also outlined.
Lesson 24: Areas and Distances, The Definite Integral (handout)Matthew Leingang
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
Lesson 24: Areas and Distances, The Definite Integral (slides)Matthew Leingang
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
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Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few good examples.
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Lesson 20: Derivatives and the Shapes of Curves (handout)Matthew Leingang
This document contains lecture notes on calculus from a Calculus I course. It covers determining the monotonicity of functions using the first derivative test. Key points include using the sign of the derivative to determine if a function is increasing or decreasing over an interval, and using the first derivative test to classify critical points as local maxima, minima, or neither. Examples are provided to demonstrate finding intervals of monotonicity for various functions and applying the first derivative test.
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
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3. **Education**: FDM plays a crucial role in education by enabling students to learn about design and engineering through hands-on 3D printing projects. It promotes innovation and practical skill development in STEM disciplines.
4. **Science**: Researchers use FDM to prototype equipment for scientific experiments, build custom laboratory tools, and create models for visualization and testing purposes. It facilitates rapid iteration and customization in scientific endeavors.
5. **Automotive**: Automotive manufacturers employ FDM for prototyping vehicle components, tooling for assembly lines, and customized parts. It speeds up the design validation process and enhances efficiency in automotive engineering.
6. **Consumer Electronics**: FDM is utilized in consumer electronics for designing and prototyping product enclosures, casings, and internal components. It enables rapid iteration and customization to meet evolving consumer demands.
7. **Robotics**: Robotics engineers leverage FDM to prototype robot parts, create lightweight and durable components, and customize robot designs for specific applications. It supports innovation and optimization in robotic systems.
8. **Aerospace**: In aerospace, FDM is used to manufacture lightweight parts, complex geometries, and prototypes of aircraft components. It contributes to cost reduction, faster production cycles, and weight savings in aerospace engineering.
9. **Architecture**: Architects utilize FDM for creating detailed architectural models, prototypes of building components, and intricate designs. It aids in visualizing concepts, testing structural integrity, and communicating design ideas effectively.
Each industry example demonstrates how FDM enhances innovation, accelerates product development, and addresses specific challenges through advanced manufacturing capabilities.
Fluttercon 2024: Showing that you care about security - OpenSSF Scorecards fo...Chris Swan
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You can do the same for your projects, and this presentation will show you how, with an emphasis on the unique challenges that come up when working with Dart and Flutter.
The session will provide a walkthrough of the steps involved in securing a first repository, and then what it takes to repeat that process across an organization with multiple repos. It will also look at the ongoing maintenance involved once scorecards have been implemented, and how aspects of that maintenance can be better automated to minimize toil.
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This brochure gives introduction of MYIR Electronics company and MYIR's products and services.
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comprehensive solutions based on various architectures such as ARM, FPGA, RISC-V, and AI. We cater to customers' needs for large-scale production, offering customized design, industry-specific application solutions, and one-stop OEM services.
MYIR, recognized as a national high-tech enterprise, is also listed among the "Specialized
and Special new" Enterprises in Shenzhen, China. Our core belief is that "Our success stems from our customers' success" and embraces the philosophy
of "Make Your Idea Real, then My Idea Realizing!"
Performance Budgets for the Real World by Tammy EvertsScyllaDB
Performance budgets have been around for more than ten years. Over those years, we’ve learned a lot about what works, what doesn’t, and what we need to improve. In this session, Tammy revisits old assumptions about performance budgets and offers some new best practices. Topics include:
• Understanding performance budgets vs. performance goals
• Aligning budgets with user experience
• Pros and cons of Core Web Vitals
• How to stay on top of your budgets to fight regressions
Scaling Connections in PostgreSQL Postgres Bangalore(PGBLR) Meetup-2 - MydbopsMydbops
This presentation, delivered at the Postgres Bangalore (PGBLR) Meetup-2 on June 29th, 2024, dives deep into connection pooling for PostgreSQL databases. Aakash M, a PostgreSQL Tech Lead at Mydbops, explores the challenges of managing numerous connections and explains how connection pooling optimizes performance and resource utilization.
Key Takeaways:
* Understand why connection pooling is essential for high-traffic applications
* Explore various connection poolers available for PostgreSQL, including pgbouncer
* Learn the configuration options and functionalities of pgbouncer
* Discover best practices for monitoring and troubleshooting connection pooling setups
* Gain insights into real-world use cases and considerations for production environments
This presentation is ideal for:
* Database administrators (DBAs)
* Developers working with PostgreSQL
* DevOps engineers
* Anyone interested in optimizing PostgreSQL performance
Contact info@mydbops.com for PostgreSQL Managed, Consulting and Remote DBA Services
For the full video of this presentation, please visit: https://www.edge-ai-vision.com/2024/07/intels-approach-to-operationalizing-ai-in-the-manufacturing-sector-a-presentation-from-intel/
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Sustainability requires ingenuity and stewardship. Did you know Pigging Solutions pigging systems help you achieve your sustainable manufacturing goals AND provide rapid return on investment.
How? Our systems recover over 99% of product in transfer piping. Recovering trapped product from transfer lines that would otherwise become flush-waste, means you can increase batch yields and eliminate flush waste. From raw materials to finished product, if you can pump it, we can pig it.
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2. Harnessing the power of GenAI for your business by Siddharth
3. Fallacies of GenAI by Raju Kandaswamy
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Quantum Communications Q&A with Gemini LLM. These are based on Shannon's Noisy channel Theorem and offers how the classical theory applies to the quantum world.
UiPath Community Day Kraków: Devs4Devs ConferenceUiPathCommunity
We are honored to launch and host this event for our UiPath Polish Community, with the help of our partners - Proservartner!
We certainly hope we have managed to spike your interest in the subjects to be presented and the incredible networking opportunities at hand, too!
Check out our proposed agenda below 👇👇
08:30 ☕ Welcome coffee (30')
09:00 Opening note/ Intro to UiPath Community (10')
Cristina Vidu, Global Manager, Marketing Community @UiPath
Dawid Kot, Digital Transformation Lead @Proservartner
09:10 Cloud migration - Proservartner & DOVISTA case study (30')
Marcin Drozdowski, Automation CoE Manager @DOVISTA
Pawel Kamiński, RPA developer @DOVISTA
Mikolaj Zielinski, UiPath MVP, Senior Solutions Engineer @Proservartner
09:40 From bottlenecks to breakthroughs: Citizen Development in action (25')
Pawel Poplawski, Director, Improvement and Automation @McCormick & Company
Michał Cieślak, Senior Manager, Automation Programs @McCormick & Company
10:05 Next-level bots: API integration in UiPath Studio (30')
Mikolaj Zielinski, UiPath MVP, Senior Solutions Engineer @Proservartner
10:35 ☕ Coffee Break (15')
10:50 Document Understanding with my RPA Companion (45')
Ewa Gruszka, Enterprise Sales Specialist, AI & ML @UiPath
11:35 Power up your Robots: GenAI and GPT in REFramework (45')
Krzysztof Karaszewski, Global RPA Product Manager
12:20 🍕 Lunch Break (1hr)
13:20 From Concept to Quality: UiPath Test Suite for AI-powered Knowledge Bots (30')
Kamil Miśko, UiPath MVP, Senior RPA Developer @Zurich Insurance
13:50 Communications Mining - focus on AI capabilities (30')
Thomasz Wierzbicki, Business Analyst @Office Samurai
14:20 Polish MVP panel: Insights on MVP award achievements and career profiling
3. Objectives
State and explain the
Fundemental Theorems of
Calculus
Use the first fundamental
theorem of calculus to find
derivatives of functions
defined as integrals.
Compute the average value
of an integrable function
over a closed interval.
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 3 / 33
4. Outline
Recall: The Evaluation Theorem a/k/a 2FTC
The First Fundamental Theorem of Calculus
The Area Function
Statement and proof of 1FTC
Biographies
Differentiation of functions defined by integrals
“Contrived” examples
Erf
Other applications
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 4 / 33
5. The definite integral as a limit
Definition
If f is a function defined on [a, b], the definite integral of f from a to b
is the number
b n
f (x) dx = lim f (ci ) ∆x
a ∆x→0
i=1
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 5 / 33
6. Big time Theorem
Theorem (The Second Fundamental Theorem of Calculus)
Suppose f is integrable on [a, b] and f = F for another function F , then
b
f (x) dx = F (b) − F (a).
a
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 6 / 33
7. The Integral as Total Change
Another way to state this theorem is:
b
F (x) dx = F (b) − F (a),
a
or the integral of a derivative along an interval is the total change between
the sides of that interval. This has many ramifications:
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 7 / 33
8. The Integral as Total Change
Another way to state this theorem is:
b
F (x) dx = F (b) − F (a),
a
or the integral of a derivative along an interval is the total change between
the sides of that interval. This has many ramifications:
Theorem
If v (t) represents the velocity of a particle moving rectilinearly, then
t1
v (t) dt = s(t1 ) − s(t0 ).
t0
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 7 / 33
9. The Integral as Total Change
Another way to state this theorem is:
b
F (x) dx = F (b) − F (a),
a
or the integral of a derivative along an interval is the total change between
the sides of that interval. This has many ramifications:
Theorem
If MC (x) represents the marginal cost of making x units of a product, then
x
C (x) = C (0) + MC (q) dq.
0
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 7 / 33
10. The Integral as Total Change
Another way to state this theorem is:
b
F (x) dx = F (b) − F (a),
a
or the integral of a derivative along an interval is the total change between
the sides of that interval. This has many ramifications:
Theorem
If ρ(x) represents the density of a thin rod at a distance of x from its end,
then the mass of the rod up to x is
x
m(x) = ρ(s) ds.
0
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 7 / 33
11. My first table of integrals
[f (x) + g (x)] dx = f (x) dx + g (x) dx
x n+1
x n dx = + C (n = −1) cf (x) dx = c f (x) dx
n+1
1
e x dx = e x + C dx = ln |x| + C
x
ax
sin x dx = − cos x + C ax dx = +C
ln a
cos x dx = sin x + C csc2 x dx = − cot x + C
sec2 x dx = tan x + C csc x cot x dx = − csc x + C
1
sec x tan x dx = sec x + C √ dx = arcsin x + C
1 − x2
1
dx = arctan x + C
1 + x2
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 8 / 33
12. Outline
Recall: The Evaluation Theorem a/k/a 2FTC
The First Fundamental Theorem of Calculus
The Area Function
Statement and proof of 1FTC
Biographies
Differentiation of functions defined by integrals
“Contrived” examples
Erf
Other applications
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 9 / 33
13. An area function
x
Let f (t) = t 3 and define g (x) = f (t) dt. Can we evaluate the integral
0
in g (x)?
0 x
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 10 / 33
14. An area function
x
Let f (t) = t 3 and define g (x) = f (t) dt. Can we evaluate the integral
0
in g (x)?
Dividing the interval [0, x] into n pieces
x ix
gives ∆t = and ti = 0 + i∆t = . So
n n
x x 3 x (2x)3 x (nx)3
Rn = · 3+ · 3
+ ··· + ·
n n n n n n3
x4
= 4 13 + 23 + 33 + · · · + n3
n
x4 1 2
= 4 2 n(n + 1)
n
0 x
x 4 n2 (n + 1)2 x4
= 4
→
4n 4
as n → ∞.
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 10 / 33
15. An area function, continued
So
x4
g (x) = .
4
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 11 / 33
16. An area function, continued
So
x4
g (x) = .
4
This means that
g (x) = x 3 .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 11 / 33
17. The area function
Let f be a function which is integrable (i.e., continuous or with finitely
many jump discontinuities) on [a, b]. Define
x
g (x) = f (t) dt.
a
The variable is x; t is a “dummy” variable that’s integrated over.
Picture changing x and taking more of less of the region under the
curve.
Question: What does f tell you about g ?
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 12 / 33
18. Envisioning the area function
Example
Suppose f (t) is the function graphed below:
y
x
2 4 6 8 10f
x
Let g (x) = f (t) dt. What can you say about g ?
0
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 13 / 33
19. Envisioning the area function
Example
Suppose f (t) is the function graphed below:
y
g
x
2 4 6 8 10f
x
Let g (x) = f (t) dt. What can you say about g ?
0
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 13 / 33
20. Envisioning the area function
Example
Suppose f (t) is the function graphed below:
y
g
x
2 4 6 8 10f
x
Let g (x) = f (t) dt. What can you say about g ?
0
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 13 / 33
21. Envisioning the area function
Example
Suppose f (t) is the function graphed below:
y
g
x
2 4 6 8 10f
x
Let g (x) = f (t) dt. What can you say about g ?
0
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 13 / 33
22. Envisioning the area function
Example
Suppose f (t) is the function graphed below:
y
g
x
2 4 6 8 10f
x
Let g (x) = f (t) dt. What can you say about g ?
0
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 13 / 33
23. Envisioning the area function
Example
Suppose f (t) is the function graphed below:
y
g
x
2 4 6 8 10f
x
Let g (x) = f (t) dt. What can you say about g ?
0
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 13 / 33
24. Envisioning the area function
Example
Suppose f (t) is the function graphed below:
y
g
x
2 4 6 8 10f
x
Let g (x) = f (t) dt. What can you say about g ?
0
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 13 / 33
25. Envisioning the area function
Example
Suppose f (t) is the function graphed below:
y
g
x
2 4 6 8 10f
x
Let g (x) = f (t) dt. What can you say about g ?
0
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 13 / 33
26. Envisioning the area function
Example
Suppose f (t) is the function graphed below:
y
g
x
2 4 6 8 10f
x
Let g (x) = f (t) dt. What can you say about g ?
0
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 13 / 33
27. Envisioning the area function
Example
Suppose f (t) is the function graphed below:
y
g
x
2 4 6 8 10f
x
Let g (x) = f (t) dt. What can you say about g ?
0
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 13 / 33
28. Envisioning the area function
Example
Suppose f (t) is the function graphed below:
y
g
x
2 4 6 8 10f
x
Let g (x) = f (t) dt. What can you say about g ?
0
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 13 / 33
29. Envisioning the area function
Example
Suppose f (t) is the function graphed below:
y
g
x
2 4 6 8 10f
x
Let g (x) = f (t) dt. What can you say about g ?
0
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 13 / 33
30. features of g from f
y
Interval sign monotonicity monotonicity concavity
of f of g of f of g
g
[0, 2] +
x
2 4 6 8 10f [2, 4.5] +
[4.5, 6] −
[6, 8] −
[8, 10] − → none
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 14 / 33
31. features of g from f
y
Interval sign monotonicity monotonicity concavity
of f of g of f of g
g
[0, 2] +
x
2 4 6 8 10f [2, 4.5] +
[4.5, 6] −
[6, 8] −
[8, 10] − → none
We see that g is behaving a lot like an antiderivative of f .
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 14 / 33
32. Another Big Time Theorem
Theorem (The First Fundamental Theorem of Calculus)
Let f be an integrable function on [a, b] and define
x
g (x) = f (t) dt.
a
If f is continuous at x in (a, b), then g is differentiable at x and
g (x) = f (x).
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 15 / 33
33. Proving the Fundamental Theorem
Proof.
Let h > 0 be given so that x + h < b. We have
g (x + h) − g (x)
=
h
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 16 / 33
34. Proving the Fundamental Theorem
Proof.
Let h > 0 be given so that x + h < b. We have
x+h
g (x + h) − g (x) 1
= f (t) dt.
h h x
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 16 / 33
35. Proving the Fundamental Theorem
Proof.
Let h > 0 be given so that x + h < b. We have
x+h
g (x + h) − g (x) 1
= f (t) dt.
h h x
Let Mh be the maximum value of f on [x, x + h], and let mh the minimum
value of f on [x, x + h]. From §5.2 we have
x+h
f (t) dt
x
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 16 / 33
36. Proving the Fundamental Theorem
Proof.
Let h > 0 be given so that x + h < b. We have
x+h
g (x + h) − g (x) 1
= f (t) dt.
h h x
Let Mh be the maximum value of f on [x, x + h], and let mh the minimum
value of f on [x, x + h]. From §5.2 we have
x+h
f (t) dt ≤ Mh · h
x
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 16 / 33
37. Proving the Fundamental Theorem
Proof.
Let h > 0 be given so that x + h < b. We have
x+h
g (x + h) − g (x) 1
= f (t) dt.
h h x
Let Mh be the maximum value of f on [x, x + h], and let mh the minimum
value of f on [x, x + h]. From §5.2 we have
x+h
mh · h ≤ f (t) dt ≤ Mh · h
x
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 16 / 33
38. Proving the Fundamental Theorem
Proof.
Let h > 0 be given so that x + h < b. We have
x+h
g (x + h) − g (x) 1
= f (t) dt.
h h x
Let Mh be the maximum value of f on [x, x + h], and let mh the minimum
value of f on [x, x + h]. From §5.2 we have
x+h
mh · h ≤ f (t) dt ≤ Mh · h
x
So
g (x + h) − g (x)
mh ≤ ≤ Mh .
h
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 16 / 33
39. Proving the Fundamental Theorem
Proof.
Let h > 0 be given so that x + h < b. We have
x+h
g (x + h) − g (x) 1
= f (t) dt.
h h x
Let Mh be the maximum value of f on [x, x + h], and let mh the minimum
value of f on [x, x + h]. From §5.2 we have
x+h
mh · h ≤ f (t) dt ≤ Mh · h
x
So
g (x + h) − g (x)
mh ≤ ≤ Mh .
h
As h → 0, both mh and Mh tend to f (x).
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 16 / 33
40. Meet the Mathematician: James Gregory
Scottish, 1638-1675
Astronomer and Geometer
Conceived transcendental
numbers and found evidence
that π was transcendental
Proved a geometric version
of 1FTC as a lemma but
didn’t take it further
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 17 / 33
41. Meet the Mathematician: Isaac Barrow
English, 1630-1677
Professor of Greek, theology,
and mathematics at
Cambridge
Had a famous student
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 18 / 33
42. Meet the Mathematician: Isaac Newton
English, 1643–1727
Professor at Cambridge
(England)
Philosophiae Naturalis
Principia Mathematica
published 1687
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 19 / 33
43. Meet the Mathematician: Gottfried Leibniz
German, 1646–1716
Eminent philosopher as well
as mathematician
Contemporarily disgraced by
the calculus priority dispute
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 20 / 33
44. Differentiation and Integration as reverse processes
Putting together 1FTC and 2FTC, we get a beautiful relationship between
the two fundamental concepts in calculus.
Theorem (The Fundamental Theorem(s) of Calculus)
I. If f is a continuous function, then
x
d
f (t) dt = f (x)
dx a
So the derivative of the integral is the original function.
II. If f is a differentiable function, then
b
f (x) dx = f (b) − f (a).
a
So the integral of the derivative of is (an evaluation of) the original
function.
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 21 / 33
45. Outline
Recall: The Evaluation Theorem a/k/a 2FTC
The First Fundamental Theorem of Calculus
The Area Function
Statement and proof of 1FTC
Biographies
Differentiation of functions defined by integrals
“Contrived” examples
Erf
Other applications
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 22 / 33
46. Differentiation of area functions
Example
3x
Let h(x) = t 3 dt. What is h (x)?
0
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 23 / 33
47. Differentiation of area functions
Example
3x
Let h(x) = t 3 dt. What is h (x)?
0
Solution (Using 2FTC)
3x
t4 1
h(x) = = (3x)4 = 1
4 · 81x 4 , so h (x) = 81x 3 .
4 0 4
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 23 / 33
48. Differentiation of area functions
Example
3x
Let h(x) = t 3 dt. What is h (x)?
0
Solution (Using 2FTC)
3x
t4 1
h(x) = = (3x)4 = 1
4 · 81x 4 , so h (x) = 81x 3 .
4 0 4
Solution (Using 1FTC)
u
We can think of h as the composition g ◦ k, where g (u) = t 3 dt and
0
k(x) = 3x.
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49. Differentiation of area functions
Example
3x
Let h(x) = t 3 dt. What is h (x)?
0
Solution (Using 2FTC)
3x
t4 1
h(x) = = (3x)4 = 1
4 · 81x 4 , so h (x) = 81x 3 .
4 0 4
Solution (Using 1FTC)
u
We can think of h as the composition g ◦ k, where g (u) = t 3 dt and
0
k(x) = 3x. Then h (x) = g (u) · k (x), or
h (x) = g (k(x)) · k (x) = (k(x))3 · 3 = (3x)3 · 3 = 81x 3 .
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50. Differentiation of area functions, in general
by 1FTC
k(x)
d
f (t) dt = f (k(x))k (x)
dx a
by reversing the order of integration:
b h(x)
d d
f (t) dt = − f (t) dt = −f (h(x))h (x)
dx h(x) dx b
by combining the two above:
k(x) k(x) 0
d d
f (t) dt = f (t) dt + f (t) dt
dx h(x) dx 0 h(x)
= f (k(x))k (x) − f (h(x))h (x)
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51. Another Example
Example
sin2 x
Let h(x) = (17t 2 + 4t − 4) dt. What is h (x)?
0
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52. Another Example
Example
sin2 x
Let h(x) = (17t 2 + 4t − 4) dt. What is h (x)?
0
Solution
We have
sin2 x
d
(17t 2 + 4t − 4) dt
dx 0
d
= 17(sin2 x)2 + 4(sin2 x) − 4 · sin2 x
dx
= 17 sin4 x + 4 sin2 x − 4 · 2 sin x cos x
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53. A Similar Example
Example
sin2 x
Let h(x) = (17t 2 + 4t − 4) dt. What is h (x)?
3
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54. A Similar Example
Example
sin2 x
Let h(x) = (17t 2 + 4t − 4) dt. What is h (x)?
3
Solution
We have
sin2 x
d
(17t 2 + 4t − 4) dt
dx 0
d
= 17(sin2 x)2 + 4(sin2 x) − 4 · sin2 x
dx
= 17 sin4 x + 4 sin2 x − 4 · 2 sin x cos x
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55. Compare
Question
Why is
sin2 x sin2 x
d 2 d
(17t + 4t − 4) dt = (17t 2 + 4t − 4) dt?
dx 0 dx 3
Or, why doesn’t the lower limit appear in the derivative?
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56. Compare
Question
Why is
sin2 x sin2 x
d 2 d
(17t + 4t − 4) dt = (17t 2 + 4t − 4) dt?
dx 0 dx 3
Or, why doesn’t the lower limit appear in the derivative?
Answer
Because
sin2 x 3 sin2 x
2 2
(17t + 4t − 4) dt = (17t + 4t − 4) dt + (17t 2 + 4t − 4) dt
0 0 3
So the two functions differ by a constant.
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57. The Full Nasty
Example
ex
Find the derivative of F (x) = sin4 t dt.
x3
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58. The Full Nasty
Example
ex
Find the derivative of F (x) = sin4 t dt.
x3
Solution
ex
d
sin4 t dt = sin4 (e x ) · e x − sin4 (x 3 ) · 3x 2
dx x3
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59. The Full Nasty
Example
ex
Find the derivative of F (x) = sin4 t dt.
x3
Solution
ex
d
sin4 t dt = sin4 (e x ) · e x − sin4 (x 3 ) · 3x 2
dx x3
Notice here it’s much easier than finding an antiderivative for sin4 .
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60. Why use 1FTC?
Question
Why would we use 1FTC to find the derivative of an integral? It seems
like confusion for its own sake.
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61. Why use 1FTC?
Question
Why would we use 1FTC to find the derivative of an integral? It seems
like confusion for its own sake.
Answer
Some functions are difficult or impossible to integrate in elementary
terms.
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62. Why use 1FTC?
Question
Why would we use 1FTC to find the derivative of an integral? It seems
like confusion for its own sake.
Answer
Some functions are difficult or impossible to integrate in elementary
terms.
Some functions are naturally defined in terms of other integrals.
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63. Erf
Here’s a function with a funny name but an important role:
x
2 2
erf(x) = √ e −t dt.
π 0
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64. Erf
Here’s a function with a funny name but an important role:
x
2 2
erf(x) = √ e −t dt.
π 0
It turns out erf is the shape of the bell curve.
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65. Erf
Here’s a function with a funny name but an important role:
x
2 2
erf(x) = √ e −t dt.
π 0
It turns out erf is the shape of the bell curve. We can’t find erf(x),
explicitly, but we do know its derivative: erf (x) =
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66. Erf
Here’s a function with a funny name but an important role:
x
2 2
erf(x) = √ e −t dt.
π 0
It turns out erf is the shape of the bell curve. We can’t find erf(x),
2 2
explicitly, but we do know its derivative: erf (x) = √ e −x .
π
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67. Erf
Here’s a function with a funny name but an important role:
x
2 2
erf(x) = √ e −t dt.
π 0
It turns out erf is the shape of the bell curve. We can’t find erf(x),
2 2
explicitly, but we do know its derivative: erf (x) = √ e −x .
π
Example
d
Find erf(x 2 ).
dx
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 30 / 33
68. Erf
Here’s a function with a funny name but an important role:
x
2 2
erf(x) = √ e −t dt.
π 0
It turns out erf is the shape of the bell curve. We can’t find erf(x),
2 2
explicitly, but we do know its derivative: erf (x) = √ e −x .
π
Example
d
Find erf(x 2 ).
dx
Solution
By the chain rule we have
d d 2 2 2 4 4
erf(x 2 ) = erf (x 2 ) x 2 = √ e −(x ) 2x = √ xe −x .
dx dx π π
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69. Other functions defined by integrals
The future value of an asset:
∞
FV (t) = π(s)e −rs ds
t
where π(s) is the profitability at time s and r is the discount rate.
The consumer surplus of a good:
q∗
CS(q ∗ ) = (f (q) − p ∗ ) dq
0
where f (q) is the demand function and p ∗ and q ∗ the equilibrium
price and quantity.
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70. Surplus by picture
price (p)
quantity (q)
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71. Surplus by picture
price (p)
demand f (q)
quantity (q)
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72. Surplus by picture
price (p)
supply
demand f (q)
quantity (q)
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73. Surplus by picture
price (p)
supply
p∗ equilibrium
demand f (q)
q∗ quantity (q)
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74. Surplus by picture
price (p)
supply
p∗ equilibrium
market revenue
demand f (q)
q∗ quantity (q)
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75. Surplus by picture
consumer surplus
price (p)
supply
p∗ equilibrium
market revenue
demand f (q)
q∗ quantity (q)
V63.0121.002.2010Su, Calculus I (NYU) Section 5.4 The Fundamental Theorem June 21, 2010 32 / 33
76. Surplus by picture
consumer surplus
price (p)
producer surplus
supply
p∗ equilibrium
demand f (q)
q∗ quantity (q)
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77. Summary
Functions defined as integrals can be differentiated using the first
FTC: x
d
f (t) dt = f (x)
dx a
The two FTCs link the two major processes in calculus: differentiation
and integration
F (x) dx = F (x) + C
Follow the calculus wars on twitter: #calcwars
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