Book Language: English. x E Var and E is a term, and applications, which have the form El E2. But strongly indirect. That the PDE (5) has only one solution that satises the initial condition (6) follows from the maximum principle: see a PDE text if you On Sequence, Series, Functions of Several variables, Rolle's Theorem and Mean Value Theorem, Integral Calculus, Improper Integrals, Beta-gamma function Part of Mathematics-I By Professor Shiv Datt Kumar, MNNIT, Allahabad .

In addition to this, the calculus of variations is an active topic of study in its own right. Value at Risk (VaR) is a financial metric that estimates the risk of an investment. Related Symbolab blog posts. Advanced Calculus of Several Variables (Dover Books on .

If X has high variance, we can observe values of X a long way from the mean. C 4. The equation (5) is called the heat equation. Use tables for means of commonly used distribution. He is the recipient of the 2013 Text and Academic Authors Association Award for CALCULUS, the 2012 William Holmes McGuffey Longevity Award for CALCULUS: AN APPLIED APPROACH, the 2011 William Holmes McGuffey Longevity Award for PRECALCULUS: REAL MATHEMATICS . CALCULUS OF VARIATIONS c 2006 Gilbert Strang 7.2 Calculus of Variations One theme of this book is the relation of equations to minimum principles. El is called Advanced Calculus of Several Variables (Dover Books on Mathematics) Revised Edition. Calculus 2nd Multi Var - Solution.pdf . 0.1 Reminder For a function of one variable, f(x), we nd the local maxima/minima by dierenti- ation.

24.

Dr. Stewart authored a best-selling calculus textbook series, including CALCULUS, CALCULUS: EARLY TRANSCENDENTALS and CALCULUS: CONCEPTS AND CONTEXTS as well as a series of successful precalculus texts. 250 101 4MB Read more.

ISBN-10: 0486683362. [1] Multivariable calculus may be thought of as an elementary part of advanced calculus. Advanced Calculus of Several Variables provides a conceptual treatment of multivariable calculus. Calc. The rst part treats analysis in one variable, and the text [44] was written to cover that material. Amar M., De Cicco V., Fusco N.: Lower semicontinuity and relaxation results in BV for integral functionals with BV integrands. Linear Algebra and Di erential Calculus of Several Variables Stanford Math 42, Calculus ACE (Additional Calculus for Engineers) . 6 Pyramid selling schemes Have you received a chain letter like this one? The quotient rule states that the derivative of f (x) is f (x)= (g (x)h (x)-g (x)h (x))/h (x). This calculus course covers differentiation and integration of functions of one variable, and concludes with a brief discussion of infinite series. 3 PHY 132/132L General Physics 3/1 PHY 133/133L General Physics 3/1 MAT 224 Diff. Calculus of Variations and Partial Differential Equations attracts and collects many of the important top-quality contributions to this field of research, and stresses the interactions between analysts, geometers, and physicists.. Coverage in the journal includes: Minimization problems for variational integrals, existence and regularity theory for minimizers and critical points, geometric . Ithasonlyonecriticalpoint(attheorigin)wherethereisanon-degeneratelocalminimum. The coordinate system shown in Figure 1.1.1 is known as aright-handed coordinate system, because it is possible, using the right hand, to point the index nger in the positive directionof thex-axis, themiddlengerinthepositivedirectionof they-axis, andthethumb in the positive direction of thez-axis, as in Figure 1.1.3. x z y 0 In this paper, we deal with the fixed-endpoint problem in the calculus of variations involving a delay in the phase coordinates. : Derivatives of multivariable functions. It includes all of the materials you will need to understand the concepts covered in this subject.

The matrix K is .

Engineering Mathematics by Ravish R. Singh and Mukul Bhatt is one of the most important book for Engineering 1st Semester Students. A 10. B 7. ISBN: 9781305272422. en. Var(X) = E (X X) 2 = E(X ) E(X) .

14, 456-477 (2008) MATH Article MathSciNet Google Scholar. You can publish your own PDF file online for free in a few minutes!

Differential calculus grade 12 questions and answers pdf Author.

A 6. The idea of di erential calculus is that you can get information about a function if you try chang-ing the independent variable by a small amount. D 9. Please note that the PDF may contain . B 3. In other words, ' ()= (). Eq. vanced calculus, whose aim is to provide a rm logical foundation for analy-sis, for students who have had 3 semesters of calculus and a course in linear algebra.

Multiple Integrals; Integral Applications. Advanced 3/1 MAT 115 Anal. 1.1 Introduction 5 Similarly, @f @y (x 0;y 0) gives the slope of the tangent Q 0Tto the curve of intersection BQ 0Cof the surface with a vertical plane x= x 0 (see gure 1.5). is given by (7) u t(x) = Ef(Wx t) = Z 1 y=1 p t(x;y)f(y)dy: Here Wx t is a Brownian motion started at x.

ESAIM Control Optim. Limit of Sum; Area under curve; Area between curves; . James Stewart. (Such a scenario is covered in Advanced Calculus.) Here are three reasons why I think that knowledge of the most basic ("entry level") calculus, and/or my question above, are relevant to an FRM candidate: Several key concepts indirectly apply such math. See why this is so. The reader should have a solid background in one- variable calculus. Var(X) = p2Var(X 1)+(1 p)2Var(X2) = p2 +(1 p) 2 : It is a nice calculus exercise to show that the minimum value is when p =1=2, which gives a variance of X of 2=2. Example: Let X be a continuous random variable with . A 2. by C. H. Edwards Jr. (Author) 4.1 out of 5 stars 25 ratings. This approach to . The moment generating function only works when the integral converges on a particular number. These articles can help understand advanced math topics like Calcul Multivariable Calculus Archive Techniques of Integration - Chapters 7 & 8 Correct solution with correct derivative Correct solution with correct derivative. gmatheuler.zip: 1k: 09-04-03: Euler on Ti-83 Plus Euler's Method . The domain of functions of two variables, z = f (x,y) z = f ( x, y), are regions from two dimensional space and consist of all the coordinate pairs, (x,y) ( x, y), that we could plug into the function and get back a real number. Just send $10 to the person whose name comes at the top of the list, and add your own name to the bottom In this section, the techniques developed in an earlier chapter will be extended to help you find these Example 1 Determine the domain of each of the following. Partial derivative and gradient (articles) : Derivatives of multivariable functions. The skewness of X is given by E[(X ) 3]=Var(X) =2 and the kurtosis of X is given by E[(X )4]=Var(X)2. In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Instructional math help video lessons online and on CD Harvard Mathematics Department : Home page 1 The Chain Rule Calculus AB/BC - 3 Calculus Tutorial Notes 2016-2017 September 13, 2016 December 21, 2017 / mattyj612 / Leave a comment For the most enjoyable viewing experience, I would recommend that you download the files and open them in a PDF reader, as opposed to reading them right off . Maxima/minima occur when f0(x) = 0. The variance of X is the expected squared distance of X from its mean.

Multivariable functions also have high points and low points. Student Solutions Manual for Stewart's Single Variable Calculus: Early Transcendentals, 8th (James Stewart Calculus) 8th Edition. Calculus is fundamental to many scientific disciplines including physics, engineering, and economics. image/svg+xml. SymPy is a Python library for symbolic mathematics. Solution: Step 1: Plug e -x in for fx (x) to get: Note that I changed the lower integral bound to zero, because this function is only valid for values higher than zero. Several years ago I mentioned this question to my advanced calculus course and asked if they could nd a polynomial example. Proof of fundamental theorem of calculus. Expected Value and Variance, Feb 2, 2003 - 3 - . I. J. Funct. This approach to . & Calculus 4 MAT 214 Calculus of Several Var. : Derivatives of multivariable functions.

: Derivatives of multivariable functions. 42 (1999) ("The Andrews Festschrift"), Article B42q, 67 pp.]. Step 2: Integrate. Am I understanding this right? That variable is X1^2, which is equal to the variance of X1 plus the mean of X1 squared.

Calculus 1c-2, Examples of Elementary Functions. 7.2. Education Stanford University . Instructions (same as always) Problems (PDF) Submission due via email on Mon Oct 19 3 , Thomas' Calculus, Pearson . It leads naturally to the Lagrangian formulation of mechanics, mentioned above, from which most dynamical equations of . View Lecture 7.pdf from MATH 104 at Kathmandu University. Var. The PDF of this extract thus shows the content exactly as it would be seen by an Open University student. Calculus 3 Notes Pdf Multivariable Calculus Lectures Online this is a link to the playlist for the lectures, from Math 231 of Spring 2016 In order to do this, we shall present a number of dierent types of integrals and sums and then the method of how to calculate them In order to do this, we shall present a number of dierent types of . 2) is the one that minimizes the arc length integral (2 Its really helpful for your assignments and homeworks ) To compute the derivative of the outer function, we just use the power rule: (u2=3)0= 2 3 u 1 2=3 = 3 u 1=3 To compute the derivative of the inner function, we also use the power rule: (1 + x4)0 Similarly, moving across the root 3 to the interval ( 4; 3 .

Figure 1 - The function gx x x x() 9 24 3 32 and its relative extrema. & Linear Alg. Ambrosio L., Dal Maso G.: On the relaxation in BV (; R m) of quasi-convex integrals. MULTIPLE CHOICE QUESTIONS (50%) All answers must be written on the answer sheet; write answers to five questions in each row, for example: 1. Long story short, E (X2)=1/2, Var (X2)= 5/12 (this part is 1/12 + 1/12 + (1/2)^2. It can accordingly be used (with omissions) as a text for a year's course in advanced calculus, or as a text for a three-semester introduction to analysis Saldelux pdf Calculus one and several variables 10th sallas, hille, etgen Hostetler, and Bruce H The precalculus model (n=964) had four significant pedagogical variables, one with positive . There may be more to it, but that is the main point. Advanced Calculus Functions of Several Variables GR Phaijoo, PhD Department of Mathematics School of Science, Kathmandu University Kavre,

Continuity of Functions of Two Var's (Denition . Title: Advanced Calculus Of Several Variables Author: nr-media-01.nationalreview.com-2022-07-04T00:00:00+00:01 Subject: Advanced Calculus Of Several Variables

Conference . Affiliation Institute for Advanced Study Visitor, September 2018 to August 2019. Now however the independent variable is the whole path, so I'll change that path by some small amount and see what happens to the value of the integral I. In other words, we have found an investment with the same expected return as X1 and X2 but with less risk / uncertainty. Intended for stu .

B 1. Search: Calculus 3 Notes Pdf. Multivariable functions also have high points and low points. Limits of 2-Variable Functions (Existence) Josh Engwer (TTU) Functions of Several Variables: Limits & Continuity 23 September 2014 11 / 17 . In Chapters 1-5 the basic ideas and techniques of partial differentiation, and of line, multiple and surface integrals are discussed. For most distributions some "advanced" knowledge of calculus is required to nd the mean. Calculus of Variations and Applications: A . Do the same thing here. Math 42 -- Introduction to Data-Driven Mathematical Modeling: Life, Universe, and Everything Algebra (25%) If X has low variance, the values of X tend to be clustered tightly around the mean value. In the present article, I share with the reader my experience of applying the methods described in the . elementary calculus courses differential and integral calculus of one and of several variables including calculus-based applications and connections with coordinate geometry, trigonometry, differential equations, and other branches of mathematics II. There are 17 problems in total to be solved. Veach's multiple importance sampling: minimizing second moment goal: choose w 1 and w 2 to minimize Var[ w 1 s.t. Bundle: Precalculus, Loose-leaf Version, 10th + WebAssign Printed Access Card for Larson's Precalculus, 10th Edition, Single-Term. Geom. Dr. Stewart most recently served as a professor of mathematics at McMaster University, and his research focused on harmonic analysis.

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. Curriculum Vitae, Chao Li, 1. List of JEE Main & JEE Advanced Books While NCERT Chemistry books are good, one can use Solomons for Organic Chemistry, O. P. Tandon for Inorganic Chemistr and P. Bahadur book for Physical Chemistry, For problems, Numerical Chemistry by R. C. Mukherjee is quite good. . Partial Di erential Equations 16 (2003), pp. Var.

From a programming language viewpoint, the \(\lambda \)-calculus formalises several features of the modern description of computation and its implementation.We present a denotational semantics for the untyped calculus that captures a basic feature of probabilistic programming languages, namely probability distributions as both the objects and the result of a computation. D 5. Gradient and directional derivatives. Calc. Ses #18-25 complete (PDF - 8.6 MB) 19 First fundamental theorem of calculus 20 Second fundamental theorem 21 Applications to logarithms and geometry (PDF - 1.4 MB) 22 Volumes by disks and shells (PDF - 1.7 MB) 23 Work, average value, probability (PDF - 2.2 MB) 24 Numerical integration (PDF - 1.1 MB) 25 Exam 3 review Let f (x)=g (x)/h (x), where both g and h are differentiable and h (x)0. However this is not the global minimum.

In addition to this, the calculus of variations is an active topic of study in its own right. Value at Risk (VaR) is a financial metric that estimates the risk of an investment. Related Symbolab blog posts. Advanced Calculus of Several Variables (Dover Books on .

If X has high variance, we can observe values of X a long way from the mean. C 4. The equation (5) is called the heat equation. Use tables for means of commonly used distribution. He is the recipient of the 2013 Text and Academic Authors Association Award for CALCULUS, the 2012 William Holmes McGuffey Longevity Award for CALCULUS: AN APPLIED APPROACH, the 2011 William Holmes McGuffey Longevity Award for PRECALCULUS: REAL MATHEMATICS . CALCULUS OF VARIATIONS c 2006 Gilbert Strang 7.2 Calculus of Variations One theme of this book is the relation of equations to minimum principles. El is called Advanced Calculus of Several Variables (Dover Books on Mathematics) Revised Edition. Calculus 2nd Multi Var - Solution.pdf . 0.1 Reminder For a function of one variable, f(x), we nd the local maxima/minima by dierenti- ation.

24.

Dr. Stewart authored a best-selling calculus textbook series, including CALCULUS, CALCULUS: EARLY TRANSCENDENTALS and CALCULUS: CONCEPTS AND CONTEXTS as well as a series of successful precalculus texts. 250 101 4MB Read more.

ISBN-10: 0486683362. [1] Multivariable calculus may be thought of as an elementary part of advanced calculus. Advanced Calculus of Several Variables provides a conceptual treatment of multivariable calculus. Calc. The rst part treats analysis in one variable, and the text [44] was written to cover that material. Amar M., De Cicco V., Fusco N.: Lower semicontinuity and relaxation results in BV for integral functionals with BV integrands. Linear Algebra and Di erential Calculus of Several Variables Stanford Math 42, Calculus ACE (Additional Calculus for Engineers) . 6 Pyramid selling schemes Have you received a chain letter like this one? The quotient rule states that the derivative of f (x) is f (x)= (g (x)h (x)-g (x)h (x))/h (x). This calculus course covers differentiation and integration of functions of one variable, and concludes with a brief discussion of infinite series. 3 PHY 132/132L General Physics 3/1 PHY 133/133L General Physics 3/1 MAT 224 Diff. Calculus of Variations and Partial Differential Equations attracts and collects many of the important top-quality contributions to this field of research, and stresses the interactions between analysts, geometers, and physicists.. Coverage in the journal includes: Minimization problems for variational integrals, existence and regularity theory for minimizers and critical points, geometric . Ithasonlyonecriticalpoint(attheorigin)wherethereisanon-degeneratelocalminimum. The coordinate system shown in Figure 1.1.1 is known as aright-handed coordinate system, because it is possible, using the right hand, to point the index nger in the positive directionof thex-axis, themiddlengerinthepositivedirectionof they-axis, andthethumb in the positive direction of thez-axis, as in Figure 1.1.3. x z y 0 In this paper, we deal with the fixed-endpoint problem in the calculus of variations involving a delay in the phase coordinates. : Derivatives of multivariable functions. It includes all of the materials you will need to understand the concepts covered in this subject.

The matrix K is .

Engineering Mathematics by Ravish R. Singh and Mukul Bhatt is one of the most important book for Engineering 1st Semester Students. A 10. B 7. ISBN: 9781305272422. en. Var(X) = E (X X) 2 = E(X ) E(X) .

14, 456-477 (2008) MATH Article MathSciNet Google Scholar. You can publish your own PDF file online for free in a few minutes!

Differential calculus grade 12 questions and answers pdf Author.

A 6. The idea of di erential calculus is that you can get information about a function if you try chang-ing the independent variable by a small amount. D 9. Please note that the PDF may contain . B 3. In other words, ' ()= (). Eq. vanced calculus, whose aim is to provide a rm logical foundation for analy-sis, for students who have had 3 semesters of calculus and a course in linear algebra.

Multiple Integrals; Integral Applications. Advanced 3/1 MAT 115 Anal. 1.1 Introduction 5 Similarly, @f @y (x 0;y 0) gives the slope of the tangent Q 0Tto the curve of intersection BQ 0Cof the surface with a vertical plane x= x 0 (see gure 1.5). is given by (7) u t(x) = Ef(Wx t) = Z 1 y=1 p t(x;y)f(y)dy: Here Wx t is a Brownian motion started at x.

ESAIM Control Optim. Limit of Sum; Area under curve; Area between curves; . James Stewart. (Such a scenario is covered in Advanced Calculus.) Here are three reasons why I think that knowledge of the most basic ("entry level") calculus, and/or my question above, are relevant to an FRM candidate: Several key concepts indirectly apply such math. See why this is so. The reader should have a solid background in one- variable calculus. Var(X) = p2Var(X 1)+(1 p)2Var(X2) = p2 +(1 p) 2 : It is a nice calculus exercise to show that the minimum value is when p =1=2, which gives a variance of X of 2=2. Example: Let X be a continuous random variable with . A 2. by C. H. Edwards Jr. (Author) 4.1 out of 5 stars 25 ratings. This approach to . The moment generating function only works when the integral converges on a particular number. These articles can help understand advanced math topics like Calcul Multivariable Calculus Archive Techniques of Integration - Chapters 7 & 8 Correct solution with correct derivative Correct solution with correct derivative. gmatheuler.zip: 1k: 09-04-03: Euler on Ti-83 Plus Euler's Method . The domain of functions of two variables, z = f (x,y) z = f ( x, y), are regions from two dimensional space and consist of all the coordinate pairs, (x,y) ( x, y), that we could plug into the function and get back a real number. Just send $10 to the person whose name comes at the top of the list, and add your own name to the bottom In this section, the techniques developed in an earlier chapter will be extended to help you find these Example 1 Determine the domain of each of the following. Partial derivative and gradient (articles) : Derivatives of multivariable functions. The skewness of X is given by E[(X ) 3]=Var(X) =2 and the kurtosis of X is given by E[(X )4]=Var(X)2. In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Instructional math help video lessons online and on CD Harvard Mathematics Department : Home page 1 The Chain Rule Calculus AB/BC - 3 Calculus Tutorial Notes 2016-2017 September 13, 2016 December 21, 2017 / mattyj612 / Leave a comment For the most enjoyable viewing experience, I would recommend that you download the files and open them in a PDF reader, as opposed to reading them right off . Maxima/minima occur when f0(x) = 0. The variance of X is the expected squared distance of X from its mean.

Multivariable functions also have high points and low points. Student Solutions Manual for Stewart's Single Variable Calculus: Early Transcendentals, 8th (James Stewart Calculus) 8th Edition. Calculus is fundamental to many scientific disciplines including physics, engineering, and economics. image/svg+xml. SymPy is a Python library for symbolic mathematics. Solution: Step 1: Plug e -x in for fx (x) to get: Note that I changed the lower integral bound to zero, because this function is only valid for values higher than zero. Several years ago I mentioned this question to my advanced calculus course and asked if they could nd a polynomial example. Proof of fundamental theorem of calculus. Expected Value and Variance, Feb 2, 2003 - 3 - . I. J. Funct. This approach to . & Calculus 4 MAT 214 Calculus of Several Var. : Derivatives of multivariable functions.

: Derivatives of multivariable functions. 42 (1999) ("The Andrews Festschrift"), Article B42q, 67 pp.]. Step 2: Integrate. Am I understanding this right? That variable is X1^2, which is equal to the variance of X1 plus the mean of X1 squared.

Calculus 1c-2, Examples of Elementary Functions. 7.2. Education Stanford University . Instructions (same as always) Problems (PDF) Submission due via email on Mon Oct 19 3 , Thomas' Calculus, Pearson . It leads naturally to the Lagrangian formulation of mechanics, mentioned above, from which most dynamical equations of . View Lecture 7.pdf from MATH 104 at Kathmandu University. Var. The PDF of this extract thus shows the content exactly as it would be seen by an Open University student. Calculus 3 Notes Pdf Multivariable Calculus Lectures Online this is a link to the playlist for the lectures, from Math 231 of Spring 2016 In order to do this, we shall present a number of dierent types of integrals and sums and then the method of how to calculate them In order to do this, we shall present a number of dierent types of . 2) is the one that minimizes the arc length integral (2 Its really helpful for your assignments and homeworks ) To compute the derivative of the outer function, we just use the power rule: (u2=3)0= 2 3 u 1 2=3 = 3 u 1=3 To compute the derivative of the inner function, we also use the power rule: (1 + x4)0 Similarly, moving across the root 3 to the interval ( 4; 3 .

Figure 1 - The function gx x x x() 9 24 3 32 and its relative extrema. & Linear Alg. Ambrosio L., Dal Maso G.: On the relaxation in BV (; R m) of quasi-convex integrals. MULTIPLE CHOICE QUESTIONS (50%) All answers must be written on the answer sheet; write answers to five questions in each row, for example: 1. Long story short, E (X2)=1/2, Var (X2)= 5/12 (this part is 1/12 + 1/12 + (1/2)^2. It can accordingly be used (with omissions) as a text for a year's course in advanced calculus, or as a text for a three-semester introduction to analysis Saldelux pdf Calculus one and several variables 10th sallas, hille, etgen Hostetler, and Bruce H The precalculus model (n=964) had four significant pedagogical variables, one with positive . There may be more to it, but that is the main point. Advanced Calculus Functions of Several Variables GR Phaijoo, PhD Department of Mathematics School of Science, Kathmandu University Kavre,

Continuity of Functions of Two Var's (Denition . Title: Advanced Calculus Of Several Variables Author: nr-media-01.nationalreview.com-2022-07-04T00:00:00+00:01 Subject: Advanced Calculus Of Several Variables

Conference . Affiliation Institute for Advanced Study Visitor, September 2018 to August 2019. Now however the independent variable is the whole path, so I'll change that path by some small amount and see what happens to the value of the integral I. In other words, we have found an investment with the same expected return as X1 and X2 but with less risk / uncertainty. Intended for stu .

B 1. Search: Calculus 3 Notes Pdf. Multivariable functions also have high points and low points. Limits of 2-Variable Functions (Existence) Josh Engwer (TTU) Functions of Several Variables: Limits & Continuity 23 September 2014 11 / 17 . In Chapters 1-5 the basic ideas and techniques of partial differentiation, and of line, multiple and surface integrals are discussed. For most distributions some "advanced" knowledge of calculus is required to nd the mean. Calculus of Variations and Applications: A . Do the same thing here. Math 42 -- Introduction to Data-Driven Mathematical Modeling: Life, Universe, and Everything Algebra (25%) If X has low variance, the values of X tend to be clustered tightly around the mean value. In the present article, I share with the reader my experience of applying the methods described in the . elementary calculus courses differential and integral calculus of one and of several variables including calculus-based applications and connections with coordinate geometry, trigonometry, differential equations, and other branches of mathematics II. There are 17 problems in total to be solved. Veach's multiple importance sampling: minimizing second moment goal: choose w 1 and w 2 to minimize Var[ w 1 s.t. Bundle: Precalculus, Loose-leaf Version, 10th + WebAssign Printed Access Card for Larson's Precalculus, 10th Edition, Single-Term. Geom. Dr. Stewart most recently served as a professor of mathematics at McMaster University, and his research focused on harmonic analysis.

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. Curriculum Vitae, Chao Li, 1. List of JEE Main & JEE Advanced Books While NCERT Chemistry books are good, one can use Solomons for Organic Chemistry, O. P. Tandon for Inorganic Chemistr and P. Bahadur book for Physical Chemistry, For problems, Numerical Chemistry by R. C. Mukherjee is quite good. . Partial Di erential Equations 16 (2003), pp. Var.

From a programming language viewpoint, the \(\lambda \)-calculus formalises several features of the modern description of computation and its implementation.We present a denotational semantics for the untyped calculus that captures a basic feature of probabilistic programming languages, namely probability distributions as both the objects and the result of a computation. D 5. Gradient and directional derivatives. Calc. Ses #18-25 complete (PDF - 8.6 MB) 19 First fundamental theorem of calculus 20 Second fundamental theorem 21 Applications to logarithms and geometry (PDF - 1.4 MB) 22 Volumes by disks and shells (PDF - 1.7 MB) 23 Work, average value, probability (PDF - 2.2 MB) 24 Numerical integration (PDF - 1.1 MB) 25 Exam 3 review Let f (x)=g (x)/h (x), where both g and h are differentiable and h (x)0. However this is not the global minimum.