File
... An ellipse is a curve on a plane surrounding two focal points such that a straight line drawn from one of the focal points to any point on the curve and then back to the other focal point has the same length for every point on the curve. As such, it is a generalization of a circle, which is a specia ...
... An ellipse is a curve on a plane surrounding two focal points such that a straight line drawn from one of the focal points to any point on the curve and then back to the other focal point has the same length for every point on the curve. As such, it is a generalization of a circle, which is a specia ...
1 Introduction and Definitions 2 Example: The Area of a Circle
... Notation 1 If I slip up, it’s likely that I’ll denote x1 ; x2 ; ::: as fxn gn=1 ; or, even shorter, as fxn g. Keep in mind that this is just notation, so you shouldn’t be scared of it. However, I’ll try to avoid building up an excessive amount of notation since that can get confusing. This seems lik ...
... Notation 1 If I slip up, it’s likely that I’ll denote x1 ; x2 ; ::: as fxn gn=1 ; or, even shorter, as fxn g. Keep in mind that this is just notation, so you shouldn’t be scared of it. However, I’ll try to avoid building up an excessive amount of notation since that can get confusing. This seems lik ...
FABER FUNCTIONS 1. Introduction 1 Despite the fact that “most
... and the sum on the right clearly does not converge as n goes to infinity, we can conclude that F 0 (x) does not exist. However, since x was chosen arbitrarliy in [0, 1] which is the interval where we defined F , we see that F is in fact differentiable nowhere on its domain. ...
... and the sum on the right clearly does not converge as n goes to infinity, we can conclude that F 0 (x) does not exist. However, since x was chosen arbitrarliy in [0, 1] which is the interval where we defined F , we see that F is in fact differentiable nowhere on its domain. ...
math318hw1problems.pdf
... Let f : Y → Z be continuous. For every open V ⊂ Z and for every g ∈ C 0 (V ) we obtain f ∗ g ∈ C 0 (f −1 V ) giiven by (f ∗ g)x = g(f (x)) for all x ∈ f −1 V . Definition 3.2. Let Y be a topological space. A subpresheaf (or a presheaf of subsets) R of CY0 consists of (1) the data: a subset R(U ) ⊂ C ...
... Let f : Y → Z be continuous. For every open V ⊂ Z and for every g ∈ C 0 (V ) we obtain f ∗ g ∈ C 0 (f −1 V ) giiven by (f ∗ g)x = g(f (x)) for all x ∈ f −1 V . Definition 3.2. Let Y be a topological space. A subpresheaf (or a presheaf of subsets) R of CY0 consists of (1) the data: a subset R(U ) ⊂ C ...
MATH 2720 Winter 2012 Assignment 4 Questions 1, 2, 6 and 8 were
... 2. Let c(t) be a flow line of a gradient field F = −∇V . Prove that V (c(t)) is a decreasing function of t. Solution. We obviously have to consider a function V : Rn → R, since c : R → Rn and we do not know what it means for a function to be decreasing except when it is scalar valued. So V (c(t)) is ...
... 2. Let c(t) be a flow line of a gradient field F = −∇V . Prove that V (c(t)) is a decreasing function of t. Solution. We obviously have to consider a function V : Rn → R, since c : R → Rn and we do not know what it means for a function to be decreasing except when it is scalar valued. So V (c(t)) is ...
Calculus - maccalc
... 6. To what new value should f(1) be changed to make f continuous at x = 1? ______ 7. What is the domain of this function? (Interval Notation) ________________________ 8. What is the range of this function? (Interval Notation) ______________________ ...
... 6. To what new value should f(1) be changed to make f continuous at x = 1? ______ 7. What is the domain of this function? (Interval Notation) ________________________ 8. What is the range of this function? (Interval Notation) ______________________ ...
Mathematical Analysis Worksheet 8
... xe−c = 1 − e−x . As 0 < e−c < 1, we get the estimate x > 1 − e−x which proves that e−x > 1 − x. Note: We could have also used the more general result (cf. Exercise Sheet 4) that if |f ′ (x)| < M for all x, then |f (b) − f (a)| < M (b − a). Another consequence of the MVT is that a function is constan ...
... xe−c = 1 − e−x . As 0 < e−c < 1, we get the estimate x > 1 − e−x which proves that e−x > 1 − x. Note: We could have also used the more general result (cf. Exercise Sheet 4) that if |f ′ (x)| < M for all x, then |f (b) − f (a)| < M (b − a). Another consequence of the MVT is that a function is constan ...
Final Review - Mathematical and Statistical Sciences
... Chapter I: Functions and Models Function, domain, codomain, range, independent and dependent variable, graph of function, Vertical Line Test, piecewise defined fns, even, odd, periodic, increasing, decreasing, linear function, slope, intercept, polynomial, coefficient, degree, power function, ration ...
... Chapter I: Functions and Models Function, domain, codomain, range, independent and dependent variable, graph of function, Vertical Line Test, piecewise defined fns, even, odd, periodic, increasing, decreasing, linear function, slope, intercept, polynomial, coefficient, degree, power function, ration ...
Block 5 Stochastic & Dynamic Systems Lesson 14 – Integral Calculus
... interval [a,b] we divide the interval into n subintervals of equal width, x, and from each interval choose a point, xi*. Then the definite integral of f(x) from a to b is ...
... interval [a,b] we divide the interval into n subintervals of equal width, x, and from each interval choose a point, xi*. Then the definite integral of f(x) from a to b is ...
3.5 Derivatives of Trigonometric Functions
... Jerk is the derivative of acceleration. If a body’s position at time t is s(t), the body’s jerk at time t is ...
... Jerk is the derivative of acceleration. If a body’s position at time t is s(t), the body’s jerk at time t is ...
THE FEBRUARY MEETING IN NEW YORK The two hundred sixty
... 7. Professor W. A. Wilson : On linear upper semi-continuous collections of continua. Let M be a bounded upper semi-continuous collection of continua {x\ such t h a t no two have common points and X =f(t) is upper semicontinuous in the interval a^t^b; a collection of this kind may be called linear an ...
... 7. Professor W. A. Wilson : On linear upper semi-continuous collections of continua. Let M be a bounded upper semi-continuous collection of continua {x\ such t h a t no two have common points and X =f(t) is upper semicontinuous in the interval a^t^b; a collection of this kind may be called linear an ...
Partial solutions to Even numbered problems in
... 32. This problem is tricky. We have to apply the MVT to the velocity function (first derivative), so that we can make a conclusion about acceleration (second derivative). Let p(t) be the position function of the car at time t. Then p0 is the velocity function of the car. Also, p0 is continuous and d ...
... 32. This problem is tricky. We have to apply the MVT to the velocity function (first derivative), so that we can make a conclusion about acceleration (second derivative). Let p(t) be the position function of the car at time t. Then p0 is the velocity function of the car. Also, p0 is continuous and d ...
4.2 Mean Value Theorem
... right? In fact, we could just take this as the definition of increasing couldn’t we? We could say that by definition, f 0 (x) > 0 means a function is increasing. But before we ever knew calculus, we probably knew what increasing meant. It meant that a second value of f was bigger than a first value. ...
... right? In fact, we could just take this as the definition of increasing couldn’t we? We could say that by definition, f 0 (x) > 0 means a function is increasing. But before we ever knew calculus, we probably knew what increasing meant. It meant that a second value of f was bigger than a first value. ...
Course Narrative
... • Corresponding characteristics of the graphs of ƒ, ƒ’ and ƒ” • Relationship between the concavity of ƒ and the sign of ƒ” • Points of inflection as places where concavity changes Applications of derivatives • Analysis of curves, including the notions of monotonicity and concavity • Optimization, bo ...
... • Corresponding characteristics of the graphs of ƒ, ƒ’ and ƒ” • Relationship between the concavity of ƒ and the sign of ƒ” • Points of inflection as places where concavity changes Applications of derivatives • Analysis of curves, including the notions of monotonicity and concavity • Optimization, bo ...
Solutions for Exam 4
... may choose any six to do. Please write DON’T GRADE on the one that you don’t want me to grade. In writing your solution to each problem, include sufficient detail and use correct notation. (For instance, don’t forget to write “=” when you mean to say that two things are equal.) Your method of solving ...
... may choose any six to do. Please write DON’T GRADE on the one that you don’t want me to grade. In writing your solution to each problem, include sufficient detail and use correct notation. (For instance, don’t forget to write “=” when you mean to say that two things are equal.) Your method of solving ...
Calculus BC Study Guide Name
... ___________ 42. When does a function have a horizontal tangent line? What is the equation of this line? ___________ 43. When does a function have a vertical tangent line? What is the equation of this line? ___________ 44. Guidelines for Solving Optimization problems ___________ 45. Local Linearizati ...
... ___________ 42. When does a function have a horizontal tangent line? What is the equation of this line? ___________ 43. When does a function have a vertical tangent line? What is the equation of this line? ___________ 44. Guidelines for Solving Optimization problems ___________ 45. Local Linearizati ...
15 - BrainMass
... A) C0 = -1, C1 = 21 B) C0 = 1, C1 = -21 C) C0 = -1, C1 = 19 D) C0 = 0, C1 = 1 17. What does du equal in ∫2x(x2 + 1)5 dx? A) 2x B) 2u du C) 2x dx D) 5u4 18. What is needed to fully determine an anti-differentiated function? A) A lot of luck. B) A boundary condition. C) What its value is at (0, 0). D) ...
... A) C0 = -1, C1 = 21 B) C0 = 1, C1 = -21 C) C0 = -1, C1 = 19 D) C0 = 0, C1 = 1 17. What does du equal in ∫2x(x2 + 1)5 dx? A) 2x B) 2u du C) 2x dx D) 5u4 18. What is needed to fully determine an anti-differentiated function? A) A lot of luck. B) A boundary condition. C) What its value is at (0, 0). D) ...
MATH M16A: Applied Calculus Course Objectives (COR) • Evaluate
... Sketch the graph of functions using the first and second derivatives to determine intervals where the functions are decreasing and increasing, maximum and minimum values, intervals of concavity and points of inflection. Solve applied problems involving tangent lines, rates of change and related rate ...
... Sketch the graph of functions using the first and second derivatives to determine intervals where the functions are decreasing and increasing, maximum and minimum values, intervals of concavity and points of inflection. Solve applied problems involving tangent lines, rates of change and related rate ...
CHEM 442 Lecture 15 Problems (see reverse) 15
... 15-3. Expand the first-order correction to the wave function as Y (1) = å Ck t Y (0) e- iEk ...
... 15-3. Expand the first-order correction to the wave function as Y (1) = å Ck t Y (0) e- iEk ...
![Mathematics 414 2003–04 Exercises 5 [Due Monday February 16th, 2004.]](http://s1.studyres.com/store/data/000084574_1-c1027704d816dc0676e3e61ce7dab3b7-300x300.png)
Mathematics 414 2003–04 Exercises 5 [Due Monday February 16th, 2004.]
... 1. Suppose f and g are meromorphic functions on an open set G ⊆ C, and let Hf be the points where f is analytic in G (Hf = G \ {poles of f }), Hg the set where g is analytic. Show that in each of the following cases, there is a unique meromorphic function h on G such that: (a) h(z) = f (z) + g(z) fo ...
... 1. Suppose f and g are meromorphic functions on an open set G ⊆ C, and let Hf be the points where f is analytic in G (Hf = G \ {poles of f }), Hg the set where g is analytic. Show that in each of the following cases, there is a unique meromorphic function h on G such that: (a) h(z) = f (z) + g(z) fo ...
Ph.D. QUALIFYING EXAM DIFFERENTIAL EQUATIONS Spring II, 2009
... (b) Set up the Picard iteration process and prove that the sequence converges uniformly on the interval [a, b] to a limit function x~ (t). (c) Show that x~o (t) is a solution to the differential equation on [a, b]. (d) Establish that the solution to the differential equation with the given initial c ...
... (b) Set up the Picard iteration process and prove that the sequence converges uniformly on the interval [a, b] to a limit function x~ (t). (c) Show that x~o (t) is a solution to the differential equation on [a, b]. (d) Establish that the solution to the differential equation with the given initial c ...
Solutions - Penn Math
... You could also do this problem without using the fundamental theorem of line integrals, just by using the definition of a line integral. ...
... You could also do this problem without using the fundamental theorem of line integrals, just by using the definition of a line integral. ...
Math 108, Final Exam Checklist
... • Elementary functions: Polynomials, rational functions, exponential functions, logarithms, trigonometric functions, inverse trigonometric functions Chapter 2: Limits, Continuity and Derivatives • Intuitive definition of limit • One-sided limits • Infinite limits and vertical asymptotes • Limits at ...
... • Elementary functions: Polynomials, rational functions, exponential functions, logarithms, trigonometric functions, inverse trigonometric functions Chapter 2: Limits, Continuity and Derivatives • Intuitive definition of limit • One-sided limits • Infinite limits and vertical asymptotes • Limits at ...