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This document contains 5 math homework problems for MATH 2090 due on January 24, 2013. The problems involve (1) verifying a solution to a differential equation, (2) verifying a family of solutions to a differential equation, (3) solving an initial value problem, (4) solving another initial value problem, and (5) finding a number k such that a family of functions are solutions to a differential equation.
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This document contains solutions to 4 problems involving ordinary differential equations:
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1. The document contains 20 multi-part math problems involving functions, equations, inequalities, exponents, and logarithms. The problems cover topics like finding domains and ranges of functions, solving equations and inequalities, finding inverse functions, and composition of functions.
2. Contact information is provided at the top for Ersingh with an email of ersingh@hotmail.com regarding higher level mathematics.
3. The problems are presented without explanations and involve advanced concepts requiring mathematical reasoning and problem solving skills to arrive at the solutions.
Jejemon
This document is a take-home exam for a mathematics course. It contains 3 sections with multiple questions each: 1) limits of multivariable functions, 2) partial differentiation, and 3) tangent planes and differentials. Students are instructed to show their work and solutions independently in a blue book and submit it by a specified deadline.
Q uiz sequence n series...stpm
This document contains two mathematics quizzes covering sequences and series. Quiz 1 has three problems: (1) expressing a fraction in partial fractions and finding the expansion and convergence of a series, (2) using the method of differences to find sums of series, (3) expressing a recurring decimal as a rational number. Quiz 2 has three problems: (1) finding terms in a binomial expansion, (2) expanding a binomial expression and stating the valid range, (3) proving an equality for small x and using it to evaluate an expression.
Cs 601
This document contains a 7 page exam for the course CS-601: Differential and Integral Calculus with Applications. The exam contains 8 questions testing a variety of calculus concepts:
1) Part a contains 6 multiple choice questions testing derivatives, integrals, limits, and monotonicity. Part b contains 6 fill in the blank questions testing derivatives, integrals, and equations of tangents.
2) Questions 2-5 contain additional multiple choice or short answer problems testing continuity, derivatives, integrals, Rolle's theorem, and partial derivatives.
3) Questions 6-8 contain free response problems on geometry, differential equations, and approximating an area using Simpson's rule. The exam tests a comprehensive understanding of
Emat 213 midterm 2 fall 2005
This document contains a midterm exam for an engineering mathematics course. It consists of 4 problems:
1. Finding the general solution to two linear ODEs.
2. Finding the complementary and particular solutions for a given non-homogeneous linear ODE, and using them to solve an IVP.
3. Repeating steps from problem 2 for another given non-homogeneous linear ODE.
4. Modeling and solving an ODE describing the motion of a damped spring-mass system subject to an external force.
Double integration
This document provides examples and explanations of double integrals. It defines a double integral as integrating a function f(x,y) over a region R in the xy-plane. It then gives three key points:
1) To evaluate a double integral, integrate the inner integral first treating the other variable as a constant, then integrate the outer integral.
2) The easiest regions to integrate over are rectangles, as the limits of integration will all be constants.
3) For non-rectangular regions, the limits of integration may be variable, requiring more careful analysis to determine the limits for each integral.
Introduction to inverse problems
This document provides an introduction to inverse problems and their applications. It summarizes integral equations like Volterra and Fredholm equations of the first and second kind. It also describes inverse problems for partial differential equations, including inverse convection-diffusion, Poisson, and Laplace problems. Applications mentioned include medical imaging, non-destructive testing, and geophysics. Bibliographic references are provided.
Calculus Final Exam
1) The student solved several integral evaluation problems and derivative problems.
2) They sketched the region bounded by two curves and found its area.
3) Several functions were analyzed, including finding their derivatives, extrema, concavity, asymptotes and sketching their graphs.
4) Some proofs and word problems involving applications of calculus like radioactive decay were also addressed.
Quadratic functions and models
The document discusses quadratic functions and models. It defines quadratic functions as functions of the form f(x) = ax^2 + bx + c. It provides examples of expressing quadratic functions in standard form and using standard form to sketch graphs and find minimum/maximum values. The document also provides examples of modeling real-world situations using quadratic functions to find things like maximum area or revenue.
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Lesson 1: Functions and their representations (slides)
Lesson 1: Functions and their representations (slides)
Hw1
- 1. MATH 2090: HOMEWORK DUE JAN. 24, 2013. (1) Verify that the function f (x) = ln x + x is a solution of the ordinary differential equation x2 y + 2xy + y = ln x + 3x + 1. Say on which interval. 3 (2) Verify that the family of functions y(x) = c1 + c2 e−x + x are solutions of the 3 ordinary differential equation y + y − x2 − 2x = 0. Say on which interval. (3) Solve the initial value problem 2 y + x(y ) = 0 y(1) = 1 y (1) = −1. Hint: try to solve for y first. (4) Solve the initial value problem xy − 2y = 0 x y(2) = 1 . e (5) Find a number k such that the family of functions y(x) = c1 sin kx + c2 cos kx are solutions of the ordinary differential equation y + 9y = 0. Say on which interval. 1