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ISU - MATH 166 - Math 166 Calculus 2 - Study Guide

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ISU - MATH 166 - Math 166 Calculus 2 - Study Guide

School: Iowa State University
Department: Math
Course: Calculus 2
Professor: Paul Barloon
Term: Summer 2016
Tags: integral, improper, type, monotonic, bounded, Upper, Lower, Trapezoidal Rule, Simpson's Rule, Error, Convergence, Divergence, Limit, Comparison, direct, Sequences, nonincreasing, and nondecreasing
Name: Math 166 Calculus 2
Description: This study guide covers what will be on the next exam. They cover sections 8.6, 8.7, and 10.1.
Uploaded: 03/20/2018
This preview shows pages 1 - 2 of a 4 page document. to view the rest of the content
background image Math 166 - Calculus 2   Chapter 8: Techniques of Integration  8.6 | Numerical Integration 
The Trapezoidal Rule:  
To approximate   use (x)dx b a f   T =  (y y y .. y ) 2 Δx 0 + 2 1 + 2 2 + . + 2 n−1 y n   The y’s are the values of   f   ​at the partition points  , where x x, Δx, ... , x   n xx x 0 a 1 1 + Δ x 2 + 2     n−1 + ( − 1   n b       = (b-a)/n x Δ    
Simpson’s Rule: 
To approximate   use (x)dx b a f   S =  (y 4 y .. y y ) 3 Δx 0 + 2 + 2 2 + . + 2 n−2 + 4 n−1 y n   The y’s are the values of   f   ​at the partition points  , the number  n x x, Δx, ... , x   n xx x 0 a 1 1 + Δ x 2 + 2     n−1 + ( − 1   n b     is even, and   = (b-a)/n x Δ    
Error Estimations: 
If  
f’’  ​is continuous and M is any upper bound for the values of   on [a,b], then the error in the f| ′′ |   approximation of the integral of  f ​ from ​a​ to ​b​ for n steps satisfies the inequality:   Trapezoidal Error:   E | | T | | ≤ 12n 2 M(ba) 3     Simpson’s Error:  E | | S | | ≤ 180n 4 M(ba) 4    
8.7 | Improper Integrals 
Integrals with infinite limits of integration are improper integrals of Type 1. 
1. If f(x) is continuous on [a,  ), then   (xdx (xdx a f = lim b→∞ b a f   2. If f(x) is continuous on (- , b], then    (xdx (xdx b −∞ f = lim a→−∞ b a f  
background image 3. If f(x) is continuous on (- ,  ), then ∞ ∞    , where  c ​ is any number (xdx (xdx  (xdx −∞ f = lim a→∞ c a f +   lim b→∞ b c f   In each case, if the limit is finite we say that the improper integral  converges ​and that the limit is  the  value​ of the improper integral. If the limit fails to exist, the improper integral ​diverges​.   
Integrals of functions that become infinite at a point within the interval of integration are 
improper integrals of Type 2. 
1. If f(x) is continuous on (a, b] and discontinuous at  a ​, then  (xdx (xdx b a f = lim ca + b c f   2. If f(x) is continuous on [a, b) and discontinuous at  b ​, then   (xdx (xdx b a f = lim cb c a f   3. If f(x) is discontinuous at c, where a < c < b, and continuous on [a,c) (c, b], then    (xdx (xdx  (xdx b a f = c a f +   b c f   In each case, if the limit is finite we say that the improper integral  converges ​and that the limit is  the  value​ of the improper integral. If the limit does not exist, the integral ​diverges​.   
Direct Comparison Test: 
Let 
​and ​g​ be continuous on [a,  ) with 0  f(x)  g(x) for all x  a. Then,   1.  converges if   converges. (xdx a f (xdx a g   2.  diverges if   diverges. (xdx a g (xdx a f    
 
Limit Comparison Test 
If the positive functions f and g are continuous on [a,  ), and if 
,  0 < L <  , then lim x→∞ f(x) g(x) L     and  both converge or both diverge. (xdx a f (xdx  a g          Chapter 10: Infinite Sequences and Series 

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School: Iowa State University
Department: Math
Course: Calculus 2
Professor: Paul Barloon
Term: Summer 2016
Tags: integral, improper, type, monotonic, bounded, Upper, Lower, Trapezoidal Rule, Simpson's Rule, Error, Convergence, Divergence, Limit, Comparison, direct, Sequences, nonincreasing, and nondecreasing
Name: Math 166 Calculus 2
Description: This study guide covers what will be on the next exam. They cover sections 8.6, 8.7, and 10.1.
Uploaded: 03/20/2018
4 Pages 33 Views 26 Unlocks
  • Better Grades Guarantee
  • 24/7 Homework help
  • Notes, Study Guides, Flashcards + More!
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