Calculus II: Week Two notes
Calculus II: Week Two notes M 145
University of Hartford
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This 7 page Class Notes was uploaded by Alison Holden on Wednesday September 21, 2016. The Class Notes belongs to M 145 at University of Hartford taught by Dr. Hadad in Fall 2016. Since its upload, it has received 11 views.
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Date Created: 09/21/16
Calculus II Week Two Notes Dean Hadad Partial Fractions, Partial fractions: integrals with fractions with a product as a denominator may be seperable into two separate integrals using the method of partial fractions. 1 ∫ ⅆx x(x+1 ) Separate to sum of two fractions: 1 A B ∫ ⅆx= + x(x+1) x x+1 Cross multiply the numerators and the denominators: A(x+1 ) B x) + x x+1 Simplify the numerators. (denominators can be ignored at this point) and set equal to numerator in original integral: Ax+A+Bx=1 Make two equations by setting like terms to each other: A+B=O (x terms) A=1 (constants) Solve the system of equations: A=1 B= -1 Going back to the partial fractions equation, substitute the new values for the variables: ∫ 1 ⅆx+∫ −1 ⅆx x x+1 Solve the integral: ln|x−ln x+1 +C Ex #2: ∫ 5 x−4 ⅆx=∫ 5x−4 ⅆx (2 x +x−1) (2 x−1)(+1 ) A + B 2x−1 x+1 Ax+A+2 Bx−B=5x−4 A+2B=5 A−B=−4 A=−1 B=3 −1 3 ∫ 2 x−1ⅆx+∫ x+1 ⅆx −1 ln|2 x−| + 3ln|x+1| 2 Linear Approximation: using shapes on graphs to estimate the area under the curve b−a Δx= n lower limit = b upper limit = a number of sections = n Right Approximation: 5 2 ∫ X ⅆx , n=4 1 Split into 4 sections, draw rectangles anchored at the right number. Find the height of each rectangle by putting n in the function x 2 f(2)=4 f (3) = 9 f (4) = 16 f (5) = 25 Use the height and distance to each new rectangle (in this case, 1) to find the area of each rectangle. Then add them all together. ∫ x ⅆx=1 4+9+16+25 ) = 54 Left approximation: Same as right approximation except the rectangles are anchored to the left Finish by finding area of each rectangle, similar to right approximation, however use the height calculated by using the left most x-value to each rectangle. Middle Approximation Split the graph the same way you would for the other approximations using the given n value. Split each section in half and use that line to draw a rectangle from both sides of the middle line Find the area of each rectangle and add them together to find the approximate value Trapezoidal Approximation: Δx General Formula: T n [ ( 0 +2 f( 1+2 f ( 2…+2 f x ( n−1) f( n] 2 Split the area from a to b under the curve using the value given as n. Find the area of each trapezoid and add them together Simpson’s Rule: Δx General Formula: sn= [f( 0+4 f ( 12 f x( 24 f x ( n−1+ f( n ] 3 Ex: y = x 2 0≤ x≤4 , n=4 4−0 ΔX= =1 4 1 Sn= (f( )+4 f( )+2 f( )+4 f ( ) f( )) 3 f(0)=0, f (1=1, f (2)=4, f 3 =9, f (4)=16 1 sn= (0+1+4+9+16 ) = 10 3
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