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The document discusses mathematical formulas and proofs. It contains: 1) Formulas for polynomials and series expansions using binomial coefficients. 2) A claim and proof about the series expansion of (1-x)-3 using binomial theorem. 3) Notations showing equivalence and equality of expressions.
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1. Find the line tangent to the curve of intersection between the surfaces F(x,y,z)=x^2+y^2+z^2=1 and g(x,y,z)=x+y+z+5 at the point P=(1,2,2).
2. Find the values of constants a and b that will make the expression zxy-zx-zy identically zero, given z=U(x,y)e(ax+by) where Uxy=0.
3. Find the points on the sphere x^2+y^2+z^2=9 whose distance from the point (4,-8,8) areSheet1 simplified
Sheet1 simplifiedmarwan a
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The document describes a damped mass-spring system and provides the equation of motion for analyzing the free vibration of the system. It then gives the general solution to the differential equation that describes the response x(t) in terms of the system's natural frequency, damping ratio, initial displacement, and initial velocity. The student is asked to:
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2. Run sample code that plots the response for different damping ratios.
3. Calculate and submit the response at two specific cases.Physical Chemistry Assignment Help
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I am Frank P. I am a Physical Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Physical Chemistry, from Malacca, Malaysia. I have been helping students with their homework for the past 6 years. I solve assignments related to Physical Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Physical Chemistry Assignments.Calculus First Test 2011/10/20
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This document contains the work of a student on a calculus test. It includes:
1) Solving limits, finding derivatives, and applying L'Hopital's rule.
2) Using induction to prove an identity.
3) Providing epsilon-delta proofs of limits.
4) Finding where a tangent line is parallel to a secant line.
5) Proving statements about limits of functions.
The student provides detailed solutions showing their work for each problem on the test.Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)
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This document provides an overview of the key points covered in a calculus lecture on derivatives of logarithmic and exponential functions:
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03 truncation errorsmaheej
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Let's analyze the remainder term R6 using the geometry series method:
|tj+1| = (j+1)¦Ð-2 ¡Ü ¦Ð-2 = k|tj| for all j ¡Ý 6 (where 0 < k = ¦Ð-2 < 1)
Then, |R6| ¡Ü t7(1 + k + k2 + k3 + ...)
= t7/(1-k)
= 7¦Ð-2/(1-¦Ð-2)
So the estimated upper bound of the truncation error |R6| is 7¦Ð-2/(1-¦Ð-2)Binomial theorem for any index
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The document summarizes the binomial theorem and properties of binomial coefficients. It provides:
1) The binomial theorem expresses the expansion of (a + b)n as a sum of terms involving binomial coefficients for any positive integer n.
2) Important properties of binomial coefficients are discussed, such as their relationship to factorials and the symmetry of coefficients.
3) Examples are given of using the binomial theorem to find coefficients and solve problems involving divisibility and series of binomial coefficients.Recently uploaded (20)
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- 1. Mathtype f ( x) = an x n + an ?1 x n ?1 + L , an ¡Ù 0 an ?1 n ?1 = (x + n x + L), exist an = ( x m + bm ?1 x m ?1 + L) ³Ìʽ¾ŽÝ‹Æ÷ claim: ¡Æ hx n 1 3 n¡Ý0 n = (1 ? x ) 13 + 23 +L + n3 =? A=B A= A1 = A2 = L =B B= B1 = B2 = L =A (1 + x ) 2 =1+2x+ x 2 1 3 1 1 1 2 2 2 Pf: (1 ? x ) = (1 ? x ) (1 ? x ) (1 ? x ) = (1+x+x +L ) (1+x+x +L ) (1+x+x +L ) = ¡Æxei ¡Ý 0 e1 x e2 x e3 = ¡Æx ei ¡Ý 0 e1 + e2 + e3 = ¡Æ (# of sols of e1 + e2 + e3 =n) x n n¡Ý 0 On the other hand,we have by newton¡¯s binomial theorem ? ?3 ? ? n + 2? n ? 2? n ) = ( x ? 1) ?3 = ¡Æ ? ? (? x ) n = ¡Æ ? ? x ? ? = ¡Æ n ¡.(1) 1 3 h xn ( 1? x n¡Ý 0 ?n? n¡Ý 0 ? 2 ? ? 1 ? n¡Ý0