By Russell L. Herman

Creation and ReviewWhat Do i must be aware of From Calculus?What i want From My Intro Physics Class?Technology and TablesAppendix: Dimensional AnalysisProblemsFree Fall and Harmonic OscillatorsFree FallFirst Order Differential EquationsThe uncomplicated Harmonic OscillatorSecond Order Linear Differential EquationsLRC CircuitsDamped OscillationsForced SystemsCauchy-Euler EquationsNumerical recommendations of ODEsNumericalRead more...

summary: creation and ReviewWhat Do i must be aware of From Calculus?What i want From My Intro Physics Class?Technology and TablesAppendix: Dimensional AnalysisProblemsFree Fall and Harmonic OscillatorsFree FallFirst Order Differential EquationsThe easy Harmonic OscillatorSecond Order Linear Differential EquationsLRC CircuitsDamped OscillationsForced SystemsCauchy-Euler EquationsNumerical ideas of ODEsNumerical ApplicationsLinear SystemsProblemsLinear AlgebraFinite Dimensional Vector SpacesLinear TransformationsEigenvalue ProblemsMatrix formula of Planar SystemsApplicationsAppendix: Diagonali

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The derivative has been moved from one function to the other. Recall that this formula was derived by integrating the product rule for differentiation. ) These two formulae can be related using the differential relations u = f (x) → du = f ( x ) dx, v = g( x ) → dv = g ( x ) dx. 63) This also gives a method for applying the Integration by Parts Formula. 8. Consider the integral x sin 2x dx. We choose u = x and dv = sin 2x dx. This gives the correct left side of the Integration by Parts Formula.

We will investigate such convergence shortly. Series expansions for many elementary functions arise in a variety of applications. 7. We still need to determine the values of x for which a given power series converges. The first five of the above expansions converge for all reals, but the others only converge for | x | < 1. n We consider the convergence of ∑∞ n=0 cn ( x − a ) . For x = a, the series obviously converges. Will it converge for other points? 2. If ∑∞ n=0 cn ( b − a ) converges for b = a, then ∞ n ∑n=0 cn ( x − a) converges absolutely for all x satisfying | x − a| < |b − a|.

2 2 Geometric series are important because they are easily recognized and summed. Other series which can be summed include special cases of Taylor series and telescoping series. Next, we show an example of a telescoping series. 28. ∑∞ n =1 1 n ( n +1) The first few terms of this series are ∞ 1 1 1 1 1 = + + + +.... n ( n + 1 ) 2 6 12 20 n =1 ∑ It does not appear that we can sum this infinite series. However, if we used the partial fraction expansion 1 1 1 = − , n ( n + 1) n n+1 then we find that the kth partial sum can be written as k sk = 1 n ( n + 1) n =1 = ∑ ∑ k n =1 = 1 1 − n n+1 1 1 − 1 2 + 1 1 − 2 3 +···+ 1 1 − k k+1 .