By Russell L. Herman

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

summary: creation and ReviewWhat Do i have to be aware of From Calculus?What i would like 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 suggestions of ODEsNumerical ApplicationsLinear SystemsProblemsLinear AlgebraFinite Dimensional Vector SpacesLinear TransformationsEigenvalue ProblemsMatrix formula of Planar SystemsApplicationsAppendix: Diagonali

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For example, consider cos3 x dx. This can be rewritten as cos3 x dx = Integration of odd powers of sine and cosine. cos2 x cos x dx. Let u = sin x. Then, du = cos x dx. Since cos2 x = 1 − sin2 x, we have cos3 x dx = cos2 x cos x dx = (1 − u2 ) du 1 = u − u3 + C 3 1 = sin x − sin3 x + C. 74) A quick check confirms the answer: d dx sin x − 1 sin3 x + C 3 = cos x − sin2 x cos x = cos x (1 − sin2 x ) = cos3 x. Integration of even powers of sine and cosine. 75) Even powers of sines and cosines are a little more complicated, but doable.

1 2 2(1) (1− x )3 2(1) 2! 2! =1 3 3(2)(1) (1− x )4 3(2)(1) 3! 3! =1 cn ∞ 1 ∼ ∑ xn . 1−x n =0 We can replace ∼ by equality if we can determine the range of x-values for which the resulting infinite series converges. 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 2. ∑∞ n=0 cn ( x − a ) may converge for all real numbers. 7: Common Mclaurin Series Expansions ex = cos x = sin x = cosh x = sinh x = 1 1−x 1 1+x tan−1 x ln(1 + x ) = = = = Series Expansions You Should Know ∞ x2 x3 x4 xn 1+x+ + + +... = ∑ 2 3! 4! n! n =0 ∞ 2 4 x x x2n 1− + −... = ∑ (−1)n 2 4! (2n)! n =0 ∞ 3 5 x x x2n+1 x− + −... = ∑ (−1)n 3! 5! (2n + 1)! n =0 ∞ 2 4 2n x x x 1+ + +... = ∑ 2 4! ( 2n )! n =0 ∞ x3 x5 x2n+1 x+ + +... = ∑ 3! 5! (2n + 1)! n =0 ∞ 1 + x + x2 + x3 + . . 2 = ∑ xn n =0 ∞ 3 1−x+x −x +...