Appendix%20Math

Appendix%20Math - Appendix Mathematical Background Review...

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1 Appendix. Mathematical Background Review of Calculus and Solution of Ordinary Differential Equations. The following covers many of the techniques used in the solution of ordinary differential equations, integration, and evaluation of limits. A.1 . Calculus A.1.1. Integration by Parts. Often a function u can be integrated by transposition of the function and the integrand dv using the following relation: udv = uv - vdu ( A . 1 . 1 ) Example A.1. Evaluate the following integral by parts sin 2 (ax)dx . Solution : To solve, make the following substitutions: u = sin(ax) dv = sin(ax)dx du = acos(ax)dx v = - 1 a cos(ax) Inserting into Eq. A.1.1, the integral becomes sin 2 (ax)dx = 1 a sin(ax)cos(ax) - cos 2 (ax)dx (A.1.1a) To simplify, note that cos 2 (ax) = 1 - sin 2 (ax) and cos 2 (ax)dx 1 sin 2 (ax)  dx . After rearrangement, the solution is sin 2 (ax)dx = - 1 2a sin(ax)cos(ax) + x 2 ( A . 1 . 1 b ) A.1.2. Evaluation of Limits L'Hôpital's Rule For two functions f(z) and g(z) that are zero at z = a and have k derivatives equal to zero at z = a, limit z a f(z) g(z) f k +1 (a) g k+1 (a) (A.1.2)
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2 where f (k+1) (a) is the k + 1 derivative of the function f(z) evaluated at z = a and g (k+1) (a) 0. Example A.2. Determine the following limit: limit z 0 1- cos(z) z 2 Solution : At z equal to zero, cos(z) is equal to 1 and both the numerator and denominator are zero. If we apply L'Hopital's rule once, we obtain sin(z) 2z , which at z = 0 is undefined. Applying L'Hopital's rule a second time, we obtain cos(z) 2 , which is equal to one-half at z = 0. Thus, limit z 0 1- cos(z) z 2 = cos(0) 2 1 2 (A.1.3) A.1.3. Taylor Series Approximations The value of a function f(z) about the point z = a can be approximated by the following series expansion: f(z) f(a) f' (a)(z a) f''(a)(z 2 2 f' '' (a)(z 3 6 ... f n (a)(z n n! n 0 (A.1.4) where the prime denotes differentiation with respect to the independent variable z (e.g. df(z) dz z a ). Such an expansion is known as a Taylor series approximation . A common application of the Taylor series approximation is to estimate the value of a function at a small perturbation away from a known value. Consider a small perturbation z + where 0 < « A. To simplify, truncate the Taylor series after the second term. f(z  ) dz z  (A.1.5) For functions of two variables y and z, this approximate formula becomes ,y  ) f(z,y) z z  y y  (A.1.6)
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3 Example A.3. Derive the Taylor series approximation for sin(x). For this case, a = 0 and the Taylor series approximation is sin(x) d n sin(x) dx n x 0 x n n! n 0 (A.1.7a) Solution: For n = 0, 2, 4, 6,. .., the derivatives are sin(x), -sin(x), sin(x), -sin(x), etc., which are equal to zero at x=0. For n = 1, 3, 5, 7,. .., the derivatives are cos(x), -cos(x), cos(x), -cos(x), etc., which are equal to 1 for x=0. Thus, sin(x) x x 3 3!
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Appendix%20Math - Appendix Mathematical Background Review...

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