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pr_midterm - ENM510 - Foundations of Engineering...

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Unformatted text preview: ENM510 - Foundations of Engineering Mathematics I (Practice Problems for the Midterm) Fall Semester, 2010 M. Carchidi –––––––––––––––––––––––––––––––––––– Problem #1 Consider all real values of λ for which there are nonzero solutions ϕ ( x ) to the problem consisting of the ordinary di f erential equation ϕ 00 ( x ) + λ 2 ϕ ( x ) = 0 , for ≤ x ≤ L , and the conditions: 1 L Z L ϕ ( x ) dx = ϕ (0) and ϕ ( L ) = 0 . a.) Show that these satisfy an equation of the form f ( z ) = 0 , with z = λ L , and you are to determine the functional form of f . b.) Show that z n = 2 n π gives a subset of the solutions to f ( z ) = 0 , and for these, determine (up to a multiplicative constant), the corresponding non- zero function ϕ n ( x ) . –––––––––––––––––––––––––––––––––––– Problem #2 Let P 3 [ x ; R ] be the vector space of all polynomials in x of degree less than or equal to 3 with the standard de f nitions of polynomial addition and scalar multiplication. Next let U [ x ; R ] be the subset of P 3 [ x ; R ] consisting of those polynomials in P 3 [ x ; R ] that satisfy the equations Z 1 p ( x ) dx = Z 2 1 p ( x ) dx and p (1) = p (1) Show that U [ x ; R ] is a subspace of P 3 [ x ; R ] by determining a basis set for U [ x ; R ] , and state the dimension of U [ x ; R ] . –––––––––––––––––––––––––––––––––––– –––––––––––––––––––––––––––––––––––– Problem #3 Determine only the natural frequencies of vibration for a string having non- constant linear mass density ρ ( x ) = ρ μ 1 + x L ¶ − 1 / 2 , if the magnitude of the tension in the string is given by T ( x ) = T μ 1 + x L ¶ 1 / 2 for constants T and ρ , and for ≤ x ≤ L . Assume also that the string is f xed at the ends, x = 0 and x = L . Your f nal answers should be in terms of ρ , T , and L . Hint : You need only consider the case that leads to the following ODE and its general solution and 2( x + L ) ϕ 00 ( x ) + ϕ ( x ) + λ 2 ϕ ( x ) = 0 and ϕ ( x ) = C 1 cos( λ q 2( x + L )) + C 2 sin( λ q 2( x + L )) . Also recall the trigonometric identity sin( α − β ) = sin( α ) cos( β ) − cos( α ) sin( β ) . –––––––––––––––––––––––––––––––––––– 2 –––––––––––––––––––––––––––––––––––– Problem #4 Let V = M 2 × 2 ( R ) be the vector space of all 2 × 2 matrices under the usual operations of matrix addition " a 11 a 12 a 21 a 22 # + " b 11 b 12 b 21 b 22 # = " a 11 + b 11 a 12 + b 12 a 21 + b 21 a 22 + b 22 # and scalar multiplication α " a 11 a 12 a 21 a 22 # = " α a 11 α a 12 α a 21 α a 22 # and let A = " 1 2 2 4 # and B = " 4 2 2 1 # be two f xed elements in...
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This note was uploaded on 02/25/2012 for the course ENM 510 taught by Professor Car during the Fall '10 term at UPenn.

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pr_midterm - ENM510 - Foundations of Engineering...

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