Seminar 01 Solutions.pdf

# X y 3 2 1 x y z 2 x y z 2 marks for the correct

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x y , 3 2 1 x y z , 2 0 x y z . (2 marks for the correct system of equations) Solving the three equations to get: 1/ 2 x y , 3/ 2 z   . (1 mark for correct solution to the equations) Therefore 1/2 3 35 / 4.05 10 P l Gh c m . (1 mark for the numerical vale.)

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2 3. [10 points] Consider the function 2 /2 2 x x xe which arises in the quantum mechanics of the simple harmonic oscillator. (a) Compute the first and second derivatives of the function. [4 points] (b) Find all the coordinates , x where the slope is zero. For each, indicate if it is a maximum or minimum and justify your answer. [2 points] (c) Find the coordinates , x of any inflection points. [1 point] (d) Use (a) through (c) and the fact that 0 x only at 0 x , and 0 x as x   , to sketch the curve on ,   . Label each axis with a suitable linear scale. [3 points] Solution: (a) Using the chain rule and product rule: 2 2 2 /2 /2 2 /2 2 2 2 1 x x x d x e xe x x e dx (2 marks) 2 2 2 2 /2 2 /2 3 /2 2 2 2 2 1 2 6 x x x d x x e x e x x x e dx 
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• Fall '15
• Mass, Kilogram, Calculus Review,  1011 N  m

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