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FM5002-HW1-1.25.12

# FM5002-HW1-1.25.12 - Fin an cial Math em atics 5002 Hom...

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Financial Mathem atics 5002 : Hom ework 1 (0037) Due on 25 January 2012 Scot Adams Solutions 0037 M 1 .Let f L x, y R E e 2 x M y sin L x P y R . a.Compute p p x l f L x, y Rr . b.Compute ± p p x l f L x, y Rr² x S 4, y S 4 . c.Compute p p y l f L x, y . d.Compute ± p p y l f L x, y Rr² x S 4, y S 4 . e. Compute the gradient of f (x, y) with respect to (x, y). f. Evaluate the gradient of f (x, y) with respect to (x, y) at the point (x, y) = ( Π / 4, Π / 4). g. Find the 1 - jet of f (x, y) with respect to (x, y) at the point (x, y) = ( Π / 4, Π / 4). h. Find the 1 st order Maclaurin approximation of f (x + Π / 4, y + Π / 4). i.Compute p 2 p x 2 l f L x, y . j.Compute ± p 2 p x 2 l f L x, y Rr² x S 4, y S 4 . k.Compute p 2 p x p y l f L x, y . l.Compute ± p 2 p x p y l f L x, y Rr² x S 4, y S 4 . m.Compute p 2 p y 2 l f L x, y . n.Compute ± p 2 p y 2 l f L x, y Rr² x S 4, y S 4 . o. Compute the 2 x2 Hessian of f (x, y) with respect to (x, y).

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p. Evaluate the 2 x2 Hessian of f (x, y) with respect to (x, y) at the point (x, y) = ( Π / 4, Π / 4). q. Find the 2 - jet of f (x, y) with respect to (x, y) at the point (x, y) = ( Π / 4, Π / 4). (Use lexicographic ordering.) r. Find the 2 nd order Maclaurin approximation of f (x + Π / 4, y + Π / 4). a. P P x L f l x, y Rr E e 2 x M y Cos L x p y r p 2 e 2 x M y Sin L x p y r . b. ± P P x L f l x, y Rr² x S 4, y S 4 E 2 e Π S 4 . c. P P y L f l x, y E e 2 x M y Cos L x p y r M e 2 x M y L x p y r . d. ± P P y L f l x, y Rr² x S 4, y S 4 E Me Π S 4 . e. D f l x, y R E ³ e 2 x M y Cos L x p y r p 2 e 2 x M y L x p y r , e 2 x M y Cos L x p y r M e 2 x M y L x p y . f. D f l Π S 4, Π S 4 R E ³ 2 e Π S 4 , Me Π S 4 ´ .
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FM5002-HW1-1.25.12 - Fin an cial Math em atics 5002 Hom...

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