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Lambda Calculus: Alpha Renaming, Beta Reduction, Eta Conversion

1. a-convert the outer-most x to y in the following calculus expressions, if possible: (a) y.(Î»y.yy) (b) Ay.(yy.yx)

-reduce the following calculus expressions, if possible: (Xx.Xy.(xy)(yw)) (xx.(xx) xx.(xx))

(c) (d)

n-reduce the following calculus expressions, if possible: (e) Xx.(xy.xx) (f) Xx.(y.yx)

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Find the partial derivatives. The variables are restricted to a domain on which the function is defined.$$f_{x} \text { and } f_{y} \text { if } f(x, y)=A^{a} x^{a+\beta} y^{1-a-\beta}$$

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Express the following derivatives in ” $\partial^{\prime \prime}$ notation.(a) $f_{x x x}$(b) $f_{x y y}$(c) $f_{y y x x}$(d) $f_{x y y y}$

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Use partial differentiation to determine expressions for $\frac{\mathrm{d} y}{\mathrm{~d} x}$ in the following cases:(a) $x^{3}+y^{3}-2 x^{2} y=0$(b) $e^{x} \cos y=e^{y} \sin x$(c) $\sin ^{2} x-5 \sin x \cos y+\tan y=0$

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Partial derivative

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Evaluate all first and second partial derivatives of the following functions:(a) $f(x, y)=x \arctan (x / y)$(b) $f(x, y)=\cos \sqrt{x^{2}+y^{2}}$(c) $f(x, y)=\operatorname{cxp}\left(-x^{2}-y^{2}\right)$

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