Logarithm 12

Calculus Level 4

Given the functions

$$f(x,y) = \frac{x}{2} e^{ - \big(\frac{x^2 + y^2}{2} \big)}$$

$$g(z) = \log_2 (\frac{z}{5})$$

and the matrices

A = $$\begin{bmatrix} \\ {{\partial f(x,y) \over \partial x} \bigg|_{(0,0)}} & {{\partial^2 f(x,y) \over \partial x \partial y} \bigg|_{(1,1)}} \\ \\ {{\partial^2 f(x,y) \over \partial y \partial x} \bigg|_{(0,0)}} & {{\partial f(x,y) \over \partial y} \bigg|_{(1,1)}} \\ \\ \end{bmatrix}$$

B = $$\begin{bmatrix} g'(1) & g'(2) \\ g''(1) & g''(2) \\ \end{bmatrix}$$

find $$k = \ln (2) . \frac{det B}{det A}$$

Notations:

• $${{\partial f(x,y) \over \partial x} \bigg|_{(a,b)}}$$ denotes the first-order partial derivative of $$f(x,y)$$

with respect to $$x$$ at the point $$(a,b,f(a,b))$$

• $${{\partial f(x,y) \over \partial y} \bigg|_{(a,b)}}$$ denotes the first-order partial derivative of $$f(x,y)$$

with respect to $$y$$ at the point $$(a,b,f(a,b))$$

• $${{\partial^2 f(x,y) \over \partial x \partial y} \bigg|_{(a,b)}}$$ denotes the second-order partial derivative of $$f(x,y)$$

with respect to $$y$$ and $$x$$ at the point $$(a,b,f(a,b))$$

• $${{\partial^2 f(x,y) \over \partial y \partial x} \bigg|_{(0,0)}}$$ denotes the second-order partial derivative of $$f(x,y)$$

with respect to $$x$$ and $$y$$ at the point $$(a,b,f(a,b))$$

• $$g'(a)$$ denotes the first derivative of $$g(z)$$ at the point $$(a, g(a))$$

• $$g''(a)$$ denotes the second derivative of $$g(z)$$ at the point $$(a, g(a))$$

• $$\log_2 (\cdot)$$ denotes the binary logarithm of $$(\cdot)$$

• $$\ln (\cdot)$$ denotes the natural logarithm of $$(\cdot)$$

• $$det A$$ denotes the determinant of matrix A

• $$det B$$ denotes the determinant of matrix B

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