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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