Chapter 3 – Quiz 2

1. How do integration and differentiation relate to each other?

 
 
 
 

2. Consider a function f:\mathbb R^2\mapsto\mathbb R and suppose that f is differentiable. Then, its derivative is:

 
 
 
 
 
 

3. How do the Jacobian and Hessian concepts relate to each other with respect to a twice differentiable function f? (Single choice)

 
 
 
 

4. Consider f:\mathbb R^2_+\mapsto\mathbb R, (x_1,x_2)'\mapsto \ln(x_1x_2) + x_1^3 + \sin(x_2). Compute the Taylor error \varepsilon(x) = T(x) - x for the first order approximation around x_0 = (1, \pi) at  x = (0.5, 2\pi). To be able to enter a natural number into the solution field, perform the following operations in the given order to your result: 1. Add  \mathbf{2\pi}, 2. Multiply with 8. (Do not use spaces, letters, etc. Hint: the correct number to enter is an element of [35,55].)

5. Consider f:\mathbb R^3\mapsto\mathbb R, (x_1, x_2, x_3)'\mapsto x_1^2x_2 + x_1x_3. The second order partial derivative \frac{\partial f}{\partial x_1\partial x_2} maps (x_1, x_2, x_3)'\in\mathbb R^3 onto:

 
 
 
 
 

6. Which relationships are always true for any function f:X\mapsto Y and a point x_0\in X? (Multiple choice)

 
 
 
 
 
 

7. Consider a function f:\mathbb R^2\mapsto\mathbb R and suppose that f is differentiable. Then, its partial derivative with respect to the first dimension evaluated at x_0 = \begin{pmatrix}2\\0\end{pmatrix}is:

 
 
 
 
 
 

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