Chapter 3 – Quiz 2 – E600 Mathematics

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

If $f$ is differentiable at $x_0$ , then all partial derivatives of $f$ at $x_0$ exist.

If $Y\subseteq \mathbb R^m$ with $m\geq 2$ , then the derivative of $f$ , if existent, is a real-valued function.

If all partial derivatives of $f$ at $x_0$ exist, then $f$ is differentiable at $x_0$ .

$f$ may be differentiable at $x_0$ even if its gradient does not exist at $x_0$ .

If $Y\subseteq \mathbb R^m$ with $m\geq 2$ , then the derivative of $f$ , if existent, is a vector-valued function.

If $Y\subseteq \mathbb R^m$ with $m\geq 2$ , then the derivative of $f$ , if existent, is a matrix-valued function.

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

The Hessian of $f$ is the Jacobian of the transposed gradient of $f$ .

The Hessian of $f$ is equal to the Jacobian of $f$ .

The Jacobian of $f$ is equal to the derivative of the Hessian of $f$ .

The Jacobian of $f$ exists only if the gradient of the Hessian of $f$ is symmetric.

3. 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:

$2x_1$ .

$x_3$ .

$0$ .

$x_1^2$ .

$2x_1x_2 + x_3$ .

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

a matrix-valued function.

a real-valued fuction.

a real vector.

a real number.

a vector-valued function.

a real matrix.

5. 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 $[35,55]$ .)

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

For any differentiable and integrable function $f$ , the function itself is the instantaneous rate of change, i.e. the differential, of the integral function.

The integral operator is the inverse function of the differential operator.

Any integrable function is differentiable.

For any differentiable and integrable function $f$ , the integral and derivative function are equal.

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:

a real number.

a real matrix.

a matrix-valued fuction.

a real-valued fuction.

a vector-valued fuction.

a real vector.

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