Chapter 3 – Quiz 2 – E600 Mathematics

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

Any integrable function is differentiable.

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

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.

2. 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 $[35,55]$ .)

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

a real matrix.

a real-valued fuction.

a matrix-valued function.

a real number.

a real vector.

a vector-valued function.

4. 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 matrix.

a real vector.

a real number.

a real-valued fuction.

a matrix-valued fuction.

a vector-valued fuction.

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

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

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

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 vector-valued function.

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

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

6. 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 equal to the Jacobian of $f$ .

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

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

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

7. 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:

$2x_1$ .

$x_1^2$ .

$0$ .

$2x_1x_2 + x_3$ .

$x_3$ .

Question 1 of 7