Chapter 3 – Quiz 1 – E600 Mathematics

1. Consider a function $f:X\mapsto Y$ and suppose that $f$ is invertible. Further, consider a set $B\subseteq Y$ and a value $b\in Y$ . Which of the following relationships are certainly true? (Multiple choice)

$f^{-1}[B]\subseteq Y$ .

$f^{-1}(b)\in Y$ .

$f^{-1}(b)\in X$ .

$f^{-1}[B]\subseteq X$ .

$f^{-1}[B]\in Y$ .

$f^{-1}[B]\in X$ .

2. Consider the function $f:\mathbb R\mapsto\mathbb R, x\mapsto\max\{x,-2\} + 1$ . This function is: (Hint: draw the function and consider the lower and upper level sets.)

quasi-concave but not quasi-convex.

neither quasi-concave nor quasi-convex.

quasi-convex but not quasi-concave.

quasi-linear, i.e. both quasi-concave and quasi-convex.

3. What is not true about Taylor’s theorem and its statement about a function $f$ around a point $x_0$ ?

The Taylor Expansion is equal to the function $f$ at any point in its domain.

The Taylor Approximation is a polynomial function aiming to give a good approximation of $f$ around $x_0$ .

A Taylor Approximation of finite degree around $x_0$ always gives a good approximation to $f$ at any point in its domain.

Approximations of higher order yield smaller errors in proximity to the point of approximation.

4. What can you say about the concepts of (quasi-)convexity and concavity? (Multiple choice)

If a function is both quasi-convex and quasi-concave, it is quasi-linear.

If a function is both quasi-convex and quasi-concave, it is linear.

Any convex function is quasi-convex.

Any quasi-concave function is concave.

A function with non-convex domain can satisfy the definition of a concave function.

If a function is both convex and concave, it is linear.

5. What does the univariate derivative of a function $f$ , evaluated at a point $x_0$ in the domain of $f$ , express?

The absolute change of $f$ around $x_0$ for fixed, non-zero changes in the argument.

The absolute change of $f$ around $x_0$ for infinitely small changes in the argument.

The change of $f$ around $x_0$ relative to variation in the argument, for infinitely small changes in the argument.

The change of $f$ around $x_0$ relative to variation in the argument, for fixed, non-zero changes in the argument.

6. Consider the function $f:\mathbb R\mapsto\mathbb R, x\mapsto 2x + 4$ . This function is:

surjective but not injective.

neither injective nor surjective.

bijective, i.e. injective and surjective.

injective but not surjective.

7. Consider a univariate, real-valued function $f$ with domain $X = \mathbb R$ and suppose that it is differentiable with derivative $f'$ , and consider a point $x\in\mathbb R$ . Which statement about derivative concepts is false? (Single choice)

$f'$ is a function.

$f'(x)$ is a function.

$f$ is an element of $D^1(\mathbb R, \mathbb R)$ .

$f'$ is equal to the derivative operator evaluated at $f$ .

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