Mathematics for Management -- Supplementary Electronic Materials

Quiz: Continuity

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1. Find a value of $k \in \mathbb{R}$ such that the following function $f(x)$ is continuous for all $x \in \mathbb{R}$ $$ f(x) \, \, = \, \left\{ \begin{array}{l c l} kx \, , & & \text{if $x \leq 3$} \\ 5 \, , & & \text{if $x > 3$} \end{array} \right. $$

$ k = \frac{3}{5} $ $ k = \frac{4}{5} $ $ k = \frac{5}{3} $ $ k = \frac{5}{4} $

2. Which of the following functions $f(x)$ is continuous for all $x \in \mathbb{R}$?

$ f(x) = \sin(x) $ $ f(x) = \log_5(x) $ $ f(x) = \tan(x) $ $ f(x) = x^{-1/2} $

3. Find a value of $k \in \mathbb{R}$ such that the following function $f(x)$ is continuous for all $x \in \mathbb{R}$ $$ f(x) \, \, = \, \left\{ \begin{array}{l c l} x \, , & & \text{if $x < 2$} \\ k + \frac{1}{2} x \, , & & \text{if $x \geq 2$} \end{array} \right.$$

$ k = \frac{1}{2} $ $ k = 1 $ $ k = \frac{3}{2} $ $ k = 2 $

4. Find all values of $k \in \mathbb{R}$ such that the following function $f(x)$ is continuous at $x = 3$ $$ f(x) \, \, = \, \left\{ \begin{array}{l c l} 2x^2 + 3 \, , & & \text{if $x \leq 3$} \\ 3x + k \, , & & \text{if $x > 3$} \end{array} \right. $$

$ k = 6 $ $ k = 9 $ $ k = 12 $ $ k = 15 $

5. Find all values of $a \in \mathbb{R}$ such that the following function $f(x)$ is continuous everywhere $$ f(x) \, \, = \, \left\{ \begin{array}{l c l} x^2 + 2x \, , & & \text{if $x < a$} \\ -1 \, , & & \text{if $x \geq a$} \end{array} \right. $$

$ a=-1 $ only $ a=-2 $ only $ a=-1 $ and $ a=1 $ $ a=-2 $ and $ a=2 $

6. The following function $f(x)$ is not continuous at $x = 2$ because $$ f(x) \, \, = \, \left\{ \begin{array}{l c l} x-3 \, , & & \text{if $x > 2$} \\ -5 \, , & & \text{if $x = 2$} \\ 3x-7 \, , & & \text{if $x < 2$} \end{array} \right.$$

$ f(2) $ is not defined $ \lim_{x \to 2} f(x) $ does not exist $ \lim_{x \to 2} f(x) = f(2) $ $ f(2) \neq -5 $

7. The following function $f(x)$ is not continuous at $x = 2$ because $$ f(x) \, \, = \, \left\{ \begin{array}{l c l} x^2 \, , & & \text{if $x > 2$} \\ 4-2x \, , & & \text{if $x < 2$} \end{array} \right. $$

$ f(2) $ is not defined $ \lim_{x \to 2} f(x) $ does not exist $ \lim_{x \to 2} f(x) \neq f(2) $ all of these

8. Which of the following is true for the function $f(x)$ given by $$ f(x) \, \, = \, \left\{ \begin{array}{l c l} 2x-1 \, , & & \text{if $x < -1$} \\ x^2 + 1 \, , & & \text{if $-1 \leq x \leq 1$} \\ x+1 \, , & & \text{if $x > 1$} \end{array} \right. $$

$ f $ is continuous everywhere. $ f $ is continuous except at $x=-1$ and $x=1$. $ f $ is continuous except at $x=-1$. $ f $ is continuous except at $x=1$.

9. Suppose $f(x)$ is a continuous function on $[-1,3]$, and $f(-1) = 4$ as welll as $f(3) = 7$. By the Intermediate Value Property, we can conclude that

that there exist a number $c \in (-1,3)$ such that $f(c) = 0$. that there exist a number $c \in (-1,3)$ such that $f(c) = 5$. that there exist a number $c \in (4,7)$ such that $f(c) = 0$. that there exist a number $c \in (4,7)$ such that $f(c) = 5$.

10. Let the function $f(x)$ be continuous on the closed interval $[0,2]$ and have the values that are given in the following table $$ \begin{array}{c || c c c} x & 0 & 1 & 2 \\ \hline f(x) & 1 & k & 2 \end{array} $$ The equation $f(x) = \frac{1}{2}$ must have at least two solutions in $(0,1)$ if

$ k = 0 $ $ k = \frac{1}{2} $ $ k = 1 $ $ k = 2 $

Your grade is: __ of 10

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Solution: 1c; 2a; 3b; 4c; 5a; 6d; 7a; 8c; 9b; 10a

Here, a, b, c, d indicate the 1st, 2nd, 3rd, and 4th answer choice, respectively, for the numbered questions.

Quiz: Continuity

1. Find a value of \(k \in \mathbb{R}\) such that the following function \(f(x)\) is continuous for all \(x \in \mathbb{R}\) $$ f(x) \, \, = \, \left\{ \begin{array}{l c l} kx \, , & & \text{if $x \leq 3$} \\ 5 \, , & & \text{if $x > 3$} \end{array} \right. $$

2. Which of the following functions \(f(x)\) is continuous for all \(x \in \mathbb{R}\)?

3. Find a value of \(k \in \mathbb{R}\) such that the following function \(f(x)\) is continuous for all \(x \in \mathbb{R}\) $$ f(x) \, \, = \, \left\{ \begin{array}{l c l} x \, , & & \text{if $x < 2$} \\ k + \frac{1}{2} x \, , & & \text{if $x \geq 2$} \end{array} \right.$$

4. Find all values of \(k \in \mathbb{R}\) such that the following function \(f(x)\) is continuous at \(x = 3\) $$ f(x) \, \, = \, \left\{ \begin{array}{l c l} 2x^2 + 3 \, , & & \text{if $x \leq 3$} \\ 3x + k \, , & & \text{if $x > 3$} \end{array} \right. $$

5. Find all values of \(a \in \mathbb{R}\) such that the following function \(f(x)\) is continuous everywhere $$ f(x) \, \, = \, \left\{ \begin{array}{l c l} x^2 + 2x \, , & & \text{if $x < a$} \\ -1 \, , & & \text{if $x \geq a$} \end{array} \right. $$

6. The following function \(f(x)\) is not continuous at \(x = 2\) because $$ f(x) \, \, = \, \left\{ \begin{array}{l c l} x-3 \, , & & \text{if $x > 2$} \\ -5 \, , & & \text{if $x = 2$} \\ 3x-7 \, , & & \text{if $x < 2$} \end{array} \right.$$

7. The following function \(f(x)\) is not continuous at \(x = 2\) because $$ f(x) \, \, = \, \left\{ \begin{array}{l c l} x^2 \, , & & \text{if $x > 2$} \\ 4-2x \, , & & \text{if $x < 2$} \end{array} \right. $$

8. Which of the following is true for the function \(f(x)\) given by $$ f(x) \, \, = \, \left\{ \begin{array}{l c l} 2x-1 \, , & & \text{if $x < -1$} \\ x^2 + 1 \, , & & \text{if $-1 \leq x \leq 1$} \\ x+1 \, , & & \text{if $x > 1$} \end{array} \right. $$

9. Suppose \(f(x)\) is a continuous function on \([-1,3]\), and \(f(-1) = 4\) as welll as \(f(3) = 7\). By the Intermediate Value Property, we can conclude that