Q no. 2 - Calculus

Question

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) A function that is decreasing on \((-\infty, 0)\), increasing on \((0, \infty)\), and undefined at \(x=0\).

(b) A function that is decreasing on \((-\infty, 0]\) and increasing on \([0, \infty)\).

(c) A function that is always positive and always decreasing, on all of \(\mathbb{R}\).

Step-by-Step Solution

Answer

(a). The example of the function that is decreasinq on $(-\infty, 0)$, increasinq on $(0, \infty)$, and undefined at $x=0$ is, $f(x)=\frac{x^{6}-1}{x^{2}}$

(b). The example of the function that is decreasing on $(-\infty, 0)$ increasing on $(0, \infty)$ is $f(x)=x^{2}$

(c). The example of the function that is always positive and always decreasing, on all of $\mathbb{R}$ is $f(x)=e^{-x}$

Part (a) Step 1: Given information

 A function that is decreasing on $(-\infty, 0)$, increasing on $(0, \infty)$, and undefined at $x=0$.

Part (a) Step 2: Simplification

Consider the function, $f(x)=\frac{x^{6}-1}{x^{2}}$

The above function is undefined at $x=0$.

Similarly, verify that the function is decreasing on $(-\infty, 0)$, increasing on $(0, \infty)$ by taking different values in those intervals.

Therefore, the example of the function that is decreasinq on $(-\infty, 0)$, increasing on $(0, \infty)$, and undefined at $x=0$ is, $f(x)=\frac{x^{6}-1}{x^{2}}$

Part (b) Step 1: Given information

A function that is decreasing on $(-\infty, 0]$ and increasing on $[0, \infty)$.

Part (b) Step 2: Simplification

Consider the function, $f(x)=x^{2}$

Verify that the above function is decreasing on $(-\infty, 0)$ and increasing on $(0, \infty)$ by taking different values in those intervals.

Therefore, the example of the function that is decreasing on $(-\infty, 0)$ increasing on $(0, \infty)$ is $f(x)=x^{2}$

Part (c) Step 1: Given information

A function that is always positive and always decreasing, on all of $\mathbb{R}$.

Part (c) Step 2: Simplification

Consider the function, $f(x)=e^{-x}$.

Verify that the above function is positive and always decreasing froe any values of $x \in \mathbb{R}$. Therefore, the example of the function that is always positive and always decreasing, on all of $\mathbb{R}$ is $f(x)=e^{-x}$.