Q. 23 - Calculus

Question

Use algebra to find the largest possible value of δ or smallest possible value of N that makes each implication is true. Then verify and support your answers with labeled graphs.

If $0<\left|x-2\right|<\delta$, then $\left|\left(3x-1\right)-5\right|<0.25$.

Step-by-Step Solution

Answer

The required value of N is $\delta =\frac{0.25}{3}$.

The required graph is given below,

Step 1. Given information.

Consider the given question,

If $0<\left|x-2\right|<\delta$, then $\left|\left(3x-1\right)-5\right|<0.25$.

Step 2. Find the value of &#948; .

It can be written,

$$\begin{gathered} \left|3x-1-5\right|<0.25 \\ \left|3x-6\right|<0.25 \\ 3\left|x-2\right|<0.25 \\ \left|x-2\right|<\frac{0.25}{3} \end{gathered}$$

As $0<\left|x-2\right|<\delta$, then,

$\delta =\frac{0.25}{3}$

Step 3. Sketching the function to illustrate the choice of &#948; .

Draw the graph of the given function and indicate on the graph that every value of x in $(1.917,2)\cup (2,2.083)$has an data-custom-editor="chemistry" $\mathrm{f}\left(\mathrm{x}\right)$ value in $(4.75,5.25)$.