Q. 17 - Calculus

Question

Determine whether the implication that follows is true or false. Use graphs to justify any implications that are true, and counter examples for any implications that are false.

If $0<\left|x+2\right|<0.075$, then $\left|x^2-4\right|<0.4$.

Step-by-Step Solution

Answer

The given implication is true.

To justify the implication is true, we use the graph,

Step 1. Given information.

Consider the given question,

$0<\left|x+2\right|<0.075$

Also, $\left|x^2-4\right|<0.4$

Step 2. Determine if the implication is true or false.

The statement is true.

Draw the graph of $y=x^2$ and show an interval around $y=4$ with $3.6<y<4.4$and draw another interval around $x=-2$ with $-2.075<x<-1.925$, excluding $x=-2$. Then indicate on the graph that every value of x in $\left(-2.075,-2\right)\cup \left(-2,-1.925\right)$ has an $f\left(x\right)$ value in $\left(3.6,4.4\right)$.