Chapter 2.2, Problem 89E

a.

To determine

Find unit length intervals when polynomial function is zero.

Answer to Problem 89E

  $\left[-1,0\right],\left[1,2\right]\text{}\text{}and\left[2,3\right]$

Explanation of Solution

Given information:

  $f\left(x\right)=x^3+3x^2+3$

Use the Intermediate Value Theorem, and the table feature of a graphing utility.

Calculation:

Consider the function,

  $f\left(x\right)=x^3+3x^2+3$

Apply the Intermediate Value Theorem,

Let $aandb$ are the real numbers such that $a<b$ . If function is polynomial such that $f\left(a\right)\ne f\left(b\right)$ ,then the interval $\left[a,b\right]$ , $f$ takes on every value between $f\left(a\right)andf\left(b\right)$

Now compute some function value,

  $\begin{array}{cccccc}x& -1& 0& 1& 2& 3\\ f\left(x\right)& -1& 3& 1& -1& 3\end{array}$

Because $f\left(-1\right)isnegativef\left(0\right)ispositive$ ,according to intermediate theorem we conclude that function has a zero between $-1and0$ so we conclude that function also have zero between $1and2,3and4$

Hence, unit length intervals are $\left[-1,0\right],\left[1,2\right]\text{}\text{}and\left[2,3\right]$

b.

To determine

Approximate the zeros of function to nearest thousand.

Answer to Problem 89E

  $-0.879,10347and2.532$

Explanation of Solution

Given information:

  $f\left(x\right)=x^3+3x^2+3$

Calculation:

Consider the function,

  $f\left(x\right)=x^3+3x^2+3$

The graph of the polynomial to find zero location,

  

Hence, zero’s lies on the graph $-0.879,10347and2.532$ .