Chapter 5.5, Problem 8QR

To determine

To calculate:

The value of $f\left(x\right)$ and $g\left(x\right)$ to complete the given table.

Answer to Problem 8QR

The value of $f\left(x\right)$ and $g\left(x\right)$ for the values is obtained in the table as,

$x$	$f\left(x\right)$	$g\left(x\right)$
0.7	-1.457	-1.7
0.8	-1.688	-1.8
0.9	-1.871	-1.9
1.0	-2	-2
1.1	-2.069	-2.1
1.2	-2.072	-2.2
1.3	-2.003	-2.3

Explanation of Solution

Given information:

The function is $g\left(x\right)$ whose graph is the tangent line.

  $f\left(x\right)=x^3-4x+1$ At $x=1$

Calculation:

Since the tangent is passing through $x=1$ and touching $f\left(x\right)$ the slope of the tangent and the curve at point must be same.

The derivation form of $f\left(x\right)$ is $f\text{'}\left(x\right)=3x^2-4$

  $m=f\text{'}\left(1\right)=3-4=-1$

The slope of the tangent at point $\left(1,-2\right)=-1$

Equation of the slope,

  $\begin{array}{l}g\left(x\right)=f\left(1\right)+m\left(x-1\right)\\ g\left(x\right)=-2-1\left(x-1\right)=-x-1\end{array}$

When, $x=0.7$ then

  $\begin{array}{l}f\left(x\right)={\left(0.7\right)}^3-\left(4\times 0.7\right)+1\\ f\left(x\right)=0.343-2.8+1\\ f\left(x\right)=-1.457\end{array}$

And,

  $\begin{array}{l}g\left(x\right)=-x-1\\ g\left(x\right)=-0.7-1\\ g\left(x\right)=-1.7\end{array}$

The table is completed by similar calculation

$x$	$f\left(x\right)$	$g\left(x\right)$
0.7	-1.457	-1.7
0.8	-1.688	-1.8
0.9	-1.871	-1.9
1.0	-2	-2
1.1	-2.069	-2.1
1.2	-2.072	-2.2
1.3	-2.003	-2.3