Chapter 1.3, Problem 36E

To determine

Tofind:The missing values in the table given below for $y=3e^x$ .

$x$	$y$	Ratio $\left(\frac{y_i}{y_{i-1}}\right)$
$1$	$?$	$?$$?$$?$
$2$	$?$
$3$	$?$
$4$	$?$

Answer to Problem 36E

The complete table for $y=3e^x$ is shown below:

$x$	$y$	Ratio $\left(\frac{y_i}{y_{i-1}}\right)$
$1$	$8.155$	$2.718$$2.718$$2.718$
$2$	$22.167$
$3$	$60.257$
$4$	$163.794$

Explanation of Solution

Given information:

The equation is $y=3e^x$ and the table is:

$x$	$y$	Ratio $\left(\frac{y_i}{y_{i-1}}\right)$
$1$	$?$	$?$$?$$?$
$2$	$?$
$3$	$?$
$4$	$?$

Calculation:

Substitute 1 for $x$ in the given equation.

  $\begin{array}{c}y=3e^x\\ y=3e^1\\ y=3\times 2.718\\ y\approx 8.155\end{array}$

Substitute 2 for $x$ in the given equation.

  $\begin{array}{c}y=3e^x\\ y=3e^2\\ y=3\times {\left(2.718\right)}^2\\ y\approx 22.167\end{array}$

Substitute 3 for $x$ in the given equation.

  $\begin{array}{c}y=3e^x\\ y=3e^3\\ y=3\times {\left(2.718\right)}^3\\ y\approx 60.257\end{array}$

Substitute $4$ for $x$ in the given equation.

  $\begin{array}{c}y=3e^x\\ y=3e^4\\ y=3\times {\left(2.718\right)}^4\\ y\approx 163.794\end{array}$

Now, calculate the ratio $\left(\frac{y_i}{y_{i-1}}\right)$ as:

  $\begin{array}{c}\frac{y_i}{y_{i-1}}=\frac{y_2}{y_1}\\ =\frac{22.167}{8.155}\\ \approx 2.718\end{array}$

And

  $\begin{array}{c}\frac{y_i}{y_{i-1}}=\frac{y_3}{y_2}\\ =\frac{60.257}{22.167}\\ \approx 2.718\end{array}$

And

  $\begin{array}{c}\frac{y_i}{y_{i-1}}=\frac{y_4}{y_3}\\ =\frac{163.794}{60.257}\\ \approx 2.718\end{array}$

So, the values of ratio are $2.718$ , $2.718$ and $2.718$ .

Therefore, the complete table is given below:

$x$	$y$	Ratio $\left(\frac{y_i}{y_{i-1}}\right)$
$1$	$8.155$	$2.718$$2.718$$2.718$
$2$	$22.167$
$3$	$60.257$
$4$	$163.794$