Question

How do I find an exponential model of the form $y=a{{e}^{kt}}$ on a $\text{TI-84}$?

Answer 1

In this question we have been told to model the expression $y=a{{e}^{kt}}$ on the $\text{TI-84}$ calculator. We will first understand how to do this calculation using basic exponential rules and then look at how to model the same using the calculator. We will look for an example for both the cases.

Complete step by step solution:
We have the given expression as $y=a{{e}^{kt}}$, which is an exponential function. Now to solve this expression we need to know the value of the variables $a$ , $k$ and $t$. Since we have not been given any values for the expression, we will assume the values. Let the value of the variables $a$ , $k$ and $t$ be $1$ , $2$ and $3$ respectively. therefore, on substituting the values in the expression, we get:
$\Rightarrow y=1\times {{e}^{2\times 3}}$
On simplifying the exponent, we get:
$\Rightarrow y=1\times {{e}^{6}}$
On simplifying, we get:
$y={{e}^{6}}$
Now on the $\text{TI-84}$ this modelling is not possible since there are two variables in the exponent which are $k$ and $t$.
Therefore, we will use the $\hat{\ }$ operator in the calculator to find the value. The steps to be followed for the same question on the $\text{TI-84}$ are:
$1)$ enter the value of $a$
$2)$ hit the multiply button which is $\text{x}$.
$3)$ press the $\left[ \text{2nd} \right]$ button and press $\div$ to get $e$
$4)$ press the raised to button which is $\hat{\ }$
$5)$ open the parenthesis by pressing $($
$6)$ enter the value of $k$
$7)$ hit the multiply button which is $\text{x}$.
$8)$ enter the value of $t$
$9)$ close the parenthesis by pressing $)$
$10)$ press the $\left[ \text{enter} \right]$ button to get the required solution.

Note: It is to be remembered that exponential functions are used in daily life to find growth and decay. The function $y=a{{e}^{kt}}$ is used to find the exponential growth. In this equation $a$ represents the initial value, $k$ is the exponential constant and $t$ represents time. The exponential growth function is used to find the growth is bacteria culture.