Question

Evaluate ${\Delta ^2}\left( {3{e^x}} \right)$.
A. $3{e^x}$
B. $3\left( {h - 1} \right){e^x}$
C. $3{\left( {{e^h} - 1} \right)^2}{e^x}$
D. None of these

Answer 1

Hint: $\Delta$ is known as a forwarded difference operator. We will apply the general formula of the forwarded difference operator to get the first order forward difference operator. Then again we will apply the general formula of the forward difference operator in the first order to get the second order forward difference operator $3{e^x}$.

Formula Used:
$\Delta f\left(x\right)= f\left(x+h\right)- f\left(x\right)$
$\Delta \left( {a{e^x}} \right) = a\Delta \left( {{e^x}} \right)$
$\Delta \left( {{e^x}} \right) = {e^{x + h}} - {e^h}$

Complete step by step solution:
Given expression is ${\Delta ^2}\left( {3{e^x}} \right)$.
A forwarded difference operator, denoted $\Delta$, defined by the equation $\Delta f\left( x \right) = f\left( {x + h} \right) - f\left( x \right)$.
First, we will calculate the first order forward difference operator of $3{e^x}$.
Here, $f\left( x \right) = 3{e^x}$
Substitute $x = x + h$ in $f\left( x \right) = 3{e^x}$
$f\left( {x + h} \right) = 3{e^{x + h}}$
Now take forwarded difference operator on both sides of $f\left( x \right) = 3{e^x}$
$\Delta f\left( x \right) = \Delta \left( {3{e^x}} \right)$
Apply the formula $\Delta f\left( x \right) = f\left( {x + h} \right) - f\left( x \right)$
$ \Rightarrow f\left( {x + h} \right) - f\left( x \right) = \Delta \left( {3{e^x}} \right)$
Substitute $f\left( x \right) = 3{e^x}$ and $f\left( {x + h} \right) = 3{e^{x + h}}$ in the above equation:
$ \Rightarrow 3{e^{x + h}} - 3{e^x} = \Delta \left( {3{e^x}} \right)$ …….(i)
Again take the forward difference operator on both sides
$ \Rightarrow \Delta \left( {3{e^{x + h}} - 3{e^x}} \right) = {\Delta ^2}\left( {3{e^x}} \right)$
Now the formula of the sum of the forwarded difference operator:
$ \Rightarrow \Delta \left( {3{e^{x + h}}} \right) - \Delta \left( {3{e^x}} \right) = {\Delta ^2}\left( {3{e^x}} \right)$ ……(ii)
Assume that, $g\left( x \right) = 3{e^{x + h}}$
Take the forward difference operator on both sides
$\Delta g\left( x \right) = \Delta \left( {3{e^{x + h}}} \right)$
Now apply the general formula of the forwarded difference operator:
$ \Rightarrow g\left( {x + h} \right) - g\left( x \right) = \Delta \left( {3{e^{x + h}}} \right)$ …..(iii)
Substitute $x = x + h$ into the equation $g\left( x \right) = 3{e^{x + h}}$
$g\left( {x + h} \right) = 3{e^{x + h + h}}$
Simplify the above equation:
$ \Rightarrow g\left( {x + h} \right) = 3{e^{x + 2h}}$
Put the value of $g\left( x \right)$ and $g\left( {x + h} \right)$ in equation (iii)
$ \Rightarrow 3{e^{x + 2h}} - 3{e^{x + h}} = \Delta \left( {3{e^{x + h}}} \right)$
Substitute the value of $\Delta \left( {3{e^{x + h}}} \right)$ in equation (ii)
$3{e^{x + 2h}} - 3{e^{x + h}} - \Delta \left( {3{e^x}} \right) = {\Delta ^2}\left( {3{e^x}} \right)$
From equation (i) we get $3{e^{x + h}} - 3{e^x} = \Delta \left( {3{e^x}} \right)$, plug the value of $\Delta \left( {3{e^x}} \right)$ in above equation:
$ \Rightarrow 3{e^{x + 2h}} - 3{e^{x + h}} - \left( {3{e^{x + h}} - 3{e^x}} \right) = {\Delta ^2}\left( {3{e^x}} \right)$
Simplify the equation:
$ \Rightarrow 3{e^{x + 2h}} - 3{e^{x + h}} - 3{e^{x + h}} + 3{e^x} = {\Delta ^2}\left( {3{e^x}} \right)$
Add the like terms:
$ \Rightarrow 3{e^{x + 2h}} - 6{e^{x + h}} + 3{e^x} = {\Delta ^2}\left( {3{e^x}} \right)$

Option ‘C’ is correct

Note: In the basic formula of forwarded difference, we get a constant $h$. Remember $h$ is a constant indicating the difference between successive points of interpolation or calculation.