Question

In Exercises 23–32 we ask you to give Lagrange’s form for the corresponding remainder, $R_4\left(x\right)$

$e^x$

Short answer

The required answer is$R_4\left(x\right)=\frac{e^c}{120}{x}^5$

Step-by-step solution

Step 1. Given Information 

The given function is $f\left(x\right)=e^x$

Step 2. Explanation

The lagrange form for the remainder $R_4\left(x\right)\mathrm{is}{R}_4\left(x\right)=\frac{f^5\left(c\right)}{5!}{x}^5$

Now, we will evaluate the fifth derivative of the function $f\left(x\right)=e^x\mathrm{which}\mathrm{is}{e}^x$

Thus, we get,

$$\begin{gathered} R_4\left(x\right)=\frac{e^c}{5!}{x}^5 \\ {R}_4\left(x\right)=\frac{e^c}{120}{x}^5 \end{gathered}$$