Q. 24 - Calculus

Question

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

$e^x$

Step-by-Step Solution

Answer

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

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,

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