Q. 23 - Calculus

Question

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

$\cos x$

Step-by-Step Solution

Answer

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

Step 1. Given Information

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

Step 2. Explanation

Using the Lagrange form of remainder, we have,

$$\begin{gathered} R_n\left(x\right)=\frac{f^{n+1}\left(c\right)}{(n+1)!}{x}^{n+1} \\ {R}_4\left(x\right)=\frac{f^5\left(c\right)}{5!}{x}^5 \end{gathered}$$

Now, we will find fifth derivative of the function.

$$\begin{gathered} f^1\left(x\right)=-\sin x \\ {f}^2\left(x\right)=-\cos x \\ {f}^3\left(x\right)=\sin x \\ {f}^4\left(x\right)=\cos x \\ {f}^5\left(x\right)=-\sin x \end{gathered}$$

Thus, we get,

$R_4\left(x\right)=-\frac{\sin c}{120}{x}^5$