Question

If \[y = {x^2}cosx\] then \[{y_8}\left( 0 \right)\]is

Answer 1

Hint: Here, we have to split the given equation in to two and assume each part with a variable, like assume \[v = {x^2}\] and \[u = cosx\]. Derivate the equation up to maximum of 8 and substitute the 0 in the derivated equations. After derivating the equation in to maximum of 8, then find the values of the u and v by substituting 0 in the variable x, then we will get the answers of u and v and after substituting those values in the final equation we will get the final answer.

Complete step-by-step answer:
Given:
The variable y is equal to \[y = {x^2}cosx\].
Let us assume that \[u = cosx\] and \[v = {x^2}\].
We know that the equation to find the value of \[{y_8}\left( 0 \right)\]is given by,
\[{y_n} = {u_n}v + n{C_1}{u_{n - 1}}{v_1} + .... + n{C_{n - 1}}{u_1}{v_n}\]
Here, n is the maximum value that we need to find, u and v are the assumed values for the $ \cos x $ and \[{x^2}\]respectively.
On substituting the values in the above equation we will obtain,
\[\begin{array}{c}
{y_8} = {u_n}v + 8{C_1}{u_7}{v_1} + .... + n{C_1}{u_1}{v_8}\\
 = 0 + 0 + 8{C_1}{u_7}{v_1} + .... + 0\\
 = 8{C_1}{u_7}{v_1}
\end{array}\]
Now, at $ x = 0 $ all the derivatives of v are equal to zero except $ {v_2} = 2 $ .
The value of $ {u_6} $ is calculated as,
\[\begin{array}{c}
{u_6} = \cos (x + 3\pi )\\
{u_6}(0) = - \cos (0)\\
 = - 1
\end{array}\]
Now, we will calculate the value of \[{y_8}(0)\] which is,
\[\begin{array}{c}
{y_8}(0) = 28 \times - 1 \times 2\\
 = - 56
\end{array}\]
Therefore, the value of \[{y_8}(0)\]is equal to $ - 56 $ and the option (d) is the correct answer.

Note: Make sure while you are deriving the values in the equation, after the derivation substitute the 0 in the variables and solve the equation to get the appropriate answer. Here \[{y_8}(0)\]says that the derivation of the y is limited to the 8 terms and after derivating, 0 should be substituted in the value of the x itself which gives the solution for the equation.