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,

{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.