Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) The third remainder $R_3\left(x\right)$ for sin x when $P_3\left(x\right)$ is the third Taylor polynomial for sin x at $\frac{\mathrm{\pi }}{4}$.

(b) The integral given by Taylor’s Theorem is equal to the third remainder $R_3\left(x\right)$ for sin x when $P_3\left(x\right)$ is the third Taylor polynomial for sin x at $\frac{\mathrm{\pi }}{4}$ .

(c) Lagrange’s form for the third remainder $R_3\left(x\right)$ for sin x when $P_3\left(x\right)$ is the third Taylor polynomial for sin x at $\frac{\mathrm{\pi }}{4}$

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