91Ó°ÊÓ

The function \(y(x)=\mathrm{e}^{x}\) may be approximated by the quadratic expression \(1+x+\frac{x^{2}}{2} .\) Find an upper bound for the error term given \(|x|<0.5\)

Calculate the first-order Taylor polynomial generated by \(y(x)=\cos x\) about (a) \(x=0\) (b) \(x=1\) (c) \(x=-0.5\)

Given \(y(x)=\sin x\), obtain the third-, fourth- and fifth-order Taylor polynomials generated by \(y(x)\) about \(x=0 .\)

(a) Find a linear approximation, \(p_{1}(t)\), to \(h(t)=t^{3}\) about \(t=2\). (b) Evaluate \(h(2.3)\) and \(p_{1}(2.3)\).

Given \(y(x)=\cos x\), obtain the third-, fourth- and fifth-order Taylor polynomials generated by \(y(x)\) about \(x=0\).

A function, \(y(x)\), satisfies the equation $$ y^{\prime}=y^{2}+x \quad y(1)=2 $$ (a) Estimate \(y(1.3)\) using a first-order Taylor. polynomial. (b) By differentiating the equation with respect to obtain an expression for \(y^{\prime \prime}\). Hence evaluate \(y^{\prime \prime}(1)\). (c) Estimate \(y(1.3)\) using a second-order Taylor polynomial.

\text { Find the Taylor series for } y(x)=\sqrt{x} \text { about } x=1 \text {. }

A function, \(x(t)\), satisfies the equation. $$ \dot{x}=x+\sqrt{t+1} \quad x(0)=2 $$ (a) Estimate \(x(0.2)\) using a first-order Taylor polynomial. (b) Differentiate the equation w.r.t. \(t\) and hence obtain an expression for \(x\). (c) Estimate \(x(0.2)\) using a second-order Taylor polynomial.

(a) Given \(y(x)=\sin (k x), k\) a constant, obtain the third-, fourth- and fifth-order Taylor polynomial generated by \(y(x)\) about \(x=0\) (b) Write down the third-, fourth-and fifth-order Taylor polynomials generated by \(y=\sin (-x)\) about \(x=0\)

The function \(y(x)=x^{5}+x^{6}\) is approximated by a third-order Taylor polynomial about \(x=1\). (a) Find an expression for the third-order error term. (b) Find an upper bound for the error term given \(0 \leqslant x \leqslant 2\)