91Ó°ÊÓ

Produce the linear and quadratic Taylor polynomials for the following cases. Graph the function and these Taylor polynomials. (a) \(f(x)=\sqrt{x}, a=1\) (b) \(\quad f(x)=\sin (x), a=\pi / 4\) (c) \(f(x)=e^{\cos (x)}, a=0\) (d) \(\quad f(x)=\log \left(1+e^{x}\right), a=0\)

Produce a general formula for the degree \(n\) Taylor polynomials for the following functions, all using \(a=0\) as the point of approximation. (a) \(1 /(1-x)\) (b) \(\sin (x)\) (c) \(\sqrt{1+x}\) (d) \(\cos (x)\) (e) \((1+x)^{1 / 3}\)

(a) Produce the Taylor polynomials of degrees \(1,2,3,4\) for \(f(x)=e^{x^{2}}\) with \(a=0\) the point of approximation. (b) Using the Taylor polynomials for \(e^{t}\), substitute \(t=x^{2}\) to obtain polynomial approximations for \(e^{x^{2}}\). Compare with the results in (a).

(a) Obtain a Taylor polynomial with remainder for \(f(t)=1 /\left(1+t^{2}\right)\), about \(a=0\). Hint: Substitute \(x=-t^{2}\) into \((1.16)\). (b) Obtain a Taylor polynomial with remainder for \(g(x)=\tan ^{-1} x .\) Do this by integrating the result in (a) and using $$ \tan ^{-1}(x)=\int_{0}^{x} \frac{d t}{1+t^{2}} $$