Practice section 6 of 7

Given $y=e^{\cos x}$, show $y^{\prime\prime}=-y\cos x-y^{\prime}\sin x$. Show that the first two nonzero terms in its expansion are $e-\tfrac12ex^2$. [4]

For $y=e^x\sin x$, show $y^{\prime\prime}=2(y^{\prime}-y)$. Differentiate this further to find the Maclaurin series through $x^3$, then verify it using the standard series for $e^x$ and $\sin x$. [7]

Given $y=\ln(1+e^x)$, show $y^{\prime\prime}=y^{\prime}-(y^{\prime})^2$. [3]

Hence find its Maclaurin series through $x^2$. [2]

Deduce the first two nonzero terms in the expansion of $e^x/(1+e^x)$. [3]

(i) For $f(x)=\ln(1+2\sin x)$, find $f(0),f^{\prime}(0),f^{\prime\prime}(0),f^{\prime\prime\prime}(0)$ and hence its first three nonzero Maclaurin terms. [7]

(ii) Its first two nonzero terms match those of $e^{ax}\sin(nx)$. Use appropriate MF27 expansions to find $a,n$ and then the third nonzero term of $e^{ax}\sin(nx)$. [5]

Given $\cos y=\sqrt{1-e^x/4}$ and $0<y<\pi/2$, show $\sin(2y)y^{\prime}=e^x/4$. By further differentiation find the Maclaurin series for $y$ through $x^2$ in exact form. [5]

Deduce the tangent to $y=\cos^{-1}\!\sqrt{1-e^x/4}$ at $x=0$. [2]

For $y=\tan^{-1}x$, prove $2xy^{\prime}+(1+x^2)y^{\prime\prime}=0$. Differentiate further to show its Maclaurin series through $x^3$ is $x+hx^3$, finding $h$. [5]

Hence expand $\sqrt{1+\tan^{-1}x}$ through $x^3$. [3]