Practice section 2 of 7

The variables x and y are related by $dy/dx=2xy-1$ and y=1 when x=0.

Show that, at x=0, $d^3y/dx^3=-4$ and $d^4y/dx^4=12$.

Find the Maclaurin series for y up to and including the term in $x^4$, and hence find an approximation to the value of y when x=0.1, giving your answer to 5 decimal places.

Let $y=\tan 2x$, show that

(i) $dy/dx=2(1+y^2)$,

(ii) $d^3y/dx^3=4(dy/dx)^2+4y\,d^2y/dx^2$.

Find the Maclaurin series for $\tan 2x$, up to and including the term in $x^3$.

Hence, find the first two nonzero terms of the Maclaurin series for $\sec^2 2x$.

It is given that $y=\ln(2+\sin2x)$.

(i) Show that $e^y\dfrac{d^2y}{dx^2}+e^y\left(\dfrac{dy}{dx}\right)^2=-4\sin2x$. [2]

(ii) By further differentiation of the above results, find the Maclaurin series for y, up to and including the term in $x^3$. [3]

(iii) Verify that the series for $\ln(2+\sin2x)$ is the same as the result obtained in part (ii), if the standard series from the List of Formulae (MF27) are used. [3]

(iv) Hence deduce the series expansion for $\dfrac{\ln(2+\sin2x)}{\sqrt{1-x}}$, up to and including the term in $x^2$. [2]

(a) It is given that $dy/dx=y+2x$ and that y=1 when x=0. Find the first three terms in the Maclaurin series for y.

(b) Show that the Maclaurin series for $\ln(\sec x)$, up to and including the term in $x^4$, is $\tfrac12x^2+\tfrac1{12}x^4$.

If x is sufficiently small for terms in x higher than $x^2$ to be negligible, obtain approximations for

(a) $\sin(x+\pi/6)$,

(b) $\cos(\pi/3+x)-\cos(\pi/3-x)$.

If x is sufficiently small for terms in x higher than $x^2$ to be negligible, obtain an approximation for $\tan(\pi/4+x/3)$.