Maclaurin Series (DP IB Analysis & Approaches (AA): HL): Exam Questions

Find the Maclaurin series for ${\mathrm{e}}^x\sin x$, up to and including the term in $x^4$.

By differentiating the Maclaurin series for $\sin x$, find the first four non-zero terms of the Maclaurin series for $\cos x$.

Use the Maclaurin series for $\sin x$ and $\cos x$ to find a Maclaurin series approximation for $2\sin x\cos x$ up until the term in $x^4.$

The double angle identity for sine tells us that

$\sin 2x=2\sin x\cos x$

Use the Maclaurin series for $\sin x$ to find the Maclaurin series for $\sin 2x$, up to and including the term in $x^4$. Hence verify your answer to part (a).

Use the Binomial theorem to find a Maclaurin series for the function$f$ defined by 

$f\left(x\right)=\sqrt{1-2x^2}$

Give the series up to and including the term in $x^6$.

State the restriction which must be placed on $x$ for this expansion to be valid.

Use the answer to part (a) to estimate the value of $\sqrt{0.5}$, and compare the accuracy of that estimated value to the actual value of $\sqrt{0.5}$.

Consider the differential equation

$y\text{'}=2y^2+x$

together with the initial condition $y\left(0\right)=1$.

(i) Show that $y\text{'}\text{'}=4yy\text{'}+1$.

(ii) Find expressions for $y\text{'}\text{'}\text{'}$, $y^{\left(4\right)}$ and $y^{\left(5\right)}$ in terms of $y$ and its lower-order derivatives.

Using the initial condition above, calculate the values of $y\text{'}\left(0\right)$, $y\text{'}\text{'}\left(0\right)$, $y\text{'}\text{'}\text{'}\left(0\right)$, $y^{\left(4\right)}\left(0\right)$ and $y^{\left(5\right)}\left(0\right)$.

Let $f\left(x\right)$ be the solution to the differential equation above with the given initial condition, so that $y=f\left(x\right)$.

Using the answers to part (b), find the first six terms of the Maclaurin series for $f\left(x\right)$.

Hence approximate the value of $y$ when $x=0.1$, giving your answer correct to four decimal places.

Consider the differential equation

$y^{\text{'}}=2x{y}^2$

with the initial condition $y\left(0\right)=1$.

(i) Find $y^{\text{'}\text{'}}$.

(ii) Hence show that $y^{\text{'}\text{'}\text{'}}=8y{y}^{\text{'}}+4x{\left(y^{\text{'}}\right)}^2+4xy{y}^{\text{'}\text{'}}$  and  $y^{\left(4\right)}=12{\left(y^{\text{'}}\right)}^2+12y{y}^{\text{'}\text{'}}+12x{y}^{\text{'}}{y}^{\text{'}\text{'}}+4xy{y}^{\text{'}\text{'}\text{'}}$

Hence find the Maclaurin series for $y$, up to and including the term in $x^4$.

Use separation of variables to show that the exact solution of the differential equation with the given initial condition is 

$y=\frac{1}{1-x^2}$

Use the binomial theorem to find an approximation for $\frac{1}{1-x^2}$ up to the term in $x^4$, and verify that it matches the answer to part (b).

Find the first four non-zero terms of the Maclaurin series for $\cos 4x$ in ascending powers of $x$.

Hence approximate the value of $\cos 3$ and compare this approximation to the exact value.

Explain how the accuracy of the Maclaurin series approximation in part (b) could be improved.

Use the general Maclaurin series formula to find the first four terms of the Maclaurin series for $\frac{1}{2+3x}$  in ascending powers of $x$.

Confirm that the answer to part (a) matches the first four terms of the binomial theorem expansion of $\frac{1}{2+3x}$ .

Find the Maclaurin series for $\ln \left(2+3x\right)$ in ascending powers of $x$, up to and including the term in $x^4$.

Find the derivative of $\ln (2+3x)$ , and confirm that the series found in parts (a) and (c) reflect the relationship between $\ln \left(2+3x\right)$ and $\frac{1}{2+3x}$ that is thereby implied.

Use the binomial theorem to find a Maclaurin series for the function $f$ defined by

$f\left(x\right)=\sqrt{4-9x^2}$ 

Give the series in ascending powers of $x$ up to and including the term in $x^6$.

State the restriction which must be placed on $x$ for this expansion to be valid.

Use the answer to part (a) to estimate the value of $\sqrt{3.91}$ , and compare the accuracy of that estimated value to the actual value of  $\sqrt{3.91}$.

Find the first four non-zero terms of the Maclaurin series for ${\mathrm{e}}^{-2x}$ in ascending powers of $x$.

Hence approximate the value of $\sqrt{e}$ and compare this approximation to the exact value.

Explain how the accuracy of the Maclaurin series approximation in part (b) could be improved.

Consider the general Maclaurin series formula

$f\left(x\right)=f\left(0\right)+xf\text{'}\left(0\right)+\frac{x^2}{2!}f\text{'}\text{'}\left(0\right)+...+\frac{x^n}{n!}{f}^{\left(n\right)}\left(0\right)+...$

(where $f^{\left(n\right)}$ indicates the $n^{th}$ derivative of $f$).

Explain why the formula cannot be used to calculate a Maclaurin expansion for $\ln x$.

Use the formula to find the first five non-zero terms of the Maclaurin series for $\ln (1+x)$.

Hence approximate the value of $\ln 2$

(i) by substituting the value $x=1$

(ii) by substituting the value $x=-\frac{1}{2}$.

(i) Compare the approximations found in part (c) to the exact value of $\ln 2$.

(ii) Explain briefly the reason for the difference in accuracy between the two approximations.

Use the general Maclaurin series formula to show that the general term of the Maclaurin series for $\ln \left(1+x\right)$ is

$\frac{{\left(-1\right)}^{n+1}{x}^n}{n},n\ge 1$

Find the Maclaurin series for ${\mathrm{e}}^{2x}\ln \left(1+x\right)$ in ascending powers of $x$, up to and including the term in $x^4$.

Hence find the first four terms of the Maclaurin series for ${\mathrm{e}}^{2x}\left(2\ln \left(1+x\right)+\frac{1}{1+x}\right)$ in ascending powers of $x$.

(i) Write down the first five non-zero terms of the Maclaurin series for $\arctan x$ in ascending powers of $x$.

(ii) Hence find an approximation for the value of the integral

${\int }_0^1\arctan x \mathrm{d}x$

Use integration by parts to show that

$\int \arctan x \mathrm{d}x=x\arctan x-\frac{1}{2}\ln \left(1+x^2\right)+c$

Hence determine the exact value of ${\int }_0^1\arctan x \mathrm{d}x$, and compare it to the approximation found in part (a)(ii).

Consider the differential equation

$y\text{'}=x^2-3y^2$ 

together with the initial condition $y\left(0\right)=1$ . 

Find expressions for $y^{\text{'}\text{'}},y^{\text{'}\text{'}\text{'}},y^{\left(4\right)}$ and $y^{\left(5\right)}$ . Each should be given in terms of $x$ and $y$ and of lower-order derivatives of $y$.

Let $f\left(x\right)$ be the solution to the differential equation above with the given initial condition, so that $y=f\left(x\right)$.

Find the first six terms in ascending powers of $x$ of the Maclaurin series for $f\left(x\right)$.

Hence approximate the value of $y$ when $x=0.1$.

Consider the differential equation

$y\text{'}=-2xy$

with the initial condition $y\left(0\right)=2$.

By first finding expressions for $y\text{'}\text{'}$, $y\text{'}\text{'}\text{'}$, $y^{\left(4\right)}$, $y^{\left(5\right)}$ and $y^{\left(6\right)}$ in terms of $x$, $y$ and lower-order derivatives of $y$, find a Maclaurin series for the solution to the differential equation with the given initial condition, in ascending powers of $x$ up to and including the term in $x^6$.

Solve the differential equation with the given boundary condition analytically to find an exact solution in the form $y=f\left(x\right)$.

Find the first four non-zero terms of the Maclaurin series for the answer to part (b), and confirm that they match those in the answer to part (a).

Find the Maclaurin series for ${\mathrm{e}}^x\left(\sin 3x+\cos \sqrt{x}\right)$ in ascending powers of $x$, up to and including the term in $x^3$.

Hence find the first three non-zero terms, in ascending powers of $x$, of the Maclaurin series for

${\mathrm{e}}^x\left(2\sin 3x+6\cos 3x+2\cos \sqrt{x}-\frac{\sin \sqrt{x}}{\sqrt{x}}\right)$

Find the Maclaurin series for ${\mathrm{e}}^{-\frac{1}{2}{x}^2}$ in ascending powers of $x$, up to and including the term in $x^8$.

The probability density function for the random variable $X~\mathrm{N}\left(0,1\right)$ is

$f\left(x\right)=\frac{1}{\sqrt{2\pi }}{\mathrm{e}}^{-\frac{1}{2}{x}^2}$

Use the result of part (a) to find an approximation for the probability $\mathrm{P}\left(0\le X\le 1\right)$.

Compare your approximation from part (b) with the exact value of $\mathrm{P}\left(0\le X\le 1\right)$.

Consider the differential equation

 $y\text{'}=\cos x+x{y}^2$ 

together with the initial condition $y\left(0\right)=1$. 

Find expressions for $y^{\text{'}\text{'}},y^{\text{'}\text{'}\text{'}},y^{\left(4\right)}$ and $y^{\left(5\right)}$.  Each should be given in terms of $x$ and $y$ and of lower-order derivatives of $y$.

Let $f\left(x\right)$ be the solution to the differential equation above with the given initial condition, so that $y=f\left(x\right)$.

Find the first six terms in ascending powers of $x$ of the Maclaurin series for $f\left(x\right)$.

Hence find an approximation for the value of $y$ when $x=0.1$.

Consider the differential equation

$y\text{'}=\frac{y}{x+1}+1,x>-1$

with the initial condition $y\left(0\right)=-1$.

By first finding expressions for $y\text{'}\text{'}$, $y\text{'}\text{'}\text{'}$ and $y^{\left(4\right)}$ in terms of $x$, $y$ and lower-order derivatives of $y$, find a Maclaurin series for the solution to the differential equation with the given initial condition, in ascending powers of $x$ up to and including the term in $x^4$.

Solve the differential equation, given that $y=-1$ when $x=0$. Give your answer in the form $y=f\left(x\right)$.

Find the first four non-zero terms of the Maclaurin series for the answer to part (b), and confirm that they match those in the answer to part (a). 

Find the first three non-zero terms of the Maclaurin series for $\tan x$ in ascending powers of $x$.

Confirm that the result from part (a) gives the same type of function – either even or odd – as $\tan x$.

Hence approximate the value of $\tan 1$

(i) by substituting the value $x=1$

(ii) by substituting another positive value of $x$ .

(i) Compare the approximations found in part (c) to the exact value of $\tan 1$.

(ii) Explain briefly the reason for the difference in accuracy between the two approximations.

Consider the function $f$ defined by $f\left(x\right)={\mathrm{e}}^{3x}\cos 2x$.

Show that $f\text{'}\text{'}\left(x\right)=pf\left(x\right)+qf\text{'}\left(x\right)$, where $p$ and $q$ are constants to be determined.

Hence find the Maclaurin series for $f\left(x\right)$  in ascending powers of $x$, up to and including the term in $x^5$.

Show that $\int f\left(x\right) \mathrm{d}x=\frac{{\mathrm{e}}^{3x}}{13}\left(2\sin 2x+3\cos 2x\right)+c$.

Hence find the first seven terms, in ascending powers of $x$, of the Maclaurin series for ${\mathrm{e}}^{3x}\left(2\sin 2x+3\cos 2x\right)$.

Consider the function $f$ defined by

 $f\left(x\right)=\frac{1}{\sqrt{1-2x^2}}$ 

By first determining the Maclaurin series of $f\left(x\right)$ in ascending powers of $x$,  up to and including the term in $x^6$ , show that

$\sin \frac{\pi }{4}\approx 0.70710675$ 

Be sure to justify that the Maclaurin series is valid for the value of $x$ used to produce your approximation.