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

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

(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 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.

Find the Maclaurin series for $e^{2x}\ln \left(1+x\right)$   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 $e^{2x}\left(2\ln \left(1+x\right)+\frac{1}{1+x}\right)$ in ascending powers of $x$.

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.

(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

${\int }_0^1\arctan xdx$

Use integration by parts to show that

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

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

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 any limitations on the validity of the series expansion found in part (a).

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  $\sqrt{3.91}$.

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 boundary 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 boundary 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 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.

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

Find the Maclaurin series for $e^x\left(\sin 3x+\cos \sqrt{x}\right)$ 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

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

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

Show that  $f^{\text{'}\text{'}}\left(x\right)=pf\left(x\right)+p{f}^{\text{'}}\left(x\right)$, 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 $\int f\left(x\right)dx=\frac{e^{3x}}{13}\left(2\sin 2x+3\cos 2x\right)+c$.

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

Find the Maclaurin series for $e^{-\frac{1}{2}{x}^2}$ 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 

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

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

Determine the percentage error of your approximation from part (b).

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

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

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

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 boundary 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 boundary condition, in ascending powers of $x$  up to and including the term in $x^4$.

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).