MacLaurin Series (DP IB Maths: AA HL)

Consider the general Maclaurin series formula

(where  indicates the  derivative of ).

Explain why the formula cannot be used to calculate a Maclaurin expansion for .

Use the formula to find the first five non-zero terms of the Maclaurin series for .

Hence approximate the value of

by substituting the value

by substituting the value .

Use the general Maclaurin series formula to show that the general term of the Maclaurin series for is

Find the first four non-zero terms of the Maclaurin series for in ascending powers of .

Hence approximate the value of 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   in ascending powers of , up to and including the term in .

Hence find the first four terms of the Maclaurin series for  in ascending powers of .

Use the general Maclaurin series formula to find the first four terms of the Maclaurin series for   in ascending powers of .

Confirm that the answer to part (a) matches the first four terms of the binomial theorem expansion of .

Find the Maclaurin series for  in ascending powers of , up to and including the term in .

Find the derivative of , and confirm that the series found in parts (a) and (c) reflect the relationship between  and  that is thereby implied.

Use integration by parts to show that

Hence determine the exact value of  , and compare it to the approximation found in part (a)(ii).

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

 

Give the series in ascending powers of  up to and including the term in .

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 , and compare the accuracy of that estimated value to the actual value of  .

Consider the differential equation

together with the initial condition  . 

Find expressions for  and  . Each should be given in terms of and  and of lower-order derivatives of .

Let  be the solution to the differential equation above with the given boundary condition, so that . 

Find the first six terms in ascending powers of  of the Maclaurin series for .

Hence approximate the value of  when .

Consider the differential equation

with the initial condition . 

By first finding expressions for and  in terms of  and lower-order derivatives of ,  find a Maclaurin series for the solution to the differential equation with the given boundary condition, in ascending powers of up to and including the term in .

Solve the differential equation with the given boundary condition analytically to find an exact solution in the form .

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 in ascending powers of .

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

Hence approximate the value of

(i)

by substituting the value 

(ii)

by substituting another positive value of .

Find the first four non-zero terms of the Maclaurin series for in ascending powers of .

Hence approximate the value of  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  in ascending powers of , up to and including the term in .

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

Consider the function  defined by  .

Show that  , where  and are constants to be determined.

Hence find the Maclaurin series for   in ascending powers of , up to and including the term in .

Show that .

Hence find the first seven terms, in ascending powers of , of the Maclaurin series for .

Find the Maclaurin series for  in ascending powers of , up to and including the term in .

The probability density function for the random variable  is 

Use the result of part (a) to find an approximation for the probability .

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

Consider the function  defined by

By first determining the Maclaurin series of  in ascending powers of ,  up to and including the term in , show that

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

Consider the differential equation

together with the initial condition . 

Find expressions for  and .  Each should be given in terms of and  and of lower-order derivatives of .

Let   be the solution to the differential equation above with the given boundary condition, so that  . 

Find the first six terms in ascending powers of  of the Maclaurin series for .

Hence find an approximation for the value of  when .

Consider the differential equation

with the initial condition  . 

By first finding expressions for  and  in terms of   and lower-order derivatives of ,  find a Maclaurin series for the solution to the differential equation with the given boundary condition, in ascending powers of  up to and including the term in .

Solve the differential equation with the given boundary condition analytically to find an exact solution in the form .

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).