5.11 MacLaurin Series

Use substitution into the Maclaurin series for

to find the first four terms of the Maclaurin series for .

Hence approximate the value of  and compare this approximation to the exact value.

Without performing any additional calculations, explain whether the answer to part (a) would be expected to give an approximation of that is more accurate or less accurate than its approximation for .

The Maclaurin series for and are

Find the Maclaurin series for up to and including the term in .

Use the Maclaurin series for , along with the fact that ,  to find the first four terms of the Maclaurin series for .

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

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

The Maclaurin series for is 

Differentiate the Maclaurin series for up to its fourth term and compare this to the answer from part (a). Give an explanation for any similarities that are found.

Use the Maclaurin series for and  to find a Maclaurin series approximation for  up until the term in

The double angle identity for sine tells us that 

Use substitution into the Maclaurin series for to find a Maclaurin series approximation for  up until the term in , and confirm that this matches the answer to part (a).

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

Give the series 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 .

Show that .

Use an equivalent method to find expressions for ,  and . Each should be given in terms of  and of lower-order derivatives of .

Using the boundary condition above, calculate the values of , , ,  and .

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

Using the answers to part (b), find the first six terms of the Maclaurin series for .

Hence approximate the value of to 4 d.p. when .

Consider the differential equation

with the initial condition .

Find .

Hence show that   and  

Use the results from part (a) along with the given initial condition to find a Maclaurin series to approximate the solution of the differential equation, giving the approximation up to the term in .

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

Use the binomial theorem to find an approximation for up to the term in , and verify that it matches the answer to part (b).

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).