Home » Approximation

# Compute π using the Taylor polynomial of arctan x

For this exercise define

1. Using the trig identity

twice, once with , and then the second time with , show that

Then use the same identity again with and to show

This establishes the identity

2. Using the Taylor polynomial approximation at prove that

3. Using the Taylor polynomial approximation at prove that

4. Using the above parts show that the value of to seven decimal places is 3.1415926.

1. Proof. Letting we have

Letting we have

Letting and we have (recalling that )

But then

2. Proof. We know the Taylor polynomial approximation for from this exercise (Section 7.8, Exercise #3):

Therefore, we can compute an approximation to ,

where

Therefore,

3. Proof. Again using the Taylor polynomial approximation to we have

4. Finally,

# Use Taylor polynomials to approximate the integral of sin x / x

Using the Taylor polynomial approximation of find an approximation for the integral

Give an estimate for the error of the approximation. [Define when .]

We know (from this exercise, Section 7.8, Exercise #1) that the Taylor polynomial approximation of is given by

This implies

where

# Use Taylor polynomials to approximate an integral of sin (x2)

1. Show that

when .

2. Using part (a) find an approximation for the integral

1. Proof. We know from this exercise that

Therefore, when we have

2. From part (a) we know

where

# Use Taylor polynomials to approximate the nonzero root of arctan x = x2

1. Show that is an approximation of the nonzero root of the equation

using the cubic Taylor polynomial approximation to .

2. Given that

prove that the number from part (a) satisfies

Determine if is positive or negative and prove the result.

1. Proof. From a previous exercise (Section 7.8, Exercise #3) we know

So, to approximate the nonzero root of we have

2. We know from the same previous exercise we used in part (a) that the error term for satisfies the inequality

Using the values for and given we have

# Use Taylor polynomials to approximate the nonzero root of x2=sin x

1. Using the cubic Taylor polynomial approximation of , show that the nonzero root of the equation

is approximated by .

2. Using part (a) show that

given that . Determine whether is positive or negative and prove the result.

1. Proof. The cubic Taylor polynomial approximation of is

This implies

Therefore, we can approximate the nonzero root by

2. Proof. We know from this exercise (Section 7.8, Exercise #1) that for we have

So, for , and using the given inequality , we have

Furthermore, since

with the absolute value of each term in the sum strictly less than the absolute value of the previous term (since and ). Thus, each pair is positive, so the whole series is positive

# Use the weighted mean value theorem to approximate an integral

Note that

and use the weighted mean value theorem (Theorem 3.16 in Apostol) to prove that for positive :

Using compute the integral to six decimal places.

Recall the weighted mean value theorem:

For functions and continuous on , if never changes sign in then there exists such that

Proof. Let

Then,

for some . Since is strictly decreasing on , we know ; thus,

Furthermore,

Thus,

Now, taking we compute,