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# Find the limit as x goes to (π / 2) of cos x / (x – (π / 2))

Evaluate the limit.

Since we use the expansion (page 287) for ,

Therefore,

# Find the limit as x goes to 0 of (sin x) / (arctan x)

Evaluate the limit.

We know (p. 287) the following expansions as ,

(Note that these are the same expansion when we use only the first order terms. This tells us that and behave similarly near 0. We would need to take higher order terms to differentiate between the two. For instance, if we wanted to include cubic terms we would have , but .) From here we compute the limit,

# Find the limit as x goes to 0 of (1 – cos2 x) / (x tan x)

Evaluate the limit.

To evaluate this we use the trig identity to simplify

We have proved the limit earlier (at least once), but let’s do it again using the techniques of this section and -notation.

# 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

# Prove an inequality for the error of the Taylor polynomial of arctan x

Prove that the error term in the Taylor expansion of satisfies the following inequality.

Proof. To prove this we will work directly from the definition of the error as an integral,

We know for we have, (we need for the expansion of to be valid),

Therefore we have

So, we can bound the error term by bounding the integral,

# Prove an inequality for the error of the Taylor polynomial of sin x

Prove that the error of the Taylor expansion of satisfies the following inequality.

Proof. Since the derivatives of are always , or we know that for we have . (In other words, the st derivative is bounded above by 1 and below by .) Therefore, we can apply Theorem 7.7 (p. 280 of Apostol) to estimate the error in Taylor’s formula at with and . For this gives us

Next, (from the second part of Theorem 7.7) if we have

# Find the Taylor polynomial of sin2 x

Show that

Using the trig identity suggested in the hint, we have

We know from Example 3 (page 275 of Apostol) that

Therefore, by Theorem 7.3 (the substitution property, page 276) we know

Then, using the linearity of the Taylor operator (Theorem 7.2) we have

# Prove an integral formula for a rational function in sine and cosine

Given constants such that , prove that

for some constants .

Proof. Define constants and by

(Since we know , so these definitions make sense.) Then

Therefore, we may write,

So, to evaluate the integral we have

For the integral on the right, we make the substitution , so . Therefore,