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Math 202
Taylor Approximations Name:
Recall that if f (x) can be written as a Taylor Series centered at a then
f (x) =
∑^ ∞
n=
f (n)(a) n!
(x − a)n .
The mth^ Taylor Polynomial is
Tm(x) =
∑^ m
n=
f (n)(a) n!
(x − a)n
and the error in approximating f (x) with Tm(x) is
Rm(x) =
∑^ ∞
n=m+
f (n)(a) n!
(x − a)n .
Taylor’s inequality can help us to bound the error.
Taylor’s Inequality: If |f (m+1)(x)| ≤ M for |x − a| < d for some numbers M and d, then
|Rm(x)| ≤
M
(m + 1)!
|x − a|(m+1)
- What kinds of functions are the derivatives of sin(x) and cos(x)? In light of this, what M can you use to apply Taylor’s Inequality for sin(x) and cos(x)?
- This problem approximates sin(2).
(a) What is the 5th^ Taylor polynomial for sin(x)? (b) Use the previous part to approximate sin(2). (c) Find a bound for the error in your approximation using Taylor’s inequality. (d) How can you use the same approximation but get a smaller bound on your error? (Hint: 5?)
- This problem will show you how to get an approximation and a bound on the error associated to your approximation for a range of x’s.
(a) What is the 7th^ Taylor polynomial for cos(x). (b) Use Taylor’s inequality to determine for which x’s your approximation will have error less than .001. (Hint: Make the right side of Taylor’s inequality less than .001. )
- In this problem you will approximate sin(1) to a specified degree of accuracy.
(a) Use Taylor’s inequality to determine the smallest Taylor polynomial (i.e. which m) will make your approximation less than .0001. (b) Use this Taylor polynomial to approximate sin(1).
- In this problem you will exam the previously undoable integral
e−t 4 dt. Despite the fact that there is not a nice function that is the anti-derivative, it is relatively easy to write down a Taylor series and hence approximate this integral.
(a) Compute the Taylor series e−t 4 .
(b) Compute the Taylor series for the indefinite integral
e−t
4 dt.
(c) Compute the Taylor series for the definite integral
∫ (^) x
0
e−t 4 dt. Notice that this integral is now a function of x with a Taylor series. (d) Calculate T 23 (x). Notice that the higher derivatives of this function are not too easy. This makes Taylor’s inequality hard to use. Thus (sigh) we will have to be satisfied without a bound on the error for now. (e) What other Taylor polynomials are equal to T 23 (x)?
- For this problem you will need to use the more technical bits of Taylor’s inequality. The number e is somewhat mysterious. In this problem you will approximate it so that the first five digits are correct. Let f (x) = ex, and notice the following important facts: - f (n)(x) = ex^ for every n - ex^ is always increasing (its derivative is always positive) - since it is always increasing, if we restrict our attention to x in the interval [−b, b] the maximum value of f (x) occurs when x = b and f (b) = eb.
(a) If we are going to approximate e using a Taylor polynomial for f (x) = ex, what is x? (b) Using the points outlined above, what would be the smallest M we could use for Taylor’s inequality? (c) What is the smallest integer we could use for M? (d) Use your answer to the previous part to determine which Taylor polynomial you need to use so that your approximation is correct up to 4 decimal places. (e) What are the first 5 digits of e? (After you answer this, you are halfway to answering the Google advertisement.) (f) You could use this method with other functions but it becomes more challenging when all the derivatives are not as well known and as well behaved as those of ex. Why would this method work for ln(b − x) for some fixed positive number b?
