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- Question : 1.1.1 - Show that the third-order Taylor polynomial for f(x) = (x+ l)-1,aboutxo = 0 , is
- Question : 1.1.2 - What is the third-order Taylor polynomial for f(x) = yjx + 1, about XQ = 0?
- Question : 1.1.3 - What is the sixth-order Taylor polynomial for/(x) = \? + a
- Question : 1.1.4 - Given that ITI6 R(x) = J|-e
- Question : 1.1.5 - Repeat the above, but this time require that the upper bound be valid only for all x S [
- Question : 1.1.6 - Given that for x e [-5, 5], where ? is between a; and 0, find an upper bound for \R\, valid for all x
- Question : 1.1.7 - Use a Taylor polynomial to find an approximate value for s/e that is accurate to within 1(T3
- Question : 1.1.8 - What is the fourth-order Taylor polynomial for f(x) = l/(x + 1), about XQ = 0?
- Question : 1.1.9 - What is the fourth-order Taylor polynomial for f(x) = l/x, about xo = 1?
- Question : 1.1.10 - Find the Taylor polynomial of third-order for sin x, using:
- Question : 1.1.11 - For each function below, construct the third-order Taylor polynomial approximation, using xo = 0, and then estimate the error by computing an upper bound on the remainder, over the given interval.
- Question : 1.1.12 - Construct a Taylor polynomial approximation that is accurate to within 10~3, over the indicated interval, for each of the following functions, using x0 = 0.
- Question : 1.1.13 - Repeat the above, this time with a desired accuracy of 10~6.
- Question : 1.1.14 - Since ?
- Question : 1.1.15 - Elementary trigonometry can be used to show that arctan(l/239) = 4arctan(l/5)
- Question : 1.1.16 - In 1896 a variation on Machin's formula was found: arctan(l/239) = arctan(l)
- Question : 1.1.17 - What is the Taylor polynomial of order 3 for f(x) = x4 + 1, using xo = 0?
- Question : 1.1.18 - What is the Taylor polynomial of order 4 for f(x) = x4 + l, using x0 = 0? Simplify as much as possible.
- Question : 1.1.19 - What is the Taylor polynomial of order 2 for /(x) = x3 + x, using xo = 1?
- Question : 1.1.20 - What is the Taylor polynomial of order 3 for f(x) = x3 + x, using XQ = 1? Simplify as much as possible.
- Question : 1.1.21 - Let p{x) be an arbitrary polynomial of degree less than or equal to n. What is its Taylor polynomial of degree n, about an arbitrary xo?
- Question : 1.1.22 - The Fresnel integrals are defined as C(x) = [ cos(nt2/2)dt Jo and S{x) = I sin(7rf2 /2)dt. Jo Use Taylor expansions to find approximations to C(x) and S(x) that are 10-4 accurate for all x with |x| < \. Hint: Substitute x = ??2/2 into the Taylor expansions for the cosine and sine.
- Question : 1.1.23 - Use the Integral Mean Value Theorem to show that the "pointwise" form (1.3) of the Taylor remainder (usually called the Lagrange form) follows from the "integral" form (1.2) (usually called the Cauchy form).
- Question : 1.1.24 - For each function in Problem 11, use the Mean Value Theorem to find a value M such that \f{xi)-f(x2)\<M\x1-x2\ is valid for all xi, x2 in the interval used in Problem 11.
- Question : 1.1.25 - A function is called monotone on an interval if its derivative is strictly positive or strictly negative on the interval. Suppose / is continuous and monotone on the interval [a, b], and /(a)/(6) < 0; prove that there is exactly one value a 6 [a,b] such that / ( a ) = 0.
- Question : 1.1.26 - Finish the proof of the Integral Mean Value Theorem (Theorem 1.5) by writing up the argument in the case that g is negative.
- Question : 1.1.27 - Prove Theorem 1.6, providing all details.
- Question : 1.1.28 - Let Cfc > 0 be given, 1 < k < n, and let Xfc 6 [a, b], 1 < k < n. Then, use the Discrete Average Value Theorem to prove that, for any function / e C([a, b]),
- Question : 1.1.29 - Discuss, in your own words, whether or not the following statement is true: "The Taylor polynomial of degree n is the best polynomial approximation of degree n to the given function near the point XQ."
- Question : 1.2.1 - Use Taylor's Theorem to show that ex
- Question : 1.2.2 - Use Taylor's Theorem to show that l~c
- Question : 1.2.3 - Use Taylor's Theorem to show thatfor x sufficiently small.
- Question : 1.2.4 - Use Taylor's Theorem to show that ( 1 + x ) " 1 = l - x + x2 + 0(x3) for x sufficiently small.
- Question : 1.2.5 - Show that sinx = x + C(x3).
- Question : 1.2.6 - Recall the summation formula
- Question : 1.2.7 - Use this to prove that fc=0 Hint: What is the definition of the O notation?
- Question : 1.2.8 - Use the above result to show that 10 terms (fc = 9) are all that is needed to compute oo * = ?e" fc fc=0 to within 10-4 absolute accuracy.
- Question : 1.2.9 - Recall the summation formula fc=l Use this to show that
- Question : 1.2.10 - State and prove the version of Theorem 1.7 that deals with relationships of the form x = x
- Question : 1.2.11 - Use the definition of O to show that if y = yh + 0{hv), then hy = hyh + 0(hp+1).
- Question : 1.2.12 - Show that if an = 0(np) and bn =
- Question : 1.2.13 - Suppose that y = yh +
- Question : 1.2.14 - Show that f'M /(z + h)- 2/(a;) + f(x - h) 2 for all h sufficiently small. Hint: Expand f(x
- Question : 1.2.15 - Explain, in your own words, why it is necessary that the constant C in (1.8) be independent of h.
- Question : 1.3.1 - In each problem below, A is the exact value, and Ah is an approximation to A. Find the absolute error and the relative error.
- Question : 1.3.2 - Perform the indicated computations in each of three ways: (i) Exactly; (ii) Using three-digit decimal arithmetic, with chopping; (iii) Using three-digit decimal arith- metic, with rounding. For both approximations, compute the absolute error and the relative error.
- Question : 1.3.3 - For each function below explain why a naive construction will be susceptible to significant rounding error (for x near certain values), and explain how to avoid this error.
- Question : 1.3.4 - For f(x) = (ex
- Question : 1.3.5 - Using single-precision arithmetic only, carry out each of the following computations, using first the form on the left side of the equals sign, then using the form on the right side, and compare the two results. Comment on what you get in light of the material in
- Question : 1.3.6 - Consider the sum m s = jry14'1--0
- Question : 1.3.7 - Using the computer of your choice, find three values a, b, and c, such that (a + b) + c^a + (b + c). Repeat using your pocket calculator.
- Question : 1.3.8 - Assume that we are using three-digit decimal arithmetic. For e = 0.0001, a\ = 5, compute 0-2
- Question : 1.3.9 - Let e < u. Explain, in your own words, why the computationis potentially rife with rounding error. (Assume that ao and a\ are of comparable size.) Hint: See Problem 8.
- Question : 1.3.10 - Using the computer and language of your choice, write a program to estimate the machine epsilon
- Question : 1.3.11 - We can compute e~x using Taylor polynomials in two ways, either using e~x
- Question : 1.3.12 - What is the machine epsilon for a computer that uses binary arithmetic, 24 bits for the fraction, and rounds? What if it chops?
- Question : 1.3.13 - What is the machine epsilon for a computer that uses octal (base 8) arithmetic, assuming that it retains eight octal digits in the fraction?
- Question : 1.5.1 - Consider the error ( 1.11 ) in approximating the error function. If we restrict ourselves to k < 3, then over what range of values of x is the approximation accurate to within 10~3?
- Question : 1.5.2 - If we are interested only in x 6 [0, | ] , then how many terms
- Question : 1.5.3 - Repeat the above for x s [0,1].
- Question : 1.5.4 - Assume that x e [0,1] and write the error in the approximation to the error function using the asymptotic order notation.
- Question : 1.5.5 - For f(x) = / t~1sintdt, x e [-?/4,?/4], Jo construct a Taylor approximation that is accurate to within 10~4 over the indicated interval.
- Question : 1.5.6 - Repeat the above for
- Question : 1.5.7 - Construct a Taylor approximation forthat is accurate to within 10~3 for all values of p in the indicated range.
- Question : 1.5.8 - Does it make a difference in Problem 7 if we restrict p to p S [0,
- Question : 1.5.9 - What is the error in the Taylor polynomial of degree 5 for f(x) = l/x, using x0 = 3/4, for x
- Question : 1.5.10 - How many terms must be taken in the above to get an error of less than 10~2? 10~4?
- Question : 1.5.11 - What is the error in a Taylor polynomial ofdegree 4 for f(x) = %/x using XQ = 9/16, for all x e [1/4,1]?
- Question : 1.5.12 - Consider the rational function l + \x r(x) 2" i-
- Question : 1.5.13 - Repeat the analysis in the previous problem for the rational functionCan you get a better error in this case?
- Question : 1.5.14 - Finally, consider the rational function11 r(x) =
- Question : 1.6.1 - Write each of the following in the form x = f x 2" for some / 6 [|, 1].
- Question : 1.6.2 - For each value in the previous problem, compute the logarithm approximation using the degree 4 Taylor polynomial from (1.12). What is the error compared to the logarithm on your calculator?
- Question : 1.6.3 - Repeat the above for the degree 6 Taylor approximation.
- Question : 1.6.4 - Repeat the above for the degree 10 Taylor approximation.
- Question : 1.6.5 - Implement (as a computer program) the logarithm approximation constructed in this section. Compare it to the intrinsic logarithm function over the interval [|, 1]. What is the maximum observed error?
- Question : 1.6.6 - Let's consider how we might improve on our logarithm approximation from this section. (a) Compute the Taylor expansions, with remainder, for ln(l + x) and ln(l
- Question : 1.6.7 - Use the logarithm expansion from the previous problem, but limited to the degree 4 case, to compute approximations to the logarithm of each value in the first problem of this section.
- Question : 1.6.8 - Repeat the above, using the degree 10 approximation.
- Question : 1.6.9 - Implement (as a computer program) the logarithm approximation constructed in Problem 6. Compare it to the intrinsic logarithm function over the interval [\, 1]. What is the maximum observed error?
- Question : 1.6.10 - Try to use the ideas from this section to construct an approximation to the reciprocal function, f(x)

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