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- Question : 1EX - Show that the following equations have at least one solution in the given intervals. a. x cos x ? 2x2 + 3x ? 1 = 0, [0.2, 0.3] and [1.2, 1.3] b. (x ? 2)2 ? ln x = 0, [1, 2] and [e, 4] c. 2x cos(2x) ? (x ? 2)2 = 0, [2, 3] and [3, 4] d. x ? (ln x)x = 0, [4, 5]
- Question : 2EX - Find intervals containing solutions to the following equations. a. x ? 3?x = 0 b. 4x2 ? ex = 0 c. x3 ? 2x2 ? 4x + 2 = 0 d. x3 + 4.001x2 + 4.002x + 1.101 = 0
- Question : 3EX - Show that f (x) is 0 at least once in the given intervals. a. f (x) = 1 ? ex + (e ? 1) sin((?/2)x), [0, 1] b. f (x) = (x ? 1) tan x + x sin ?x, [0, 1] c. f (x) = x sin ?x ? (x ? 2) ln x, [1, 2] d. f (x) = (x ? 2) sin x ln(x + 2), [?1, 3]
- Question : 4EX - Find maxa?x?b |f (x)| for the following functions and intervals. a. f (x) = (2 ? ex + 2x)/3, [0, 1] b. f (x) = (4x ? 3)/(x2 ? 2x), [0.5, 1] c. f (x) = 2x cos(2x) ? (x ? 2)2, [2, 4] d. f (x) = 1 + e? cos(x?1), [1, 2]
- Question : 5EX - Use the Intermediate Value Theorem 1.11 and Rolle
- Question : 6EX - Suppose f ? C[a, b] and f (x) exists on (a, b). Show that if f (x) = 0 for all x in (a, b), then there can exist at most one number p in [a, b] with f (p) = 0.
- Question : 7EX - Let f (x) = x3. a. Find the second Taylor polynomial P2(x) about x0 = 0. b. Find R2(0.5) and the actual error in using P2(0.5) to approximate f (0.5). c. Repeat part (a) using x0 = 1. d. Repeat part (b) using the polynomial from part (c).
- Question : 8EX - Find the third Taylor polynomial P3(x) for the function f (x) = ?x + 1 about x0 = 0. Approximate ?0.5, ?0.75, ?1.25, and ?1.5 using P3(x), and find the actual errors.
- Question : 9EX - Find the second Taylor polynomial P2(x) for the function f (x) = ex cos x about x0 = 0. a. Use P2(0.5) to approximate f (0.5). Find an upper bound for error |f (0.5) ? P2(0.5)| using the error formula, and compare it to the actual error. b. Find a bound for the error |f (x) ? P2(x)| in using P2(x) to approximate f (x) on the interval [0, 1]. c. Approximate 01 f (x) dx using 01 P2(x) dx. d. Find an upper bound for the error in (c) using 01 |R2(x) dx|, and compare the bound to the actual error.
- Question : 10EX - Repeat Exercise 9 using x0 = ?/6.
- Question : 11EX - . Find the third Taylor polynomial P3(x) for the function f (x) = (x ? 1) ln x about x0 = 1. a. Use P3(0.5) to approximate f (0.5). Find an upper bound for error |f (0.5) ? P3(0.5)| using the error formula, and compare it to the actual error. b. Find a bound for the error |f (x) ? P3(x)| in using P3(x) to approximate f (x) on the interval [0.5, 1.5]. c. Approximate 0.5 1.5 f (x) dx using 0.5 1.5 P3(x) dx. d. Find an upper bound for the error in (c) using 0.5 1.5 |R3(x) dx|, and compare the bound to the actual error.
- Question : 12EX - Let f (x) = 2x cos(2x) ? (x ? 2)2 and x0 = 0. a. Find the third Taylor polynomial P3(x), and use it to approximate f (0.4). b. Use the error formula in Taylor
- Question : 13EX - Find the fourth Taylor polynomial P4(x) for the function f (x) = xex2 about x0 = 0. a. Find an upper bound for |f (x) ? P4(x)|, for 0 ? x ? 0.4. b. Approximate 00.4 f (x) dx using 00.4 P4(x) dx. c. Find an upper bound for the error in (b) using 00.4 P4(x) dx. d. Approximate f (0.2) using P4 (0.2), and find the error.
- Question : 14EX - Use the error term of a Taylor polynomial to estimate the error involved in using sin x ? x to approximate sin 1?.
- Question : 15EX - Use a Taylor polynomial about ?/4 to approximate cos 42? to an accuracy of 10?6.
- Question : 16EX - Let f (x) = ex/2 sin(x/3). Use Maple to determine the following. a. The third Maclaurin polynomial P3(x). b. f (4)(x) and a bound for the error |f (x) ? P3(x)| on [0, 1].
- Question : 17EX - Let f (x) = ln(x2 + 2). Use Maple to determine the following. a. The Taylor polynomial P3(x) for f expanded about x0 = 1. b. The maximum error |f (x) ? P3(x)|, for 0 ? x ? 1. c. The Maclaurin polynomial P
- Question : 18EX - Let f (x) = (1 ? x)?1 and x0 = 0. Find the nth Taylor polynomial Pn(x) for f (x) about x0. Find a value of n necessary for Pn(x) to approximate f (x) to within 10?6 on [0, 0.5].
- Question : 19EX - Let f (x) = ex and x0 = 0. Find the nth Taylor polynomial Pn(x) for f (x) about x0. Find a value of n necessary for Pn(x) to approximate f (x) to within 10?6 on [0, 0.5].
- Question : 20EX - Find the nth Maclaurin polynomial Pn(x) for f (x) = arctan x.
- Question : 21EX - The polynomial P2(x) = 1 ? 2 1x2 is to be used to approximate f (x) = cos x in [? 1 2, 1 2]. Find a bound for the maximum error
- Question : 22EX - The nth Taylor polynomial for a function f at x0 is sometimes referred to as the polynomial of degree at most n that
- Question : 23EX - Prove the Generalized Rolle
- Question : 24EX - In Example 3 it is stated that for all x we have | sin x| ? |x|. Use the following to verify this statement. a. Show that for all x ? 0 we have f (x) = x?sin x is non-decreasing, which implies that sin x ? x with equality only when x = 0. b. Use the fact that the sine function is odd to reach the conclusion.
- Question : 25EX - A Maclaurin polynomial for ex is used to give the approximation 2.5 to e. The error bound in this approximation is established to be E = 1 6. Find a bound for the error in E.
- Question : 26EX - The error function defined by erf(x) = 2 ?? 0 x e?t2 dt gives the probability that any one of a series of trials will lie within x units of the mean, assuming that the trials have a normal distribution with mean 0 and standard deviation ?2/2. This integral cannot be evaluated in terms of elementary functions, so an approximating technique must be used. a. Integrate the Maclaurin series for e?x2 to show that erf(x) = 2 ?? ? k=0 (?1)kx2k+1 (2k + 1)k! . b. The error function can also be expressed in the form erf(x) = 2 ?? e?x2 ? k=0 2kx2k+1 1
- Question : 27EX - A function f : [a, b] ? R is said to satisfy a Lipschitz condition with Lipschitz constant L on [a, b] if, for every x, y ? [a, b], we have |f (x) ? f (y)| ? L|x ? y|. a. Show that if f satisfies a Lipschitz condition with Lipschitz constant L on an interval [a, b], then f ? C[a, b]. b. Show that iff has a derivative that is bounded on[a, b] by L, thenf satisfies a Lipschitz condition with Lipschitz constant L on [a, b]. c. Give an example of a function that is continuous on a closed interval but does not satisfy a Lipschitz condition on the interval.
- Question : 28EX - Suppose f ? C[a, b], that x1 and x2 are in [a, b]. a. Show that a number ? exists between x1 and x2 with f (?) = f (x1) + f (x2) 2 = 1 2 f (x1) + 1 2 f (x2). b. Suppose that c1 and c2 are positive constants. Show that a number ? exists between x1 and x2 with f (?) = c1f (x1) + c2f (x2) c1 + c2 . c. Give an example to show that the result in part b. does not necessarily hold when c1 and c2 have opposite signs with c1 = ?c2
- Question : 29EX - Let f ? C[a, b], and let p be in the open interval (a, b). a. Suppose f (p) = 0. Show that a ? > 0 exists with f (x) = 0, for all x in [p ? ?, p + ?], with [p ? ?, p + ?] a subset of [a, b]. b. Suppose f (p) = 0 and k > 0 is given. Show that a ? > 0 exists with |f (x)| ? k, for all x in [p ? ?, p + ?], with [p ? ?, p + ?] a subset of [a, b].
- Question : 1EX - Compute the absolute error and relative error in approximations of p by p?. a. p = ?, p? = 22/7 b. p = ?, p? = 3.1416 c. p = e, p? = 2.718 d. p = ?2, p? = 1.414 e. p = e10, p? = 22000 f. p = 10?, p? = 1400 g. p = 8!, p? = 39900 h. p = 9!, p? = ?18?(9/e)9
- Question : 2EX - Find the largest interval in which p? must lie to approximate p with relative error at most 10?4 for each value of p. a. ? b. e c. ?2 d. ? 3 7
- Question : 3EX - Suppose p? must approximate p with relative error at most 10?3. Find the largest interval in which p? must lie for each value of p. a. 150 b. 900 c. 1500 d. 90
- Question : 4EX - Perform the following computations (i) exactly, (ii) using three-digit chopping arithmetic, and (iii) using three-digit rounding arithmetic. (iv) Compute the relative errors in parts (ii) and (iii). a. 4 5 + 1 3 b. 4 5
- Question : 5EX - Use three-digit rounding arithmetic to perform the following calculations. Compute the absolute error and relative error with the exact value determined to at least five digits. a. 133 + 0.921 b. 133 ? 0.499 c. (121 ? 0.327) ? 119 d. (121 ? 119) ? 0.327 e. 13 14 ? 67 2e ? 5.4 f. ?10? + 6e ? 3 62 g. 2 9
- Question : 6EX - Repeat Exercise 5 using four-digit rounding arithmetic.
- Question : 7EX - Repeat Exercise 5 using three-digit chopping arithmetic.
- Question : 8EX - Repeat Exercise 5 using four-digit chopping arithmetic.
- Question : 9EX - The first three nonzero terms of the Maclaurin series for the arctangent function are x ? (1/3)x3 + (1/5)x5. Compute the absolute error and relative error in the following approximations of ? using the polynomial in place of the arctangent: a. 4 arctan 2 1 + arctan 1 3 b. 16 arctan 1 5 ? 4 arctan 239 1
- Question : 10EX - The number e can be defined by e = ? n=0(1/n!), where n! = n(n ? 1)
- Question : 11EX - Let f (x) = x cos x ? sin x x ? sin x . a. Find limx?0 f (x). b. Use four-digit rounding arithmetic to evaluate f (0.1). c. Replace each trigonometric function with its third Maclaurin polynomial, and repeat part (b). d. The actual value is f (0.1) = ?1.99899998. Find the relative error for the values obtained in parts (b) and (c)
- Question : 12EX - Let f (x) = ex ? e?x x . a. Find limx?0(ex ? e?x)/x. b. Use three-digit rounding arithmetic to evaluate f (0.1). c. Replace each exponential function with its third Maclaurin polynomial, and repeat part (b). d. The actual value is f (0.1) = 2.003335000. Find the relative error for the values obtained in parts (b) and (c).
- Question : 13EX - Use four-digit rounding arithmetic and the formulas (1.1), (1.2), and (1.3) to find the most accurate approximations to the roots of the following quadratic equations. Compute the absolute errors and relative errors. a. 1 3 x2 ? 123 4 x + 1 6 = 0 b. 1 3 x2 + 123 4 x ? 1 6 = 0 c. 1.002x2 ? 11.01x + 0.01265 = 0 d. 1.002x2 + 11.01x + 0.01265 = 0
- Question : 14EX - Repeat Exercise 13 using four-digit chopping arithmetic.
- Question : 15EX - Use the 64-bit long real format to find the decimal equivalent of the following floating-point machine numbers. a. 0 10000001010 1001001100000000000000000000000000000000000000000000 b. 1 10000001010 1001001100000000000000000000000000000000000000000000 c. 0 01111111111 0101001100000000000000000000000000000000000000000000 d. 0 01111111111 0101001100000000000000000000000000000000000000000001
- Question : 16EX - Find the next largest and smallest machine numbers in decimal form for the numbers given in Exercise 15.
- Question : 17EX - Suppose two points (x0, y0) and (x1, y1) are on a straight line with y1 = y0. Two formulas are available to find the x-intercept of the line: x = x0y1 ? x1y0 y1 ? y0 and x = x0 ? (x1 ? x0)y0 y1 ? y0 a. Show that both formulas are algebraically correct. b. Use the data (x0, y0) = (1.31, 3.24) and (x1, y1) = (1.93, 4.76) and three-digit rounding arith metic to compute the x-intercept both ways. Which method is better and why?
- Question : 18EX - The Taylor polynomial of degree n for f (x) = ex is n i=0(xi/i!). Use the Taylor polynomial of degree nine and three-digit chopping arithmetic to find an approximation to e?5 by each of the following methods. a. e?5 ? 9 i=0 (?5)i i! = 9 i=0 (?1)i5i i! b. e?5 = 1 e5 ? 1 9 i=0 5 i! i . c. An approximate value of e?5 correct to three digits is 6.74
- Question : 19EX - The two-by-two linear system ax + by = e, cx + dy = f , where a, b, c, d, e, f are given, can be solved for x and y as follows: set m = c a , provided a = 0; d1 = d ? mb; f1 = f ? me; y = f1 d1 ; x = (e ? by) a . Solve the following linear systems using four-digit rounding arithmetic. a. 1.130x ? 6.990y = 14.20 1.013x ? 6.099y = 14.22 b. 8.110x + 12.20y = ?0.1370 ?18.11x + 112.2y = ?0.1376
- Question : 20EX - Repeat Exercise 19 using four-digit chopping arithmetic.
- Question : 21EX - a. Show that the polynomial nesting technique described in Example 6 can also be applied to the evaluation of f (x) = 1.01e4x ? 4.62e3x ? 3.11e2x + 12.2ex ? 1.99. b. Use three-digit rounding arithmetic, the assumption that e1.53 = 4.62, and the fact that enx = (ex)n to evaluate f (1.53) as given in part (a). c. Redo the calculation in part (b) by first nesting the calculations. d. Compare the approximations in parts (b) and (c) to the true three-digit result f (1.53) = ?7.61.
- Question : 22EX - A rectangular parallelepiped has sides of length 3 cm, 4 cm, and 5 cm, measured to the nearest centimeter. What are the best upper and lower bounds for the volume of this parallelepiped? What are the best upper and lower bounds for the surface area?
- Question : 23EX - Let Pn(x) be the Maclaurin polynomial of degree n for the arctangent function. Use Maple carrying 75 decimal digits to find the value of n required to approximate ? to within 10?25 using the following formulas. a. 4 Pn 2 1 + Pn 1 3 b. 16Pn 1 5 ? 4Pn 239 1
- Question : 24EX - Suppose that f l(y) is a k-digit rounding approximation to y. Show that y ? f l(y) y ? 0.5
- Question : 25EX - The binomial coefficient m k = k! (m m? ! k)! describes the number of ways of choosing a subset of k objects from a set of m elements. a. Suppose decimal machine numbers are of the form
- Question : 26EX - Let f ? C[a, b] be a function whose derivative exists on (a, b). Suppose f is to be evaluated at x0 in (a, b), but instead of computing the actual value f (x0), the approximate value, f (
- Question : 27EX - The following Maple procedure chops a floating-point number x to t digits. (Use the Shift and Enter keys at the end of each line when creating the procedure.) chop := proc(x, t); local e, x2; if x = 0 then 0 else e := ceil (evalf (log10(abs(x)))); x2 := evalf (trunc (x
- Question : 28EX - The opening example to this chapter described a physical experiment involving the temperature of a gas under pressure. In this application, we were given P = 1.00 atm, V = 0.100 m3, N = 0.00420 mol, and R = 0.08206. Solving for T in the ideal gas law gives T = PV NR = (1.00)(0.100) (0.00420)(0.08206) = 290.15 K = 17?C. In the laboratory, it was found that T was 15?C under these conditions, and when the pressure was doubled and the volume halved, T was 19?C. Assume that the data are rounded values accurate to the places given, and show that both laboratory figures are within the bounds of accuracy for the ideal gas law
- Question : 1EX - a. Use three-digit chopping arithmetic to compute the sum 10 i=1(1/i2) first by 1 1 + 1 4 +
- Question : 2EX - The number e is defined by e = ? n=0(1/n!), where n! = n(n ? 1)
- Question : 3EX - The Maclaurin series for the arctangent function converges for ?1 < x ? 1 and is given by arctan x = lim n?? P n(x) = lim n?? n i=1 (?1)i+1 x2i?1 2i ? 1 . a. Use the fact that tan ?/4 = 1 to determine the number of n terms of the series that need to be summed to ensure that |4Pn(1) ? ?| < 10?3. b. The C++ programming language requires the value of ? to be within 10?10. How many terms of the series would we need to sum to obtain this degree of accuracy?
- Question : 4EX - Exercise 3 details a rather inefficient means of obtaining an approximation to ?. The method can be improved substantially by observing that ?/4 = arctan 1 2 + arctan 1 3 and evaluating the series for the arctangent at 2 1 and at 1 3. Determine the number of terms that must be summed to ensure an approximation to ? to within 10?3
- Question : 5EX - Another formula for computing ? can be deduced from the identity ?/4 = 4 arctan 1 5 ? arctan 239 1 . Determine the number of terms that must be summed to ensure an approximation to ? to within 10?3.
- Question : 6EX - Find the rates of convergence of the following sequences as n ? ?. a. lim n?? sin 1 n = 0 b. lim n?? sin 1 2n = 0 c. lim n?? sin n 1 2 = 0 d. nlim ??[ln(n + 1) ? ln(n)] = 0
- Question : 7EX - Find the rates of convergence of the following functions as h ? 0. a. lim h?0 sin h h = 1 b. lim h?0 1 ? cos h h = 0 c. lim h?0 sin h ? h cos h h = 0 d. lim h?0 1 ? eh h = ?1
- Question : 8EX - a. How many multiplications and additions are required to determine a sum of the form n i=1 i j=1 aibj? b. Modify the sum in part (a) to an equivalent form that reduces the number of computations.
- Question : 9EX - Let P(x) = anxn + an?1xn?1 +
- Question : 10EX - Equations (1.2) and (1.3) in Section 1.2 give alternative formulas for the roots x1 and x2 of ax2 + bx + c = 0. Construct an algorithm with input a, b, c and output x1, x2 that computes the roots x1 and x2 (which may be equal or be complex conjugates) using the best formula for each root.
- Question : 11EX - Construct an algorithm that has as input an integer n ? 1, numbers x0, x1, . . . , xn, and a number x and that produces as output the product (x ? x0)(x ? x1)
- Question : 12EX - Assume that 1 ? 2x 1 ? x + x2 + 2x ? 4x3 1 ? x2 + x4 + 4x3 ? 8x7 1 ? x4 + x8 +
- Question : 13EX - . a. Suppose that 0 < q < p and that ?n = ? + O n?p . Show that ?n = ? + O n?q . b. Make a table listing 1/n, 1/n2, 1/n3, and 1/n4 for n = 5, 10, 100, and 1000, and discuss the varying rates of convergence of these sequences as n becomes large.
- Question : 14EX - a. Suppose that 0 < q < p and that F(h) = L + O (hp). Show that F(h) = L + O (hq). b. Make a table listing h, h2, h3, and h4 for h = 0.5, 0.1, 0.01, and 0.001, and discuss the varying rates of convergence of these powers of h as h approaches zero.
- Question : 15EX - Suppose that as x approaches zero, F1(x) = L1 + O(x?) and F2(x) = L2 + O(x?). Let c1 and c2 be nonzero constants, and define F(x) = c1F1(x) + c2F2(x) and G(x) = F1(c1x) + F2(c2x). Show that if ? = minimum {?, ?}, then as x approaches zero, a. F(x) = c1L1 + c2L2 + O(x?) b. G(x) = L1 + L2 + O(x?).
- Question : 16EX - The sequence {Fn} described by F0 = 1, F1 = 1, and Fn+2 = Fn+Fn+1, if n ? 0, is called a Fibonacci sequence. Its terms occur naturally in many botanical species, particularly those with petals or scales arranged in the form of a logarithmic spiral. Consider the sequence {xn}, where xn = Fn+1/Fn. Assuming that limn?? xn = x exists, show that x = (1 + ?5)/2. This number is called the golden ratio.
- Question : 17EX - The Fibonacci sequence also satisfies the equation Fn ? F
- Question : 18EX - The harmonic series 1 + 1 2 + 1 3 + 4 1 +

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