Let / be defined on [a, b] and let the nodes a xq < x\ < xi = b he given. A quadratic

Determine the natural cubic spline S thatinterpolates the data /(0) = 0, /(I) = 1, and /(2) = 2

Determine the clamped cubic spline s that interpolates the data /(0) = 0, /(I) = 1 and /(2) = 2 and satisfies ^'(O) = s'(2) = 1

Construct the natural cubic spline for the following data. a. X fix) b. X fix) 8.3 17.56492 0.8 0.22363362 8.6 18.50515 1.0 0.65809197 c. X fix) d. X fix) -0.5 -0.0247500 0.1 0.62049958 -0.25 0.3349375 0.2 0.28398668 0 1.1010000 0.3 0.00660095 0.4 0.24842440

Constructthe natural cubic pline for the following data. a. X fix) b. X fix) 0 1.00000 -0, 25 1.33203 0.5 2.71828 0.25 0.800781 c. X fix) d. X fix) 0.1 0.29004996 -1 0.86199480 0.2 0.56079734 -0.5 0.95802009 0.3 0.8I40I972 0 1.0986123 0.5 1.2943767

The data in Exercise 3 were generated using the following functions. Use the cubic splines constructed in Exercise 3 for the given value ofx to approximate f(x) and f'(x) and calculate the actual error. a. /(x) = x In x; approximate /(8.4) and /'(8.4). b. f(x) = sin(eA ' 2); approximate /(0.9) and /'(0.9). c. /(x) = x 3 + 4.001x2 -f 4.002.V + 1.101; approximate/(|) and/'(5). d. f(x) = x cosx 2x2 + 3x 1; approximate /(0.25) and /'(0.25).

The data in Exercise 4 were generated using the following functions. Use the cubic splines constructed in Exercise 4 for the given value of x to approximate /(x) and f'(x) and calculate the actual error. a. f(x) = e 2x; approximate/(0.43) and/'(0.43). b. /(x) = x 4 x 3 + x 2 x + 1; approximate /(0) and /'(0). c. f(x) = x 2 cos x 3x; approximate/((). 18) and/'(0.18). d. f(x) = ln(ex + 2); approximate/(0.25) and/'(0.25).

Construct the clamped cubic spline using the data of Exercise 3 and the fact that a. /'(8.3) = 1.116256 and /'(8.6) = 1.151762. b. /'(0.8) = 2.1691753 and /'(1.0) = 2.0466965. c. /'(0.5) = 0.7510000 and /'(0) = 4.0020000. d. /'(0.1) = 3.58502082 and /'(0.4) = 2.16529366.

Construct the clamped cubic spline using the data of Exercise 3 and the fact that a. /'(0) = 2 and /'(0.5) = 5.43656. b. /'(0.25) = 0.437500 and /'(0.25) = -0.625000. c. /'(0.I) = -2.8004996 and /'(0) = -2.9734038. d. /'(-I) = 0.15536240 and /'(0.5) = 0.45186276.

Repeat Exercise 5 using the clamped cubic splines constructed in Exercise 7.

Repeat Exercise 6 using the clamped cubic splines constructed in Exercise 8.

A natural cubic spline S on [0, 2] is defined by S(x) - 5o(x) = ] + 2x x 3 , if 0 < x < I, SKx) = 2 + b(x- l) + c(x - \)2 + dix - I)3 , if 1 < x < 2. Find b, c, and d

A natural cubic spline S is defined by S(x) = f 5o(x) = I + B(x - 1) - D(x - I)3 , if 1 < x < 2, \5i(x) = 1 +bix-2) - jix-2)2 + dix-2)\ if 2 < x < 3. If S interpolates the data (1, 1), (2, 1), and (3. 0), find B, D, b, and d.

A clamped cubic spline ,v for a function /is defined on [ 1, 3J by ,y(x) = .v0(x) 3(x 1) + 2(x I)2 (x I)3 , if I < x < 2, 5i(x) a + b(x 2) + c(x 2)2 +

A clamped cubic spline s for a function /is defined by s(x) = fi'o(x) = 1 + 5x + 2x2 2x3 , if 0 < x < 1, |.V|(x) = 1 + b(x - 1) - 4(x - I)2 + 7(x - I)3 , if 1 < x < 2. Find /'(0) and /'(2)

Given the partition xq = 0, xi = 0.05, and X2 = 0.1 of[0, 0.11, find the piecewise linear interpolating function F for /(x) = e 2 *. Approximate 1 e 2 * dx with /0 0 ' F(x) dx and compare the results to the actual value.

Given the partition xq = 0, xj 0.3, and X2 = 0.5 of|0, 0.5J, find the piecewise linear interpolating function F for fix) sin 3x. Approximate /n" 5 sin 3x dx with Fix) dx and compare the results to the actual value.

Construct a natural cubic spline to approximate fix) costtx by using the values given by fix) at x = 0,0.25,0.5,0.75, and 1.0. Integrate the spline over [0, 11 and compare the resultto f0 cosjrxdx = 0. Use the derivativesofthe spline to approximate /'(0.5) and /"(0.5). Compare these approximations to the actual values.

Construct a natural cubic spline to approximate/(x) = e - -'by using the values given by/(x)atx = 0, 0.25, 0.75, and 1.0. Integrate the spline over |0, 1J and compare the result to f0 e~x dx \ \/e. Use the derivatives ofthe spline to approximate /'(0.5) and /"(0.5). Compare the approximations to the actual values.

Repeat Exercise 17, constructing instead the clamped cubic spline with /'(0) = /'(I) = 0.

Repeat Exercise 18, constructing instead the clamped cubic spline with /'(0) = 1, /'(I) = e~{ .

Given the partition xy = 0, X| = 0.05, X2 = 0.1 of [0, 0.1J and fix) e 2x\ a. Find the cubic spline s with clamped boundary conditions that interpolates /. b. Find an approximation for /0 0 ' e 2x dx by evaluating /0 0 ' ,v(x) dx. c. Use Theorem 3.13 to estimate maxy<,

Given the partition xy = 0, X| = 0.3, X2 = 0.5 of [0. 0.5] and /(x) = sin 3x: a. Find the cubic spline 5 with clamped boundary conditions that interpolates /. b. Find an approximation for * sin 3x dx with six) dx and compare the results to the actual value.

A car traveling along a straight road is clocked at a number ofpoints. The data from the observations are given in the following table, where the time is in seconds, the distance is in feet, and the speed is in feet per second. Time 0 3 5 8 13 Distance 0 225 383 623 993 Speed 75 77 80 74 72 Use a clamped cubic spline to predict the position ofthe car and its speed when t 10 seconds. Use the derivative of the spline to determine whether the car ever exceeds a 55-mile-per-hour speed limit on the road; if so, what is the first time the car exceeds this speed? What is the predicted maximum speed for the car?

The introduction to this chapter included a table listing the population ofthe United States from 1960 to 2010. Use natural cubic spline interpolation to approximate the population in the years 1950, 1975,2014, and 2020. The population in 1950 was approximately 150,697,360, and in 2014 the population was estimated to be 317,298,000. How accurate do you think your 1975 and 2020 figures are?

It is suspected that the high amounts of tannin in mature oak leaves inhibit the growth ofthe winter moth (Operophtera hromata L, Geometridae) larvae that extensively damage these trees in certain years. The following table lists the average weight of two samples of larvae at times in the first 28 days after birth. The first sample was reared on young oak leaves, whereas the second sample was reared on mature leaves from the same tree. a. Use a natural cubic spline to approximate the average weight curve for each sample. b. Find an approximate maximum average weight for each sample by determining the maximum of the spline. Day 0 6 10 13 17 20 28 Sample 1 average weight (mg) 6.67 17.33 42.67 37.33 30.10 29.31 28.74 Sample 2 average weight (mg) 6.67 16.11 18.89 15.00 10.56 9.44 8.89

The 2014 Kentucky Derby was won by a horse named California Chrome (5:2 odds favorite) in a time of 2:03.66 (2 minutes and 3.66 seconds) for the I |-mile race. Times at the quarter-mile, half-mile, and mile poles were 0:23.04, 0:47.37, and 1:37.45. a. Use these values together with the starting time to construct a natural cubic spline for California Chrome's race. b. Use the spline to predict the time at the three-quarter-mile pole and compare this to the actual time of 1:11.80. c. Use the spline to predict California Chrome's starting speed and speed atthe finish line.

The upper portion ofthis noble beast is to be approximated using clamped cubic spline interpolants. The curve is drawn on a grid from which the table is constructed. Use Algorithm 3.5 to construct the three clamped cubic splines.

Repeat Exercise 27, constructing three natural splines using Algorithm 3.4.

Suppose that f(x) is a polynomial of degree three. Show that fix) is its own clamped cubic spline but thatit cannot be its own natural cubic spline

Suppose the data [x,, f(x/)))"=1 lie on a straight line. What can be said about the natural and clamped cubic splines for the function /? [Hint: Take a cue from the results of Exercises I and 2.J

Extend Algorithms 3.4 and 3.5 to include as output the first and second derivatives ofthe spline at the nodes.

Extend Algorithms 3.4 and 3.5 to include as output the integral ofthe spline over the interval [aq, x,,].

Let / C 2 ta, b) and let the nodes a = x < x\ < < x b he given. Derive an error estimate similar to that in Theorem 3.13 for the piecewise linear interpolating function F. Use this estimate to derive error bounds for Exercise 15.

Let / be defined on [a, b] and let the nodes a xq < x\ < xi = b he given. A quadratic spline interpolating function S consists of the quadratic polynomial 5o(a) = no + boix - xq) + co(x - xq)2 on [xq, Xi] and the quadratic polynomial S\ (x) = , + Mx - X|) + C\(x - x,)- on [xi,x2],such that i. 5(xo) = f(xo), S(xi) = f(xi), and Sfe) = /fe), ii. 5eCl [xo,x2l. Show that conditions (i) and (ii) lead to five equations in the six unknowns Aq, cq, a\, h\, and C|. The problem is to decide what additional condition to impose to make the solution unique. Does the condition S e C 2 [xo, X2J lead to a meaningful solution?

Determine a quadratic spline s that interpolates the data /(0) 0, /(I) = 1, and /(2) = 2 and satisfies ,v'(0) 2.