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About 3307 results. 1294 free access solutions
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| m53460 | Construct and graph the cubic Bézier polynomials given the following points and guide points.
a. Point (1, 1) with guide point (1.5, 1.25) to point (6, 2) with guide point (7, 3)
b. Point (1, 1) with guide point (1.25, 1.5) to point (6, 2) with guide point (5, 3)
c. Point (0, 0) with guide point (0.5, 0.5) to point (4, 6) with entering guide point (3.5, 7) and exiting guide point (4.5, 5) to point (6, 1) with guide point (7, 2)
d. Point (0,0) with guide point (0.5, 0.5) to point (2,1) with entering guide point (3, 1) and exiting guide point (3, 1) to point (4, 0) with entering guide point (5, 1) and exiting guide point (3,−1)
to point (6,−1) with guide point (6.5,−0.25) |
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| m53466 | Construct the clamped cubic spline using the data of Exercise 3 and the fact that
a. f (8.3) = 3.116256 and f (8.6) = 3.151762
b. f (0.8) = 2.1691753 and f (1.0) = 2.0466965
c. f (−0.5) = 0.7510000 and f (0) = 4.0020000
d. f (0.1) = 3.58502082 and f (0.4) = 2.16529366 |
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| m53467 | Construct the clamped cubic spline using the data of Exercise 4 and the fact that
a. f (0) = 2 and f (0.5) = 5.43656
b. f (−0.25) = 0.437500 and f (0.25) = −0.625000
c. f (0.1) = −2.8004996 and f (0) = −2.9734038
d. f (−1) = 0.15536240 and f (0.5) = 0.45186276 |
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| m53468 | Construct the Lagrange interpolating polynomials for the following functions, and find a bound for the absolute error on the interval [x0, xn].
a. f (x) = e2x cos 3x, x0 = 0, x1 = 0.3, x2 = 0.6, n = 2
b. f (x) = sin(ln x), x0 = 2.0, x1 = 2.4, x2 = 2.6, n = 2
c. f (x) = ln x, x0 = 1, x1 = 1.1, x2 = 1.3, x3 = 1.4, n = 3
d. f (x) = cos x + sin x, x0 = 0, x1 = 0.25, x2 = 0.5, x3 = 1.0, n = 3 |
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| m53485 | Define A = PDP−1 for the following matrices D and P. Determine A3.
a.
b.
c.
d. |
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| m53495 | Derive a method for approximating f (x0) whose error term is of order h2 by expanding the function f in a fourth Taylor polynomial about x0 and evaluating at x0 ± h and x0 ± 2h. |
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| m53496 | Derive an O(h4) five-point formula to approximate f (x0) that uses f (x0 − h), f (x0), f (x0 + h), f (x0 + 2h), and f (x0 + 3h). [Consider the expression Af (x0 − h) + Bf (x0 + h) + Cf (x0 + 2h) + Df (x0 + 3h). Expand in fourth Taylor polynomials, and choose A, B, C, and D appropriately.] |
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| m53498 | Derive Milne s method by applying the open Newton-Cotes formula (4.29) to the integral |
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| m53499 | Derive Simpson s method by applying Simpson s rule to the integral |
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| m53500 | Derive Simpson s rule with error term by using
Find a0, a1, and a2 from the fact that Simpson s rule is exact for f (x) = xn when n = 1, 2, and 3. Then find k by applying the integration formula with f (x) = x4. |
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| m53501 | Derive Simpson s three-eighths rule (the closed rule with n = 3) with error term by using Theorem 4.2. |
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| m53502 | Derive the Adams-Bashforth Three-Step method by the following method. Set
y(ti+1) = y(ti) + ahf (ti , y(ti)) + bhf (ti−1, y(ti−1)) + chf (ti−2, y(ti−2)).
Expand y(ti+1), f (ti−2, y(ti−2)), and f (ti−1, y(ti−1)) in Taylor series about (ti , y(t=)), and equate the coefficients of h, h2 and h3 to obtain a, b, and c. |
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| m53503 | Derive the Adams-Moulton Two-Step method and its local truncation error by using an appropriate form of an interpolating polynomial. |
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| m53516 | Determine a quadratic spline s that interpolates the data f (0) = 0, f (1) = 1, f (2) = 2 and satisfies S (0) = 2. |
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| m53518 | Determine a singular value decomposition for the matrices in Exercise 1.
In exercise
a.
b.
c.
d. |
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| m53519 | Determine a singular value decomposition for the matrices in Exercise 2.
In exercise
a.
b.
c.
d. |
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| m53520 | Determine A4 for the matrices in Exercise 3.
In exercise
a.
b.
c.
d. |
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| m53521 | Determine all degree 2 Pade approximations for f (x) = e2x. Compare the results at xi = 0.2i, for i = 1, 2, 3, 4, 5, with the actual values f (xi). |
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| m53522 | Determine all degree 3 Padé approximations for f (x) = x ln(x+1). Compare the results at xi = 0.2i, for i = 1, 2, 3, 4, 5, with the actual values f (xi). |
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| m53526 | Determine the discrete least squares trigonometric polynomial Sn(x) on the interval [−π, π] for the following functions, using the given values of m and n:
a. f (x) = cos 2x, m = 4, n = 2
b. f (x) = cos 3x, m = 4, n = 2
c. f (x) = sin x/2 + 2 cos x/3 , m = 6, n = 3
d. f (x) = x2 cos x, m = 6, n = 3 |
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