1. Derive the two-step Adams-Moulton formula yi+1 = yi + h 12 (5fi+1 + 8fi − fi−1). 2. Use Gauss-Chebyshev quadrature to show that Z 1 0 (2t − 1)4 √ 4t − 4t 2 dt = 3π 16 . 3. The bessel function of the first kind for the integer order n has the integral representation Jn(x) = 1 π Z π 0 cos(nt − x sin t)dt. Use a 6 × 6 Romberg quadrature to draw the graph of Jn(x) on 0 ≤ x ≤ 10, for n ∈ {0, 1, 2, 3, 4}. Set ∆x = 0.1. Hint: The bessel function usually has a built-in command. You may use this to check your work. 4. In a circuit with impressed voltage E(t) and inductance L, Kirchoff’s first law gives E(t) = L di dt + Ri, where R is the resistance in the circuit and i is the current. Suppose we measure the current for several values of t and obtain t 0.98 0.99 1 1.01 1.02 i 3.1 3.12 3.14 3.18 3.28 where t is measured in seconds, i is in amperes, the inductance L is constant 0.98 henries, and the resistance R is 0.142 ohms. Use a five-point formula to approximate E(1). 5. The irreversible chemical reaction in which two molecules of solid potassium dichromate (K2Cr2O7), two molecules of water (H2O), and three atoms of solid sulfur (S) combine to yield three molecules of the gas sulfur dioxide (SO2), four molecules of solid potassium hydroxide (KOH), and two molecules of solid chromic oxide (Cr2O3) can be represented symbolically by the stoichiometric equation: 2K2Cr2O7 + 2H2O + 3S → 4KOH + 2Cr2O3 + 3SO2. If n1 molecules of K2Cr2O7, n2 molecules of H2O, and n3 molecules of S are originally available, the following differential equation describes the amount x(t) of KOH after time t: dx dt = k ? n1 − x 2 ?2? n2 − x 2 ?2? n3 − 3x 4 ?3 , where k is the velocity constant of the reaction. If k = 6.22 × 10−19, n1 = n2 = 2 × 103 , and n3 = 3 × 103 , use the implicit Euler method to determine how many units of KOH will have been formed after 0.2 second. Set h = 0.0001. 6. Consider two bodies of masses µ = 0.012277471 and µˆ = 1 − µ (earth and sun) in a planar motion, and a third body of ”negligible” mass (moon) moving in the same plane. The motion is governed by