Using python use the one and two-point Gaussian method to calculate the following integralCalculate the first three time steps ( and using the Forward Euler method, Mid-Point method and Heun’s method on the following 1st order differential equation using .

(Numerical analysis) Here’s a challenging problem for those who know a little calculus. The Newton-Raphson method can be used to find the roots of any equation y(x)=0. In this method, the (i+1)stapproximation,xi+1,toarootofy(x)=0 is given in terms of the ith approximation, xi, by the following formula, where y’ denotes the derivative of y(x) with respect to x: xi+1=xiy(xi)/y(xi) For example, if y(x)=3x2+2x2,theny(x)=6x+2 , and the roots are found by making a reasonable guess for a first approximation x1 and iterating by using this equation: xi+1=xi(3xi2+2xi2)/(6xi+2) a. Using the Newton-Raphson method, find the two roots of the equation 3x2+2x2=0. (Hint: There’s one positive root and one negative root.) b. Extend the program written for Exercise 6a so that it finds the roots of any function y(x)=0, when the function for y(x) and the derivative of y(x) are placed in the code.

(Thermodynamics) The work, W, performed by a single piston in an engine can be determined by this formula: W=Fd F is the force provided by the piston in Newtons. d is the distance the piston moves in meters. a. Determine the units of W by calculating the units resulting from the right side of the formula. Check that your answer corresponds to the units for work listed in Table 1.1. b. Determine the work performed by a piston that provides a force of 1000 N over a distance of 15 centimeters.

(Civil eng.) The maximum load that can be placed at the end of a symmetrical wooden beam, such as the rectangular beam shown in Figure 2.20, can be calculated as the following: L=S1dc L is the maximum weight in lbs of the load placed on the beam. S is the stress in lbs/in2. I is the beam’s rectangular moment of inertia in units of in4. d is the distance in inches that the load is placed from the fixed end of the beam (the “moment arm”). c is one-half the height in inches of the symmetrical beam. For a 2” × 4” wooden beam, the rectangular moment of inertia is given by this formula: I=baseheight3=12=24312=10.674 c=(4in)=2in a. Using this information, design, write, compile, and run a C++ program that computes the maximum load in lbs that can be placed at the end of an 8-foot 24 wooden beam so that the stress on the fixed end is 3000lb/in2. b. Use the program developed in Exercise 9a to determine the maximum load in lbs that can be placed at the end of a 3” × 6” wooden beam so that the stress on the fixed end is 3000lb/in2.

(Oceanography) The pressure, P, exerted on an underwater object can be determined by this formula: P=gh is the density of water, which is 1.94slug/ft3 . g is the acceleration caused by Earth’s gravity, which is 32.2ft/sec2. h is the object’s depth in the water in feet. a. Determine the units of P by calculating the units resulting from the right side of the formula. Check that your answer corresponds to the units for pressure listed in Table 1.1. b. Determine the pressure on a submarine operating at a depth of 2500 feet.

(Statics) An annulus is a cylindrical rod with a hollow center, as shown in Figure 6.7. Its second moment of inertia is given by this formula: I4(r24r14) I is the second moment of inertia (m4). r2 is the outer radius (m). r1 is the inner radius (m). a. Using this formula, write a function called annulusMoment ( ) that accepts two double-precision numbers as parameters (one for the outer radius and one for the inner radius), calculates the corresponding second moment of inertia, and displays the result. b. Include the function written in Exercise 5a in a working program. Make sure your function is called from main(). Test the function by passing various data to it.

(Practice) Determine the value of the following expressions, assuming a=5,b=2,c=4,d=6,ande=3: a.abb.a!=bc.db==cbd.ac!=dbe.db==cef.!( ab)g.!( abc)h.!( cba)i.bca

(Automotive) a. An automobile engine’s performance can be determined by monitoring its rotations per minute (rpm). Determine the conversion factors that can be used to convert rpm to frequency in hertz (Hz), given that 1rotation=1cycle,1minute=60seconds,and1Hz=1cycle/sec. b. Using the conversion factors you determined in Exercise 7a, convert 2000 rpm into hertz.

(Chemistry) a. Determine the final units of the following expression, which provides the molecular weight of 1.5 moles of hydrogen peroxide: 1.5moles34.0146grams/mole b. Determine the final units of the following expression, which provides the molecular weight of 5.3 moles of water: 5.3moles18grams/mole

(Mechanics) The deflection at any point along the centerline of a cantilevered beam, such as the one used for a balcony (see Figure 5.15), when a load is distributed evenly along the beam is given by this formula: d=wx224EI(x2+6l24lx) d is the deflection at location x (ft). xisthedistancefromthesecuredend( ft).wistheweightplacedattheendofthebeam( lbs/ft).listhebeamlength( ft). Eisthemodulesofelasticity( lbs/f t 2 ).Iisthesecondmomentofinertia( f t 4 ). For the beam shown in Figure 5.15, the second moment of inertia is determined as follows: l=bh312 b is the beam’s base. h is the beam’s height. Using these formulas, write, compile, and run a C++ program that determines and displays a table of the deflection for a cantilevered pine beam at half-foot increments along its length, using the following data: w=200lbs/ftl=3ftE=187.2106lb/ft2b=.2fth=.3ft