Quantitative Methods Answers

The force F on a beam of length 5m is given by

𝐹= ∫⁵₀ 𝑒𝑥/1+ √𝑥

Find the value of F using the trapezium rule, Simpsons rule and the mid ordinate rule with a suitable number of intervals.

4. The equation for the velocity of a car is given as

Y = √𝑥 + 1𝑥

Use the trapezium rule, Simpson’s rule and the mid ordinate rule ( with a suitable number of intervals ) to find the distance travelled between 1 and 6 seconds.

Compare your answers to the value given by the definite integral and comment on the accuracy of the numerical methods.

Given sin250=0.42262, sin260=0.43837, sin270=0.45399, sin280=0.46947,

sin290=0.48481, sin300=0.5.

Using Newton interpolation formula find sin28024′. Estimate the error. 

Find the real root of the equation 𝑥𝑒 𝑥 = 𝑠𝑖𝑛𝑥 correct to three decimal places by regula-falsi method

Given the nodes x0 < x1 < · · · < xn, let V be the vector space of functions that are twice continuously differentiable at each node xi and cubic polynomial on (−∞, x₀), (x_n, ∞), and each of the intervals (xi , xi+1). Show that any function s(x) ∈ V can be uniquely represented as

s(x) = a0 + a₁x + a₂x^2 + a₃x^3 + "summation" (i=0 to n) c_i (x − xi) ^3 +

where (x − xi)^3₊ is 0 for x ≤ xi and (x − xi)^3 otherwise. Conclude that this vector space has dimension n + 5.

Use Simpson's Rule to estimate  ∫_0^2▒1/8 e^(x^2 ) dx with a maximum error of 0.1

Given that f(2)=4,f(2.5)=5.5 obtain the linear interpolation polynomial using Newton's Divided Difference

A prototype automotive tire has a design life of 38,500 miles with a standard deviation of 2,500 miles. Five such tires are manufactured and tested. On the assumption that the actual population mean is 38,500 miles and the actual population standard deviation is 2,500 miles, find the probability that the sample mean will be less than 36,000 miles. Assume that the distribution of lifetimes of such tires is normal.

Given a function described as equation y = 3x + 4, what is y when x is 1, 2, and 3?

(a) Solve the following equation correct to 5 significant figures using Newton’s Method. 4x−sin(x 2 ) +e −2x −x 3 −x 2 +2 = 0 Take the initial guess as x0 = 1.75.

(b) Solve the following equation correct to 5 significant figures using Secant Method. e 2x −x 2 −x−7 = 0 The root must be in the interval [1,1.3].

(c) Solve the following equation correct to 5 significant figures using Bisection Method. x sin(2x) +e 2x +3x−3 = 0 The root must be in the interval [0,0.5].

(d) [ The Newton-Raphson Method ] Perform two iterations to find a real root of the equations y 2 −5y+4 = 0 3yx2 −10x+7 = 0 using the Newton-Rhaphson method and let x0 = y0 = 0.5.

(e) Find a recurrence formula for solving √7 6 and hence approximate its value correct to 5 significant figure. HINT: Apply Newton-Raphson Method