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MDB 3053: Numerical Methods (May 2016 Sem.)Assignment #1 Due: 15 Jun 2016 (Wed)

Q1

Newtons law of cooling says that the temperature of a body changes at a rate proportional to the difference between itstemperature and that of the surrounding medium (the ambient temperature),

where T = the temperature of the body (C), t = time (min),

k = the proportionality constant (per minute), and Ta = the ambient temperature (C).

Suppose that a cup of coffee originally has a temperature of 68C. Use Eulers method to compute the temperature from t = 0 to

10 min using a step size of 1 min if Ta = 21C and k = 0.1/min.

Eulers method:

Q2

Determine the number of terms necessary to approximate to 4 significant figures using the Maclaurin series approximation

Calculate the approximation using a value ofx = 0.3.

Q3

(a) Evaluate the polynomialy=x36.2x

2+ 8x1.5 atx= 1.37. Use 3-digit arithmetic (3 S.F.) with chopping. Evaluate the true

percent relative error t.

(b) Repeat (a) by using 3-digit arithmetic (3 S.F.) with rounding. Evaluate the error t. Comment your results between part (a)and (b).

(c) Repeat (a) but express y in nested form as y = ((x6.2)x+ 8)x1.5 . Evaluate the error t and compare with part(a).

Comment your results.Ans: (a) 0.47 (b) 0.37 (c) 0.40

Q4

Use zero- through third-order Taylor series expansions to predictf(4) forf(x) = 20x36x2+ 7x50 using a base point at x=1.Compute the tfor each approximation.

Ans: 0thorder: -29, 2ndorder: 622

Q5

Given the function: f(x) = 25x36x2+ 7x- 88

(a) Use the forward, backward and centered difference approximations to evaluate the first derivative of the function at xi= 2

using a step size of h = 0.25. Compute the tand interpret your results.

(b) Calculate the remainder term O(h) for forward and backward difference approximation and remainder term O(h2) for centered

difference approximation (recall Eq. 4.15 & 4.16 in textbook). Compare the values the remainder term with the true error |E t|

(recall Eq.3.2) for each case and comment your results.

Ans: (b) forward: O(h)=36,|Et|=37.6 ; centered: O(h2)=1.562, |Et|=1.56

Q6

Estimate the uncertainty for the following case:

(a) A wall has a measured heightH= 2.00 0.03 m. The wall has a light switch at a height h= 0.88 0.04 measured from thebase of the wall. Find the distance,Dfrom the switch to the top of the wall and its uncertainty.

(b) A car travels a distance d= 1120 3 m during a time t= 20.0 1.2 s. Evaluate the average speed Vof the car and its

uncertainty.

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(c) The period of an oscillating pendulum is measured to be T = 0.20 0.01 s. Find the frequencyf of the oscillation and its

uncertainty.(d) Use ideal gas law to calculate the temperature in Kelvin (with its uncertainty) of 1 mol of an ideal gas at P = 275.00 kPa andV = 12.500 L. The uncertainties in the pressure and volume measurements are 0.04 kPa and 0.005 L, respectively. Constant R =8.314 L.kPa/(mol .K)

Q7

Determine the real root of (a) Graphically using hand or excel plot.

(b) Using bisection to determine the root to s= 10%. Employ initial guesses of = 0.5 and = 1.0.

Ans: (b) after i=5, xr=0.5312

Q8

Determine the real root of .(a) analytically, and

(b) with the false-position method to within = 2.5% and 4 significant figures. Use initial guesses of 2.0 and 5.0.

Ans: (b) after i=4, xr=3.4435