BT-401 Mathematics III - Nov 2023 Question Paper

Under each question, How to answer lists the points an answer should cover, outlined by CampusPrep from the syllabus. Check the details against your textbook.

1. 7 Marks
   aEvaluate $\sqrt{12}$ to four decimal places by Newton Raphson Method.
   How to answer

   • Define f(x)=x^2-12 and Newton-Raphson iteration x_{n+1}=x_n-f/f'
   • Choose initial approximation near 3.464
   • Iterate till stable to 4 decimal places
   • Obtain sqrt(12) approx 3.4641
2. 7 Marks
   bWhat is the rate of convergence of bisection method?
   How to answer

   • Define rate of convergence via error relation e_{n+1} ~ C e_n^p
   • Show bisection halves interval each step
   • Conclude order p=1 linear convergence
   • Give error bound factor 1/2
3. 7 Marks
   aProve that i) $\Delta = \frac{1}{2}\delta^{2} + \delta\sqrt{1+\left(\frac{\delta^{2}}{4}\right)}$ ii) $\Delta + \nabla = \frac{\Delta}{\nabla} - \frac{\nabla}{\Delta}$
   How to answer

   • Recall definitions of forward, backward, central differences
   • Express shift operator E in terms of delta/del
   • Derive operator identities algebraically
   • Verify both given relations
4. 7 Marks
   b

   Construct a backward difference table for $y = \log x$ given that

   $x$	10	20	30	40	50
   $y$	1	1.3010	1.4771	1.6021	1.6990

   And find values of $\nabla^{3} \log 40$ and $\nabla^{4} \log 50$

   How to answer

   • Tabulate y=log x for x=10,20,30,40,50
   • Compute successive backward differences nabla, nabla^2, etc.
   • Form backward difference table
   • Read off nabla^3 log40 and nabla^4 log50
5. 7 Marks
   aUsing following table, by Lagrange's Method find $f(x)$ as a polynomial in $x$:

   How to answer

   • Write Lagrange basis polynomials for given nodes
   • Multiply by f-values and sum
   • Simplify to polynomial in x
   • Verify at given points
6. 7 Marks
   bUsing Newton's divided difference formula, calculate the value of $f(6)$ from the following data:

   How to answer

   • Form divided difference table for given data
   • Write Newton divided difference polynomial
   • Substitute x=6
   • Compute f(6)
7. 7 Marks
   aFind $f'(1.1)$ and $f''(1.1)$ from the following table:

   How to answer

   • Use numerical differentiation formulas for equally spaced data
   • Apply first and second derivative formulas at x=1.1
   • Substitute table values with h=0.2
   • Compute f'(1.1) and f''(1.1)
8. 7 Marks
   bFind missing values in the following table:

   How to answer

   • Assume polynomial and higher differences zero
   • Build difference table with unknowns
   • Solve for missing y at x=10 and x=20
   • Verify consistency
9. 7 Marks
   aFind $\int_{0}^{6} \frac{e^{x}}{1+x} dx$ approximately using simpson's $\frac{3}{8}$th rule on integration.
   How to answer

   • Identify integrand e^x/(1+x) on [0,6]
   • Apply Simpson's 3/8 rule with suitable n multiple of 3
   • Compute ordinates and weighted sum
   • Obtain approximate integral value
10. 7 Marks
    bSolve the equation $\frac{dy}{dx} = 1 - y$ with initial condition $y(0) = 0$ using Euler's modified method and tabulate the solution at $x = 0.1, 0.2, 0.3$.
    How to answer

    • Write Euler modified predictor-corrector formulas
    • Start with y(0)=0, h=0.1
    • Iterate for x=0.1,0.2,0.3
    • Tabulate results
11. 7 Marks
    aUsing Lagrange's interpolation formula to fit a polynomial to the data:

    How to answer

    • Write Lagrange interpolation formula
    • Substitute nodes -1,0,2,3 and values
    • Simplify to polynomial
    • State fitted polynomial
12. 7 Marks
    bUsing Regula-Falsi method, find the real root of $x \log_{10} x = 1.2$, correct to four decimal places.
    How to answer

    • Define f(x)=x log10 x -1.2
    • Choose bracketing initial guesses
    • Apply Regula-Falsi iteration formula
    • Iterate to four decimal accuracy
13. 7 Marks
    aSolve $\frac{dy}{dx} = x + y^{2}$ by using Runge-Kutta method of fourth order to find an approximate value of $y$ for $x = 0.2$, given that $y = 1$ when $x = 0$. (Take $h = 0.1$)
    How to answer

    • Identify f(x,y)=x+y^2, x0=0,y0=1,h=0.1
    • Compute k1,k2,k3,k4 for two steps
    • Apply RK4 update to reach x=0.2
    • Report approximate y
14. 7 Marks
    bEvaluate $$L^{-1}\left[\frac{s+7}{s^{2}+4s+8}\right]$$
    How to answer

    • Write denominator s^2+4s+8 as (s+2)^2+4
    • Split (s+7) into (s+2)+5
    • Use shifting and sine-cosine inverse formulas
    • Obtain e^{-2t}(cos2t + ... sin2t)
15. 7 Marks
    aFind the inverse Laplace transform of $\frac{s+4}{s(s-1)(s^{2}+4)}$.
    How to answer

    • Do partial fraction decomposition of (s+4)/[s(s-1)(s^2+4)]
    • Invert each term using standard L^{-1} formulas
    • Combine exponential, constant, sine-cosine terms
    • State final f(t)
16. 7 Marks
    bProve that if $L\{f(t)\} = F(s)$ then $L\left\{\frac{1}{t}f(t)\right\} = \int_{s}^{\infty} F(s) ds$ hence, evaluate $\frac{e^{at} - \cos bt}{t}$.
    How to answer

    • Start from definition L{f(t)}=∫ e^{-st}f(t)dt
    • Interchange order to prove L{f(t)/t}=∫_s^∞ F(u)du
    • Apply to e^{at}-cos bt
    • Evaluate integral to get log expression