repeated-linear fractions

Partial Fractions

The Strange exceptions of partial fraction

(i.e. repeated-linear fractions is illustrated here. 

For a repeated linear fraction, i.e. (1 + x)2

the format will look something like:

              A              B               C

fraction = ------------ + ---------   +   ---------

            expression    (1 + x)         (1 + x)²

Example: 

Express as a partial fraction.

           x

    ---------------

    (1-x)(x + 1)²

      x             A         B         C

--------------- = ------- + ------- + --------

(1-x)(x + 1)²     (1 - x)   (x + 1)   (x + 1)²

multiply by the denominator.

x = A(x + 1)² + B(1 - x)(x + 1) + C(1 - x)

multiply out (in this case its only the 'A' term).

x = A(x +1)(x +1) + B(1 - x)(x + 1) +   C(1 - x)

x = Ax2 + 2Ax + A + Bx – Bx2 + B –Bx + C – Cx

solve for the coefficient:

A = 1/4,  B = 1/4, C = -1/2 

   x                1           1          1

--------------- = -------- - --------- - ---------

(1 - x)(x + 1)²   4(1 - x)    4(x + 1)   2(x + 1)²

More

Engineering math 1 General equation

Below is a list of general equation that might help in sketching the curve or surfaces.

1) Elliptic paraboloid x^2 / a^2 + y^2/b^2 = z/c
where z determine the axis upon which the paraboloid opens up.

The sign of c determine the direction that paraboloid opens up,

2) Hyperboloid paraboloid ---- x^2/a^2 - y^2/b^2 = z/c
Agian c will determine the direction that the surface opens up.







3) Ellipsoid --- x^2/a^2 + y^2/b^2 + z^2/c^2 = 1
Note: if a = b = c then it becomes a sphere.




4) Cone ---- x^2/a^2 + y^2/b^2 = z^2 / c^2
where the variable at RHS, z in this case, determine the axis that the cone opens up along.

5) Ellipse --- a(x-b)^2 + c(y-d)^2 = r^2
Note; If a = c, It become circle with center at (b,d) and radius, r



Hope this will help you to solve your tutorial question on level curves and surfaces.

Tutorial solution for Q17i g

Below is the solution for Q17i part g.

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Tutorial solution for Q17 i abcd

Below is the solution for Q17 i part a, b, c and d.

Tutorial Solution for Q13 to Q16

Click here for the tutorial Q13 to Q16 solution.

Tutorial Answer for Q1 to Q12

Please click here to download the answer for Q1 to Q12. The remaining will be uploaded soon.

Good luck for your mid term test tmrw. :)

Tutorial Q17ii_f

Solution for Q17ii_f

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Tutorial Solution

Solution for Q17ii (d)
Solution for Q17ii (e)

Click here for the common integrals Table

Tutorial solution for Q17ii part c

Below is the solution for Q17 ii part c

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Tutorial solution for Q17 ii b

Below is the solution for Q17ii part b.

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Tutorial Solution for Q17 part II (a)

Below is the solution for Q17 part ii, (a).

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Tutorial Solution for Q17_f

Below is the solution for Q17 part f.

Tutorial solution for Q16 part a

Below is the solution for the tutorial Q16 part a.


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Tutorial Solution for Q14 part v

Below is the solution for Q14 part v. Please feel free to drop me an e-mail or my office if you have any problem in this solution. Thank You.






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Engineering Mathematics I

Engineering Mathematics I


Objective

This course covers calculus of one variable, techniques of differentiation and integration and their applications. Sequences and series and some of the convergence tests. Extension of the concepts of differentiation and integration to two and more variables by the study of partial differentiation. Taylor series, Maclaurin series and Fourier series. Concepts of complex functions and vectors. Concepts of probability, distributions, mean, variance.


Syllabus

1. Calculus of One Variable ( 7 hrs)
Functions, inverse trigonometric functions and principal values, hyperbolic and inverse hyperbolic functions, graphs.
Concepts of continuity and differentiability. Mean-value theorem. Taylor’s series expansion. Integration by parts.

2. Sequences and Series ( 6 hrs)
Sequences of real numbers, monotone sequence, convergence and limits.
Infinite series, convergence tests, addition and multiplication of series. Power series, radius of convergence, term-by-term integration and differentiation.

3. Partial Differentiation ( 6hrs)
Functions of several variables, continuity and partial derivatives. Total differentials, approximate calculations using differentials. Chain rule. Implicit differentiation. Series representation of functions, Taylor’s Theorem.
Extremum problems, without and with constraints, Lagrange multipliers, global extremum.

4. Complex Function and Vector Algebra ( 6 hours)
DeMoivre’s theorem, powers and roots of complex numbers. Euler formula. Elementary functions of a complex variable, polynomials, rational, exponential, trigonometric, hyperbolic, logarithmic, inverse trigonometric and inverse hyperbolic functions.
Dot and cross products, triple products, lines and planes.
Vectors in Rn space, addition and scalar multiplication, linear combination of vectors, idea of linear dependence and independence.

5. Fourier Series ( 5 hours)
Periodic functions, trigonometric series. Fourier series, functions of period 2p, Fourier coefficients, Parsevals theorem, Functions of arbitrary period, even and odd functions. Half range expansion. Complex form of Fourier series.

6. Probability ( 5 hours)
Probability space. Probability theorems. Conditional probability and independence. Random variables, discrete and continuous distributions, mean and variance. Bernouli, Binomial, Poisson, hypergeometric, exponential, normal distributions and their characteristics. Examples involving experimental measurement and reliability.


Grading Policy
Final exam:
60%

Coursework:
Test 1 20%
Test 2 20%


References
1. Lecture Notes Series – Engineering Mathematics Volume 1 (2nd edition), Pearson Prentice Hall, 2006.( main reference)

2. Calculus - 6th edition, James Stewart, Thomson Learning - Brooks/Cole, 2008.

3. Thomas' Calculus - 10th edition, Finney, Weir and Giordano, Addison-Wesley Publishing, 2001.

4. Calculus - A New Horizon - 6th edition, Howard Anton, John Wiley & Sons Inc, 1999.

5. Modern Engineering Mathematics - 3rd edition, Glyn James, Prentice Hall, 2001.

6. Advanced Engineering Mathematics - 8th edition, Erwin Kreysig, John Wiley & Sons Inc, 1999.

7. Probability and Statistics for Engineers and Scientists - 7th edition, R.E.Walpole, R.H.Myers, S.L.Myres and K. Ye, Prentice Hall, 2002.

8. Schaum's 3000 Solved Problems in Calculus, E. Mendelson, McGraw Hill, 1990.