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.

Click here to download.

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

Click here to learn more about partial fraction.