Math 1432 Section 12881

James West

jdwest@math.uh.edu 620 PGH

Office Hours in 222 Garrison (CASA):

Tuesday 2:30 – 4:30 p.m.

Thursday 2:30 – 4:30 p.m.

Rational Functions and

Partial Fraction Decomposition

Section 8.5

Rational functions are defined as functions in the form

( ) ( )( )

=F x

R xG x

, where F(x) and G(x) are polynomials.

Rational functions are said to be proper if the degree of the numerator is less then the degree of the denominator (otherwise they are improper).

Theorem: If F(x) and G(x) are polynomials and the degree of F(x) is larger than or equal to the degree of G(x), then there are polynomials q(x) (quotient) and r(x) (remainder) such that

where the degree of r(x) is smaller than the degree of G(x).

( )( ) ( ) ( )

( )F x r x

xG x G x

q= +

Example:

Write +

− −

5

3 2

x 1

x x 2x in terms of its quotient and remainder.

Integrate

��� + 3�� − � − 1� − �� + � − 1 �

Section 8.7 Numerical Integration

(an introduction)

Recall: 1. The left hand endpoint method. 2. The right hand endpoint method. 3. The midpoint method. New: 4. The trapezoid method. 5. Simpson’s method

Left Hand Endpoint Method Using n subdivisions, approximate . The width of each rectangle is . The height of each rectangle is the function value of the x value on the LEFT side of the rectangle. If the function is positive, the top left corner of the rectangle is ON the curve.

( )b

ax dxf∫

b a

n

−

1111 2222 3333 4444 5555

5555

10101010

15151515

20202020Using n = 4, approximate using the Left Hand Endpoint Method.

( )5 2

1x 1 dx−∫

Right Hand Endpoint Method Using n subdivisions, approximate . The width of each rectangle is . The height of each rectangle is the function value of the x value on the RIGHT side of the rectangle. If the function is positive, the top right corner of the rectangle is ON the curve.

( )b

ax dxf∫

b a

n

−

1111 2222 3333 4444 5555

5555

10101010

15151515

20202020Using n = 4, approximate using the Right Hand Endpoint Method.

( )5 2

1x 1 dx−∫

Midpoint Method Using n subdivisions, approximate . The width of each rectangle is . The height of each rectangle is the function value of the x value at the MIDPOINT of the width of the rectangle. The midpoint of the side of the rectangle is ON the curve.

( )b

ax dxf∫

b a

n

−

1111 2222 3333 4444 5555

5555

10101010

15151515

20202020Using n = 4, approximate using the Midpoint Method.

( )5 2

1x 1 dx−∫

General Formulas to approximate

( )f∫b

ax dx

Left Hand Endpoint Method:

( ) ( ) ( )f f f −− = + + ⋅ ⋅ ⋅ + n 0 1 n 1

b aL x x x

n

Right Hand Endpoint Method:

( ) ( ) ( )f f f− = + + ⋅ ⋅ ⋅ + n 1 2 n

b aR x x x

n

Midpoint Method:

f f − + + −= + ⋅ ⋅ ⋅ +

0 1 n 1 nn

x x x xb aM

n 2 2

Trapezoid Method

Area of a trapezoid =�� ��� + ��� In this method, instead of using rectangles, we use trapezoids. The heights are the parallel sides. Their lengths are the function values. In our graphs, the base is going to be the length of the subdivision .

b a

n

−

1111 2222 3333 4444 5555

5555

10101010

15151515

20202020Using n = 4, approximate using the Trapezoid Method.

( )5 2

1x 1 dx−∫

Use the trapezoid method with n = 2 to approximate and give the result.

1

20

1dx

x 1+∫

The trapezoid method (trapezoidal rule) General Formula to approximate

Note: This is the average of the left hand estimate and the right hand estimate.

( )b

ax dxf∫

Trapezoid Rule Error Estimate From your book: As is shown in texts on numerical analysis, if f is continuous on [a, b]

and twice differentiable on (a, b), then the theoretical error of the

trapezoidal rule,

( )bTn na

E x dx Tf= −∫ ,

Can be written

(8.7.1) ( ) ( )

3

Tn 2

b aE c

12nf ''

−= − ,

where c is some number between a and b. Usually we cannot pinpoint c

any further. However, if f ’’ is bounded on [a, b], say ( )x Mf '' ≤ for

a x b≤ ≤ , then

8.7.2 ( )3

Tn 2

b aE M

12n

−≤ .

Example: Give a value of n that will guarantee the Trapezoid

method approximates ( )sinπ

∫ 20

2x dx within −410 .

( )3

Tn 2

b aE M

12n

−≤

Give a value of n that will guarantee the Trapezoid method

approximates +∫

1

0

1dx

x 1 within −210 .

( )3

Tn 2

b aE M

12n

−≤

Simpson’s Method Fit a parabola to every section.

Which is just (one third the trapezoidal estimate plus two thirds the midpoint estimate)

n n n

1 2T M S

3 3+ =

Use Simpson’s method with n = 3 to approximate .

1

0

1dx

x 1+∫

Give a value of n that will guarantee Simpson’s method

approximates ( )sinπ

∫ 2

02x dx within 410− .

( )−≤

5

Sn 4

b aE M

2880n where

( )( )4x Mf ≤ for a x b≤ ≤ .

( )( )( )( )

( )( )

f sin

f ' cos

f '' sin

f ''' cos

f sin4

x 2x

x 2 2x

x 4 2x

x 8 2x

x 16 2x

=== −= −

=