Math 1432 Notes Week 6

Important: Second exam is March 2, 3 or 5. Schedule under proctored exams on CASA. If you are more than 100 miles from campus, you should have contacted DEServices

about taking your exam remotely. If you are taking your exam in CSD, you should have already sent me your RITA form.

(if not, you must do so ASAP)

You have 75 minutes to complete the exam. Calculators are not permitted

Practice Test 2 is open now and closes 3/5. This will be a quiz. You will have 20 attempts on this.

Test 2 Review sheet is posted on week 6 (on online.math.uh.edu/courses) Bonus information is posted on the discussion board.

Lec Pop 6_2

1. Have you signed up for a time for your exam 2 yet?

a. YES b. NO (then go do it NOW and come back and choose A)

Section 8.7 Numeric Integration

Sometimes there are integrals you cannot compute by any method. In those cases we

need to use numeric integration.

Methods from Calc I:

Left endpoints: Right endpoints: Midpoints:

Summary:

New methods:

Trapezoids:

Example: Approximate 3

2

1

x dx using the Trapezoid Rule with n=4

Simpsons rule (parabolic estimate)

Approximate 3

2

1

x dx using Simpsons Rule with n=4

Error Estimates:

Since all of the methods above give estimates of the integrals, we need to know how close

we are to the real answer. We will face two types of errors: theoretical error (the error

that is inherent in the method we use) and round-off error.

The theoretical error for the trapezoid rule is )(''12

)(2

3

cfn

abE Tn

====

where c is some number between a and b. If f is bounded on [a, b], Mxf )('' for

bxa then Mn

abE Tn 23

12)(

=

Estimate the error if the Trapezoid rule is used to find 3

1

sin xdx using n=10.

The theoretical error for Simpsons rule is )(2880

)( )4(4

5

cfn

abE Sn

= where c is some

number between a and b. If f(4) is bounded on [a, b], Mxf )()4( for bxa then

Mn

abE Sn 45

2880)(

=

Summary of Error Estimates:

Trapezoid Rule: Mn

abE Tn 23

12)(

= , Mxf )('' for bxa

Simpsons rule Mn

abE Sn 45

2880)(

= , Mxf )()4( for bxa

Example:

Determine the values of n which guarantee a theoretical error of less than .001 for 3

1

sin xdx if the integral is estimated using Trapezoid rule then using Simpsons rule.

Determine the values of n which guarantee a theoretical error less than = 0.01 if the integral is estimated by the trapezoidal rule.

1 2 3 4 5

1

2

3

4

x

yLec Pop 6_2

2. If the graph of f (x) is given below and you use numeric integration to determine 4

1

( )f x dx with n=4, which of the following is true? a. Ln > Rn

b. Ln >4

1

( )f x dx

c. Tn > Ln

d. Using n = 4, Simpson's rule would have

a larger error than the trapezoid rule.

Section 9.3 & 9.4

Changing from polar form to rectangular form:

Formulas: sincos ryrx ==

Example : Change

3,2 pi to rectangular form

Changing from rectangular to polar form:

Formulas: 222 ryx =+ For , can use formulas above or 0,arctan = xx

y

Example: Change ( )3,1 to polar form.

More examples:

1. Write 422 = yx in polar form.

2. Write in rectangular form:

a. 4sin =r

b. pi31

=

c. cos3=r

Testing for Symmetry

If r r, , then the graph is symmetric about the x axis.

If r r, ,pi then the graph is symmetric about the y axis

If r r, ,pi +

then the graph is symmetric about the origin.

Circles Circle centered at (0, 0) with radius a.

Cartesian:

Polar:

Circle centered at (a, 0) with radius a.

Cartesian:

Polar:

Circle centered at (0, a) with radius a.

Cartesian:

Polar:

Lines Horizontal Lines:

Vertical Lines:

Lines through the origin:

Arbitrary Lines:

Sketch a graph of r = 2sin (3)

Polar graphs that produce flowers

r = a cos(m ) and r = a sin(m ) where a > 0 and m is a positive integer

Polar Curves of the form r = a b cos() and r = a b sin() Cardiods, Limaons with dimples and Limaons with inner loops

Cardioid Limaon with dimple Limaon with loop |a| = |b| |a| > |b| |a| < |b|

Graph: r 2 2cos= +

Graph: r 1 3cos=

2222 1111 1111 2222 3333 4444 5555 6666 7777

5555

4444

3333

2222

1111

1111

2222

3333

4444

2222 1111 1111 2222 3333 4444 5555 6666 7777

5555

4444

3333

2222

1111

1111

2222

3333

4444

2222 1111 1111 2222 3333 4444 5555 6666 7777

5555

4444

3333

2222

1111

1111

2222

3333

4444

Review

Lec Pop 6_2

3. Write the equation in polar coordinates. 2 2( 4) 16x y + =

4. Describe the curve: r = 2 - 4 sin a. Limacon with inner loop

b. Cardioid c. Limacon with no loop

d. Circle on the y- axis

5. What should you bring with you to your exam? a. Pencil and ID card b. Lunch c. Textbook d. Calculator