Prep Precalculus E 2006-07
An interactive log for students and parents in Precalculus Section E. This site is only as rich as YOU make it.
Wednesday, May 23, 2007
Tuesday, May 22, 2007
Friday's Chapter 15 Test Topics
Here’s a list of topics for Friday’s test. Can you believe it’s our last one?
Precalculus Chapter 15 Test Topics:
Identify the degree, number of real and complex zeros and the leading coefficient of a polynomial from its graph
Sketch the graph of a rational function.
Identify transformations of functions.
Simplify rational functions.
Determine the zeros, the sum of the zeros, the product of the zeros, the sum of the pairwise product of the zeros, and a possible equation from the graph of a cubic function.
Sketch the graph of a given polynomial.
Identify the zeros of the polynomial.
Factor the polynomial.
Prove that a quadratic has no real zeros.
Show that a value is a zero of a polynomial.
Find zeros of a polynomial.
Discuss the implications of nonreal zeros.
Determine an average rate of change.
Provide a formula for an average rate of change.
Determine an instantaneous rate of change.
13 questions on the non-calculator portion, 11 on the calculator portion. That’s it! I’ll be around after school on Thursday afternoon, and in early on Friday.
Mathematicians are like Frenchmen: whatever you say to them, they translate it into their own language, and forthwith it means something entirely different.
- Johann Wolfgang von Goethe
Here’s a problem to consider:
Driving problem: Hezzy Tate drives through an intersection. At time t = 2 sec she crosses the stripe at the beginning of the intersection. She slows down a bit, but does not stop, and then speeds up again. Hezzy is good at mathematics, and she figures that her displacement, , in feet, from the first stripe is given by
Use the results of the synthetic substitution and the quadratic formula to find the other two zeros of d(t).
How do the zeros of confirm the fact that Hezzy does not stop and go back across the stripe?
What is Hezzy’s average velocity from t = 3 to 3.01 sec?
Write the equation for the rational algebraic function equal to Hezzy’s average velocity from 3 sec to t sec.
By appropriate simplification of the fraction in Problem 16, calculate Hezzy’s instantaneous velocity at time t = 3 .
Monday, May 21, 2007
Wednesday, May 16, 2007
15-2 Graphs and Zeros of Polynomial Functions
Alright....APs are over so now it's time to buckle down and focus on Chapter 15!
This section shows how to recognize the degree of a polynomial function and how to find the zeros.
Stuff you should know:
Zero of a Function: A zero of a function f is an x-value c for which f(c)=0.
Example: f(x)=x^2 -6x+5 = (x-5)(x-1) *by factoring*
So...the zeros equal 5 and 1.
The degree of a one-variable polynomial is the same as the greatest exponent of the variable.
Synthetic substitution is a quick way to evaluate a polynomial function. (It's easier than long division of polynomials!!)
Synthetic substitution can be used to find out if the zeros are accurate.
~Put the zero that we want to try into a little box.
~Write the coefficients of the equation next to the box.
~Write a line under the coefficients, leaving enough room for another number.
~Drop the first coefficient below the line.
~Multiply the number in the box with the number you just put below the line.
~Put the product below the second coefficient.
~Add the coefficient and the product and put the sum below the line.
~Multiply the sum with the number in the box and repeat the steps.
If the last number behind the line is a zero, then the number in the box is a zero of the function.
The remaining numbers behind the line are the coefficients of the other factor of the equationThings to know for Synthetic Substitution:
*If the last number under the line is not 0, then the x-value is not a zero for the function. The number that you get in the last spot is called the remainder.
*The remaining numbers below the line are the coefficients for the other factor of the function. The factor is always one degree less than the degree of the function you started with. So, if you start with a cubic function, the factor you get with synthetic division is quadratic.
*The Remainder Theorem: If p(x) is a polynomial, then p(c) equals the remainder when p(x) is divided by the quantity (x-c).
*The Factor Theorem: (x-c) is a factor of polynomial p(x) if and only if p(x)=0.
The Fundamental Theorem of Algebra and Its Corollaries: A polynomial has at least one zero in teh set of complex numbers.
Corallary: An nth-degree polynomial function has exactly n zeros in the set of complex numbers, counting multiple zeros.
Corollary: If a polynomial only has only real coefficients, then any nonreal complex zeros appear in conjugate pairs.Translation: You always have the same number of zeros as the degree of the polynomial (if you count both complex and real numbers). If you have an imaginary zero, it will have a pair.
A cubic function will always have three zeros, but two of them could be complex.
Sums and Products of Zeros:
Sums of the Products of the Zeros of a cubic function:
If p(x)=ax^3+bx^2+cx+d has zeros z1, z2, and z3, then--
Sum of the zeros: z1+z2+z3=-b/a
Sum of the pariwise products of the zeros: z1(z2)+z1(z3)+z2(z3)=c/a
Product of the zeros: z1*z2*z3=-d/a
The sum of the zeros usually fills the B spot of a cubic function, the sum of the pairwise usually fills the C spot, and the product usually fills the D spot of a cubic function.
The values of the coefficients for the cubic functions are opposite for the sum and product of the zeros.
Example: 1)Find the sum of the zeros, the sum of the pairwise products, and the products of all three zeros if the z1=.6, z2=2 and z3=4.
2)Find the particular equation of the cubic function with values you find. 3)Prove the sum, sum of pairwise products, and products are correct by using the sum and products property of the zeros of a cubic function.
Sum of Pairwise: (.6)(2)+(.6)(4)+(2)(4)=11.6=58/5
2) Because you know what places the values take in the cubic function, you can figure out that the particular equation of one of the possible functions is y=x^3-(33/5)x^2+(58/5)x-(24/5).
Since the coefficients of the values are all over 5, we can change the coefficients to be integers.
So: f(x)=5(x^3)- 33(x^2)+58x-24
3) To prove the values we got in number 1 are correct, we can use the properties of this section:
Sum: (-b/a)=(33/5) ~it checks out
Sum of Pairwise: (c/a)=(58/5) ~it checks out
Product: (-d/a)=(24/5) ~it checks out
(thank Leo for this part)
~Low point = critical point = extreme point = vertex
~Number of extreme points = the power – 1
Example: Cubic functions have 2 extreme points
~Odd powers go in two different directions
~In even number powers, the -/+ on the first term determines the up and the down
~At the Point of inflection, the concavity changes.
Definition of concavity:
Concave down = tangent lines are above
Concave up = tangent lines are below
That's pretty much it!
For extra help: http://www.purplemath.com/modules/synthdiv.htm
**Gina....you're up next!
Everyone should be excited for two really cool movies that are coming out in the next two weeks:
Thursday's Quiz Topics
Now that we’re through the worst of the AP season, we can finally get back on track (after the Junior retreat on Friday, anyway)! Here’s a list of topics for Thursday’s quiz:
As a reminder, your Weekly Challenge will be due on Monday, and the test for Chapter 15 will be next Friday.
Precalculus Quiz 15.1-3 Topics
Given a graph of a function:
Determine the degree
Determine the leading coefficient
Determine the number of real and complex (nonreal) zeros
Identify an extreme point and a point of inflection.
Given a cubic function:
Determine the product of the zeros
Determine the sum of the pairwise product of the zeros
Determine the sum of the zeros
Perform synthetic substitution.
Interpret the results of your synthetic substitution.
Given the factored form of a cubic, determine the zeros of the function.
Given data for a cubic function:
Prove the constant third difference property
Determine the equation for the function algebraically (matrices)
Verify the equation with regression techniques
Analyze the equation
Given partial information about the zeros of a cubic function:
Determine the remaining zeros
Determine the product of the zeros
Determine the sum of the pairwise product of the zeros
Determine the sum of the zeros
Determine an equation for the function
Analyze a function (word problem!)
That’s it! The format’s the same as always – ½ Non-calculator, ½ Calculator. There are 20 questions on the Non-calculator portion and 20 questions on the Calculator section (all worth 1 point each). If you know your stuff, you should be able to finish quickly. I’ll be around after school on Wednesday (today), and on campus by 7:00 AM on Thursday.
"I never did very well in math - I could never seem to persuade the teacher that I hadn't meant my answers literally."
- Calvin Trillin
Trashing math and country music - ouch!
Wednesday, May 02, 2007
Friday's Quest Topics
Here’s a list of topics for Friday’s test:
Precalculus Chapter 14 Test Topics:
Find terms – arithmetic sequence (determine pattern)
Find terms – geometric sequence (determine pattern)
Definitions of sequences and series
Find terms – arithmetic series (use information)
Determine type of sequence and justify
Determine terms of a sequence – unknown type
Determine terms of a series – unknown type
Explain your method of determination/pattern of series
Determine terms of a sequence (determine pattern)
Determine terms of a series (given pattern)
Determine value of n for given term in a series/sequence
Determine term(s) of a binomial expansion
Word problem – analysis of arithmetic and geometric sequences
Arithmetic series – determine formula, apply formula
Geometric series – determine formula, apply formula
Partial sums of arithmetic and geometric series
Infinite sums of geometric series
11 questions on the non-calculator portion, 14 on the calculator portion. Since this chapter was so short, each question will be worth 3 points instead of 5, making this a “quest” instead of a test. That’s it! I’ll be around after school on Thursday afternoon, and in early on Friday.
"Knowledge is of no value unless you put it into practice."
- Anton Chekhov
This strategy won't work on me:
Tuesday, May 01, 2007
14-3: Series and Partial Sums
This chapter is all about formulas. The sequence formulas are from 14-2 (they appear in Katie's post), but I will put them here so you can see them in relation to the series formulas.
First of all, a series is the sum of the values of a sequence.
A partial sum is the sum of n terms in the series.
Things to keep in mind:
- n is the term number
- tn is the term value
- Sn is the nth partial sum
- "lim" stands for limit. This means that in a geometric series, if the constant ratio is less than 1, the partial sum of that series will approach a limit. In the geometric partial sum equation, as n gets larger and larger, r^n gets smaller and smaller, closer and closer to 0. Therefore, r^n becomes more insignificant and is "eliminated" from the lim equation. This situation is known as a convergent series because the series converges to a particular value as a limit.
- If in a geometric series |r| > 1, it is called a divergent series because the terms do not go to zero and thus the series diverges.
Q: Find the 10th partial sum of the series 1, 1/2, 1/4, 1/8...
To what limit does the series converge?
r = 1/2 (a common difference), so this is a geometric series
Use the formula for the partial sum of a geometric series.
Then use the limit formula.
Binomial Series Formulas
A binomial series (binomial expansion) is of the form (a+b)^n.
2 Ways to get the coefficients of the expanded series:
- Pascal's Triangle
- Binomial Theorem
Things to keep in mind (in a binomial series) - these can come in helpful when checking your work:
- There are (n+1) terms
- Each term has degree n.
- Powers of a start at a^n and decrease by 1. Power of b start at b^0 and increase by 1.
- Sum of exponents in each term is n.
- Coefficients symmetrical w/ respect to series' ends.
Q: Given (2a-3b)^7, find the term with b^4.
- Write down (-3b)^4, because you know this is your b^4 term.
- Multiply that by (2a)^3, because 2a is your "a" term and the exponents of the entire term must add up to 7. --> 7 - 4 = 3.
- Multiply everything by 12C7, the coefficient of the term.
- 35 * 8 * 81(a^3)(b^4)
Q: In the binomial series (a-b)^17, find the 8th term.
A & Solution: This requires one more step in your thought process. You must figure out what b term the "8th term" will be. You know the first b term is b^0, the second is b^1, the third is b^2, etc. So go forward and the 8th b term is b^7.
- Write down (-b)^7.
- Multiply by a^10 because the exponents of the term must add up to 17.
- Multiply by 17C7, the coefficient of the term.
Formulas, Worked Examples, Pascal's Triangle