UHS AP Calculus AB and AP Calculus BC

 

Calculus AB covers one full semester of college-level calculus (equivalent to Math 120 at U of I).  Calculus BC covers two full semesters of college calculus (at U of I, Math 120 and 230).  Many of you (maybe all) will earn university credit for one or two semesters of calculus after taking one of these courses.  This will save you time and money in college.  To avoid wasting a lot of time reviewing math that you should already have learned, I want you to do some work this summer.  Doing math might not be your favorite summer pastime, but think of your calculus class as free college education!  I encourage you to work together but each person will turn in his or her own work on the first day of school in the fall.  The work you turn in will count for points, and you will be tested on this material after we go over it in class.  If you have questions, please email me at:  lbeuschlein at usd116.org   Here are your summer assignments (for both AB and BC students):

 

1.      Textbook: Chapter 1 Review, pages 52-53, #1-67.  There are lots of problems, but the assignment should all be fairly easy precalc review material.  We won’t cover this material directly in class (except to answers questions you may have), but you will have no chance of succeeding in calculus without being able to do this stuff.

 

2.    Textbook: Chapter 2 Review, pages 91-93, #1-53.  Much of this should be review, but some of it will be new.  We will cover this chapter when school starts, but we’ll go through it fairly quickly.  If you and the group with whom you are working really get stuck, email me or contact other calc students.

3.    Problem set: matrices and linear algebra (see below).  This should be review too, but I included reference links in case you need help.

 

Don’t procrastinate on the problems!

 

 

Matrices and Linear Algebra

  1. See http://www.sosmath.com/matrix/matrix0/matrix0.html for help on this one.  Suppose you’re the owner of Acme Nail Guns.  You’ve got 4 customers, each of whom have made 3 purchases.  Set up a 4 ´ 3 matrix to display this data.  (Each student should make up his/her own sales data, even if you’re working together, which should discourage copying.)  How large a purchase did customer #3 make on his 2nd purchase?  Let’s call this matrix P, for purchase prices.

 

  1. The last one was too easy—my apologies.  Ok, now make another matrix that displays how much tax each customer paid for each purchase.  Let’s say sales tax is 7.5%, and call this matrix T, for taxes.  Also, write an equation that relates T and P.

 

3.        Let’s suppose we want a matrix E that displays total expenditures (tax plus purchase price) for each customer for each purchase.  Write E in two ways: as the sum of matrices; and as a multiple of one matrix.

 

4.        http://www.sosmath.com/matrix/matrix1/matrix1.html will help for this.  Let’s say your buddy is the owner of a business as well—Acme Staple Guns.  Your current monthly profit is X and hers is Y.  Do to the number of nail/staple gun accidents in your town, you decide to invest in a third business together—Acme High Powered Fastener Insurance Corporation.  You both agree to the following investment strategy: you’ll invest 35% of your current monthly profits into promoting the new business, and she’ll invest 46%; you’ll invest 10% of your profits into office space rental, and she’ll invest 5%.  Let Ix be your monthly investment and Iy be hers.  Write a matrix equation that relates all quantities involved.

5.        Suppose there are two fish bowls side by side, labeled A and B.  Each is loaded with gold fishies…bowl A starts out with a0 fishies, and B starts out with b0.  Here’s what happens every hour: 10% of the fishies in A jump into bowl B, but 5% of the fish in B jump into A.  After n hours, bowl A has an fish, and B has bn.  part 1: Write a matrix equation that relates a1 and b1 to a0 and b0.  part 2: Write a matrix equation that relates an and bn to an-1 and bn-1.  part 3: Write a matrix equation (with a matrix to a power) that relates an and bn to a0 and b0.  part 4: Each student should make up his/her own starting numbers for a0 and b0 and compute a4 and b4 using a calculator.

 

6.        http://www.sosmath.com/matrix/matinv/matinv.html  Let M = the matrix used in the last problem to go from one hour to the next.  What matrix would allow you to go back an hour?  Compute this matrix in two ways, using a calculator and using row ops.  For a review of row operations, see http://www.sosmath.com/soe/SE3/SE3.html and scroll down to the matrix section.  What do you get when you multiply your answer by the original matrix?

7.        http://www.sosmath.com/matrix/system1/system1.html http://www.sosmath.com/matrix/determ2/determ2.html
http://www.sosmath.com/matrix/determ1/determ1.html Schmedrick’s about to crack open his piggy bank which is full of nickels, dimes, and quarters only.  He counts 60 coins for a total of $6.30.  He also happens to notice that if he has as many nickels as dimes and quarters combined.  Write a system of equations to determine how many of each coin he has.  Solve in three ways.  Show all work.
a. row ops on an augmented matrix
b. matrix inverse and multiplication
c. Cramer’s rule (but just solve for the number of dimes)

8.        Each student should make up his/her own system of two linear equations with two unknowns.  (Make it a system that has exactly one solution.)  Solve it in each of following ways.
a. substitution          b. elimination of variables     c. graphing    d. row ops on augmented matrix    
e. matrix inverse       f. Cramer’s rule