Below is a letter I intend to give to my all my potential enrollees in next year's AB and BC calculus classes. What other common mathematical misconceptions and common errors, or fundamental knowledge should be included. Do you know of any other college placement tests available on the web? Please respond to the listserve or to me personally at the address at the bottom. Thanks
June 8, 1998 Dear Incoming Calculus Students,
During the summer I would like you to make sure you have the skills necessary to succeed in Calculus. Among those skills are graphing calculator usage, algebra skills, trig skills and analysis of functions and their graphs. To help you improve those skills over the summer, I would like you to be able to contact me via the Internet (firstname.lastname@example.org or Warren.Block@moa.net) and use the Internet for mathematical purposes. All of the following information will be available at my website soon, so visit http://edcen.ehhs.cmich.edu/~warblock/ If you do not have the Internet at home, use the Troy Public Library, Oakland University, or OCC. If you have a modem at home, but no Internet access, you can use Juno for free e-mail access. It does not give you Internet access, but it does let you communicate with others. If you do not have an e-mail address, visit me before leaving school and we can set you up with one at one of the following websites. l. http://www.hotmail.com (slow but great features) 2. http://www.juno.com/ (free, which is great if you don't have the Internet access at home) 3. http://edit.my.yahoo.com/config/form?.form=main_mail&.src=ym&.done=http%3a//edit.my.yahoo.com/config/mail%3f.intl= (similar to hotmail but I haven't tried it yet.)
The AP Test in calculus requires that you have these skills: 1. produce the graph of a function within an arbitrary viewing window, 2. find the zeros of a function, 3. compute the derivative of a function numerically, and 4. compute definite integrals numerically. If you do not believe me visit this site for more details: http://www.collegeboard.org/ap/math/html/exam002.html While we will learn how to do 3 and 4 during the year you should come prepared to do the following: 1. Graph the following functions: y=x, y=x^2 y= x^3, y= sin x, y = cos x, y= tan x, y= sec x, y=csc x, y =ln x , y= log10 x, y = 2^x, y = e^x, y=arctan x, y = 1/x, y= |x| and y = 1/x^2 (During the course of the year you will be expected to produce these graphs without a calculator, but to practice your skills, use one of the AP recommended calculators, a new list will be coming out soon with the new casio's and TI-89, etc. For the current list, check this site. http://www.collegeboard.org/ap/math/html/exam003.html When you graph, see the difference in the trig graphs using degrees and radians, try creating windows that cover one period, two periods, three periods etc for the trig functions. As you look at the graph, determine whether the functions are increasing, decreasing, concave up or concave down. State the domain and range of the functions. Find the roots (zero's) of each function. Discuss the asymptotic behavior, and symmetry of each graph. Graph some of these graphs simultaneously and find their points of intersection. (If any of these directions are confusing to you e- mail me at email@example.com or ask Dr. Math at: http://forum.swarthmore.edu/dr.math/tocs/about.math.high.html or Mr Calculus at http://www.geocities.com/Athens/Acropolis/6432/
2. Graph y= (1 + 1/x)^x and zoom in at x= 0 many times and see what value the calculator gives you for x= .001, .0000001,.0000000000000001 Also see what values are given for x large such as, x=100, x=10000000, x=100000000000000. See if you can get our calculator to make a table or list of the x and y values you used. 3. You should be able to write the equation of a line given 2 points (in slope intercept form y=mx+b and Taylor's form y=m(x-x1) + y1). Find the slope and equation of the line going through the following points: a. (0,0) (3,4) b. (1,-2) (-1,2) c. (2,3) (-8,-7) d. (x,c) (x^2,c^2) e. (x,x+h) (f(x), f(x+h) f. (1 hour, 10 miles) (2 hours, 50 miles) What does the slope represent? For more information on slope visit this page: http://math.rice.edu/~lanius/Algebra/stress.html
4. Use a calculator in degree and radian mode to tell if the following identities are true or false, choose any number you like for x, a and b, for trig functions try 30 degrees or pi/6, pi/4 etc. sin ^2 x + cos ^2 x = 1 ln(ab)=ln a + ln b | a + b| = |a| + |b| sin 2x = 2 sin x cos x ( a + b)^2 = a^2 + b^2 1/a + 1/b= 1/(a + b) (x^2 - a^2)/(x - a) = x + a For the false statements can you find some numbers that will make them true? In other words, are they sometimes true and sometimes false. Is there some way to change them to make them always true? Are there any that are always true? Why? E-mail me your answers. Later on in the summer, the answers will be posted.