Dr. Waller's Math 2401 Home Page

Current Semester: Fall 2009
Sections:  11123, TR 5:00-6:45 pm, 320-C


Hello Calculus Students,

Welcome to the home page designed to accompany Dr. Waller's Math 2401-Calculus I section. As the semester progresses, this page will contain various types of information important to the course, including:

- Urgent messages from Dr. Waller
- Text homework assignments
- Test dates
- Test answer sheets*
- The course syllabus*
- Computer locations on campus (and places to get help)
- My schedule (email me here)

*These files are in PDF format and require the Adobe Acrobat Reader plug-in to view and print. Click here to freely download the Adobe Acrobat Reader. When printing a PDF document, be sure to choose the option "Shrink to Fit" or "Fit to Page" from the print dialog box.

You may want to check this home page on a regular basis.

Cheers,
Dr. Waller


Urgent messages from Dr. Waller:

Posted 8/25: Three definitions of function:
0. A set of ordered pairs where each first coordinate appears exactly once.
1. A rule f that assigns each element x in a set D exactly one element, called f(x), in a set E.
2. A rule, method, or system that can be used to predict or determine an output quantity based on a known or measureable input quantity.
We say the output is a function of the input.

Posted 8/25: (as discussed in class)
Important Properties of Functions (Algebraic)
1. The input quantity and the output quantity.
2. The input variable and the output variable, such as x and y.
3. The units of measure for the input quantity and the output quantity, like lb, miles, dollars, elapsed years, etc.
4. The domain of the function.
5. The x- and y-intercepts of the function.

(Analytic or Calculus)
6. The range of the function.
7. Where in the domain (as we move from left to right) the function (output) is increasing.
8. Where in the domain (as we move from left to right) the function (output) is decreasing.
9. Turning points of the function. (Peak turning points are called local maxima and valley turning points are called local minima.)
10. The maximum or minimum outputs (similar to #6).
11. Where in the domain (as we move from left to right) the function is concave up.
12. Where in the domain (as we move from left to right) the function is concave down.
13. Inflection points of the function.
14. The signed area of the region bounded by the graph of the function and the x-axis, from a given number on the x-axis to another given number on the x-axis. 

Posted 8/25: I will begin taking attendance on Tuesday, Sept. 8.

Posted 8/25:  Here is the link to the online homework site. Remember, homework counts 16% of the course grade (the same as a regular test). You will not be permitted to continue attending class if you have not created your online homework account by the official day of record (Sept. 8).
The Class Key is  uhd 3087 0109 

Posted 9/2:  Most of you know by now that you must save your work after each session by using the Save Work button at the bottom of each question (which is not the same as submitting the work for grading). However, some of you have been confused about retrieving your saved work. This is done by choosing View Saved Work from the drop-down box next to the label About this Assignment in the top right-hand portion of the screen.

Posted 9/2:  In order to receive the email messages that I send via WebAssign through your external email address, you will need to activate this address in WebAssign. To do this, choose Notifications from the top right-hand part of the screen, then choose the Notification Contact Info tab from the resulting pop-up window. Follow the directions given there to activate your address.

Posted 9/3: (as discussed in class) A linear function can be written in the form y = f(x) = mx + b. Its graph is a straight line. Let f be a linear function whose graph contains the points (a, c) and (b, d). Then the slope or rate of change of f  is denoted by m and is computed by the formula

m = (d - c)/(b - a) = ( f(b) - f(a) ) / ( b - a )

Keep in mind that m  is a value. The proper interpretation of this value is: The outputs of  f change by  m  units per unit change in the input.

Posted 9/3: (as discussed in class) Let f be a function whose domain includes all real numbers between x=a and x=b, inclusive (assume a and b are real numbers where a < b). Then the average rate of change of f from a to b is denoted by f[a, b] and is computed by the formula

f[a, b] = ( f(b) - f(a) ) / ( b - a )

Keep in mind that  f[a, b]  is a value. The proper interpretation of this value is: On average, from the input a to the input b, the outputs of  f change by  f[a, b]  units per unit change in the input.

Posted 9/3: GraphCalc 4.0.1 Graphing Calculator (free graphing calculator simulator - download and install)

Posted 9/8: Graph Tool Applet (free online grapher)   If the applet doesn't work in your browser, you may need to download and install the Sun Java Runtime Environment.

Posted 9/14: Let f be a function whose domain includes all real numbers between x=a and x=b, inclusive. Then  f[a, b] is the same as the slope of the secant line to the graph of f connecting the points (a,f(a)) and (b,f(b)).

Posted 9/14: The average rate change of a linear function is steady and is the same as the slope m of the line. In other words,  f[a, b] =  m for all possible choices of a and b. Note: Recall that parallel lines have the same slope.

Posted 9/14: Let f be a function whose domain contains the number a. Then the instantaneous rate of change of f at a is denoted by f '(a). Keep in mind that f '(a)  is a value. The proper interpretation of this value is: At the input a, the outputs of  f change by  f '(a)  units per unit change in the input.

Posted 9/16: M.W. Milton Graphing Calculator Emulator  Download this zipped file onto your flash drive and open and extract the files using WinZip into the root directory of your flash drive. Be sure to check "Use folder names" when extracting. This program does not work well in Windows Vista.

Posted 9/22:  Here is a Flash movie that shows how to start the M.W. Milton graphing calculator emulator software unzipped onto your flash drive. The video refers to a CD but the same instructions work for a flash drive. This video has sound, so turn up the volume on your computer.

Posted 9/22:  Here is a Flash movie that shows how to print from the graphing calculator emulator software. It's not mentioned in the movie, but it works best to change the printer orientation to Landscape before printing the screen snapshot page. This video has sound, so turn up the volume on your computer.

Posted 9/22: Let f be a function whose domain contains the number a. Then  f '(a) is the same as the slope of the tangent line to the graph of f at the point (a,f(a)).

Posted 9/22:  Here is a sample test* for Test #1. Disclaimer - This review is comprehensive but should not be the only material used to study for the exam. It should not be considered a preview of the exam. It does not substitute for studying previous homework, class notes, text discussions, etc. There may be questions on the exam unlike questions on this review, and vice versa. No question on this review will be exactly duplicated on the exam. This review is longer than the exam. Answers will be posted shortly.

Posted 9/24: Let f be a function whose domain contains the number x. Then  f '(x) is the limit as h approaches 0 of the quotient ( f(x+h) - f(x) ) / ( h ). Hint: In practice, usually this quotient must be simplified before the limit value can be determined. Sometimes, the limit value may not exist. The function f ' is called the rate of change or derivative function of the function f. The process of determining  f '(x) is called differentiation. If the domain of f ' includes the number a, then we say that f is differentiable at a.

Posted 9/29: We say the limit of f(x), as x approaches a, equals L if we can make the values of f(x) arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to a (on either side of a) but not equal to a.

Posted 10/1: Here are the answers* to the Test #1 sample and homework handout #1.

Posted 10/4: Here are the answers to homework handout #3. This homework will not be collected, but remember homework handout #2 is due at the test on Tuesday.

Posted 10/5: Here is a blank copy of homework handout #2.

Posted 10/15:
Rules for Differentiation
Let f be a function. Then:
1) If the formula for f is of the type f(x) = c, where c is any real number, then f '(x) = 0. (The function f is called a constant function in this case.)
2) If the formula for f is of the type f(x) = mx + b, where m and b are any real numbers, then f '(x) = m. (Note that the function f is a linear function in this case.)
3) [Power Rule] If the formula for f is of the type f(x) = xn, where n is any real number except 0, then
f '(x) = nxn-1.
4) [Constant Multiple Rule] If the formula for f is of the type f(x) = cg(x), where c is a real number and g is a function, then  f '(x) = cg'(x).
5) [Sum Rule] If the formula for f is of the type f(x) = g(x) + s(x), where s and g are functions, then
f '(x) = g'(x) + s'(x).
6) [Product Rule] If the formula for f is of the type f(x) = g(x) s(x), where s and g are functions, then
f '(x) = g(x) s'(x) + g'(x) s(x).
7) [Quotient Rule] If the formula for f is of the type f(x) = g(x) / s(x), where s and g are functions, then
f '(x) = (s(x) g'(x) - g(x) s'(x)) / (s(x)^2).

Posted 10/22:
Rules for Differentiation (cont.)
Let f be a function. Then:
8) [Chain Rule I] If the formula for f is of the type f(x) = c(g(x))n, where c and n are any real numbers except 0 and g is a function, then f '(x) = cn(g(x))n-1(g'(x)).
9) [Chain Rule II] If the formula for f is of the type f(x) = h(g(x)), where g and h are functions, then
f '(x) = h'(g(x)) g'(x).
10) [Trig Rules]
a. sin'(x) = cos(x)
b. cos'(x) = -sin(x)
c. tan'(x) = sec2 (x)
d. csc'(x) = -csc(x) cot(x)
e. sec'(x) = sec(x) tan(x)
f. cot'(x) = -csc2(x)

Posted 10/22:
Rule for Locating Possible Turning Points
Let f be a function and suppose f ' is its rate of change (or derivative) function. Then the possible turning points or of f are located at the input numbers in the domain of f where f ' has an x-intercept or where f ' is  not defined. These are called the critical numbers.

First Derivative Test for Determining if Possible Turning Points are Actual Turning Points
Let f be a function and suppose f ' is its rate of change function. Moreover suppose that a possible turning point of f occurs at the input x = p. Choose convenient input numbers a and b such that a is slightly less than p and b is slightly more than p. Then if:
1)  f '(a) < 0 and f '(b) > 0, this implies (p, f(p)) is valley turning point on the graph of f. (The book calls f(p) a local minimum.)
2)  f '(a) > 0 and f '(b) < 0, this implies (p, f(p)) is peak turning point on the graph of f. (The book calls f(p) a local maximum.)

Posted 10/29:
Rule for Determining Maximum and Minimum Outputs on a Closed Interval
Let f be a continuous function on the closed interval [a, b]. Then the maximum and minimum outputs of f occur at either a, b, or c, where c is a critical number of f and a < c < b.

Posted 11/3:
Rule for Locating Possible Inflection Points
Let f be a function and suppose f '' is its second rate of change (or second derivative) function. (This is the rate of change function of the first rate of change function.) Then the possible inflection points of f are located at the input numbers in the domain of f where f '' has an x-intercept or where f '' is  not defined.

Test for Determining if Possible Inflection Points are Actual Inflection Points
Let f be a function and suppose f '' is its second rate of change function. Moreover suppose that a possible inflection point of f occurs at the input x = p. Choose convenient input numbers a and b such that a is slightly less than p and b is slightly more than p. Then if:
1)  f ''(a) < 0 and f ''(b) > 0, this implies (p, f(p)) is an inflection point on the graph of f.
2)  f ''(a) > 0 and f ''(b) < 0, this implies (p, f(p)) is an inflection point on the graph of f.

Second Derivative Test for Determining if Possible Turning Points are Actual Turning Points
Let f be a function and suppose f '' is its second rate of change function. Moreover suppose that a possible turning point of f occurs at the input x = p and f '' is continuous near p. Then if:
1)   f ''(p) > 0, this implies (p, f(p)) is valley turning point on the graph of f. (The book calls f(p) a local minimum.)
2)   f ''(p) < 0, this implies (p, f(p)) is peak turning point on the graph of f. (The book calls f(p) a local maximum.)

Posted 11/9:  Here is a sample test* for Test #2. Disclaimer - This review is comprehensive but should not be the only material used to study for the exam. It should not be considered a preview of the exam. It does not substitute for studying previous homework, class notes, text discussions, etc. There may be questions on the exam unlike questions on this review, and vice versa. No question on this review will be exactly duplicated on the exam. This review is longer than the exam. Answers will be posted shortly.

Posted 11/10: We say the function f is continuous at the input a if the limit of f(x), as x approaches a, equals f(a). We say the function f is continuous on an interval provided f is continuous at every input a contained in the interval. The following types of functions are continuous at every number in their domains: polynomials, rational functions, root functions, trigonometric functions.

Posted 11/10: The Intermediate Value Theorem.  Suppose that a function f is continuous on the closed interval [a,b] and let N be any number between f(a) and f(b), where f(a) is not equal to f(b). Then there exists a number c in (a,b) such that f(c) = N.

Posted 11/15: Here are the answers* to the Test #2 sample.

Posted 11/17:
Fundamental Theorem of Calculus

Let f be a function that is continuous on the interval [a,b]. Then the signed area of the region bounded by the graph of the function and the x-axis, from x=a to x=b, is given by:

F(b) - F(a)

where F is any antiderivative of f. That is to say, F '(x)=f(x).

Posted 11/17:
Rules for Antidifferentiation
Let f be a function. Then:
1) If the formula for f is of the type f(x) = xn, where n is any real number except -1, then  F(x) = xn+1/(n+1) + k  is an antiderivative of f (k can be any real number). [Anti-Power Rule]
2) If the formula for f is of the type f(x) = cg(x), where c is any real numbers except 0 and g is some function, then
F(x) = cG(x) is an antiderivative for f provided that G is antiderivative for g.
3) If the formula for f is of the type f(x) = g(x) + h(x), where h and g are functions, then
F(x) = G(x) + H(x) is an antiderivative for f provided that G and H are antiderivatives for g and h, respectively.
4) If the formula for f is of the type f(x) = h(g(x))g'(x), where h and g are functions, then
F(x) = H(g(x)) is an antiderivative for f provided that H is antiderivative for h. [Anti-Chain Rule]

Posted 11/18: The answer to Problem 6b on the Test #2 sample should be (iii), because the function has a turning point at x = -1.

Posted 11/18: The answer to Problem 1c on the Test #2 sample should be [-3.5,3].

Text homework assignments and due dates:

Homework #1.4, p. 51, 4,6,8,10,14,15,31,32. Print graphs for Problems 4 thru 14.
Homework Handout #1 will be due on Thursday, Sept. 24.
Homework Handout #2 will be due on Tuesday, Oct. 6.
Homework Handout #5 will be due on Thursday, Dec. 3.


Test dates:

Test #1: Tuesday Oct. 6.
Test #2: Tuesday Nov. 17. - POSTPONED TO NOV. 19.
Test #3: Thursday Dec. 3.
Final Exam: Thursday Dec. 10.


Test answer sheets:

Test #1 Answer Sheet
Test #2 Answer Sheet


Computer locations on campus (and places to get help):

Academic Support Center (925-N): Click here for schedule. The Math Lab is located here, so you can ask questions about the course material/homework.

PLTL Lab (738-S): Click here for schedule. The lab monitors can also answer your questions about the course material/homework.

Academic Computing Labs (800-S, 300-C, B-200): Click here for schedule and information.