Good folks! The final exam is Wednesday, December 16, 7:30 -- 9:30 pm. Please go to Final Exam Info and print out a a review sheet and check further info regarding the final. Also listed is roughly what we have coverd in class from the book and some last minute advice for the final exam.

Lost flash drive. A student in our class lost a USB Flash drive last Monday, October 26th, in the lecture hall. If you have it or have turned it in somewhere, please let me know so I can tell the student where it is. This is an important item for the student.

NOTE! There will be designated hours in the Math Tutoring Center (JC 344) specifically for Math 108 students. They are T, R 10:30 am -- 12:00 noon; This will be a problem solving session open to all sections of Math 108.

NOTE! When turning in home work, please make sure to staple together the solution sheets. I will not take resposibility for unstabled and lost solutions sheets.

Also NOTE! For those of you who feel you need review in algebra and the prerequisites for this course, you could check the Math resource page. This page is currently under construction.

For the interested: In case you are interested in mathematical contests, two are coming up, the Virginia Tech Math Competition (Saturday, October 24, 2009, 9 -- 11:30 am) and later the national one, the Putnam Competition (Saturday, December 5, 2009, 10 am -- 1 pm and 3 pm -- 6 pm). In case you are interested in participating and showing up for practice sessions, please contact Prof. Daniel Anderson, his email is danders1@gmu.edu. Note that registration deadline for Putnam is October 8th.

Here below is a pdf version of the Syllabus. It outlines the overlall plan of the course and should answer all practical questions, like when tests are, what the grading policies are etc. If you didn't get a hard copy from me, then please print one for your records.

• Syllabus.

NOTE! All handouts for this class will be labeled with an itemized icon as with the Syllabus above.

Here below are the lectures and homework assignments (HW) so far in the course. The HW should be handed in, in the beginning of the class.

1. Lectures and Homework (HW) from Week 1.
   • Lectures of week 1.
   HW for Wednesday, September 9. :
   • Chapter-1:
     • Sec 1.1: 1, 2, 3, 14, 25, 30, 34, 36, 40, 49, 61, 66, 71.
   When defining a function, be sure that it is well defined. This means (1) all possible values (the domain) must make sense when you apply the function to them, and (2) the function itself cannot have any ambiguities. The way the function operates should be clear and yield a unique number. -- Be sure to know how to evaluate composite functions. Note that the order when you apply the functions is important.
2. Lectures and Homework (HW) from Week 2.
   • Lectures of week 2.
   HW for Wednesday, September 16. :
   • Chapter-1:
     • Sec 1.2: 1, 2, 8, 12, 13, 14, 19, 20, 26, 28, 31, 32, 39, 48.
     • Sec 1.3: 1, 2, 17, 18, 22, 27, 33, 36, 39, 51. Note! updated
   A curve in the coordinate system is a graph if it satisfies the vertical line test. Important functions are polynomials and rational functions. In particular, the quadratic functions are important. Their graphs are called parabolas. Every parabola has a vertex. One can determine the vertex by a formula given in class. This vertex is either the top point of the parabola (if A is negagive) or the bottom point of the parabola (if A is positive). The x coordinate of the vertex is the value at which the quadratic function takes its largest(least) value. This completely determines the graph of every parabola. Be sure to review how to find the x intercepts of a parabolas (that is, how to solve a quadratic equation.) -- One of the most important functions are the linear functions. These are polynomials of degree one. In order to know the equation for a line you need two forms of info, either a point and a slope, or two points (from which you can compute the slope and then you can use either point). Be sure to know that parallel lines have the same slope, perpendicular lines have the product of their slopes equal -1.
3. Lectures and Homework (HW) from Week 3.
   • Lectures of week 3.
   HW for Wednesday, September 23. :
   • Chapter-1:
     • Sec 1.4: 6, 7, 12, 13, 14, 37, 38.
     • Sec 1.5: 1, 2, 3, 7, 8, 17, 18, 24, 25, 27, 31.
   Functions in one variable are very important. Many practical problems involve expressions in terms of two or more variables, but then the given conditions can relate the variables so that the expression is in terms of just one variable. -- The first important thing of calculus is that of a limit. Note that a function can have a limit when x tends to a number c without being defined at c. Be aware of limits of the form "0/0"! This means MORE WORK. Then you must manipulate the expression before you plug in the value c to compute the limit. Pretty much all of calculus is built on the notion of limits.
4. Lectures and Homework (HW) from Week 4.
   • Lectures of week 4.
   HW for Wednesday, September 30. :
   • Chapter-1:
     • Sec 1.5: 32, 33, 35, (See Examples 1.5.7 - 1.5.10.), 45, 46, 48, 56.
     • Sec 1.6: 1, 2, 3, 4, 11, 12, 14, 15, 17, 20, 23, 25, 27, 40, 44, 57.
   First we defined what it means for a function to be continuous at a single point in terms of a limit. Then we definied what it means for a function to be continuous on a whole interval. That condition depends on whether the interval is open (neither endpoints included), half-open (one endpoint included and the other not) or closed (both endpoints included). An important thing about a function that is continuous on a closed interval is that it has the Intermediate Value Property. This means that if the closed interval is [a;b] then f(x) can take any value between f(a) and f(b). This principle can be very useful to approximate solutions to certain equations that we are not able to solve directly.
5. Lectures and Homework (HW) from Week 5.
   • Lectures of week 5. Please note the snafu on page 56; the slope of the tangent should be 1/4, etc.
   Suggested HW for Wednesday, October 7. (You don't need to turn this in!):
   • Chapter-2:
     • Sec 2.1: 1, 3, 5, 6, 10, 17, 18, 22.
     • Sec 2.2: 11, 12, 13, 16, 24, 26, 33, 54, (See example 2.2.6)
     • Sec 2.3: 1, 2, 5, 7, 9, 12, 18.
   The derivative f'(c) of a function f(x) is defined in terms of a specific limit as x tends to c. This limit will always have the form "0/0". This means that the limit is never trivial and always demands MORE WORK! However, once we have done that work we have the derivative f'(c) of the function at the point x = c. This limit f'(c) (if it exists as a number) is then the derivative of the function at the given point x = c. It is the slope of the tangent line of the graph y = f(x) at the point (c,f(c)), so the tangent line has the equation y - f(c) = f'(c)(x-c). Note that f(x) is necessarily continuous at x=c if it differentiable at x=c. Hence, if f(x) is not continuous, then it cannot possibly be differentable. -- By viewing the point c as a variable, we obtain the function f'(x). This function describes the slope of the graph y = f(x) at each point x. We have so far discussed some rules on how to take the derivative of polynomials (sums of powers of x) and so we know exactly how to find the equations of tangent lines of y = P(x) for any polynomial P(x). The Product Rule and the Quotient Rule for differentiation tell us how to take the derivative of the product of two or more functions and the derivative of the quotient of two functions. As long as we take care and proceed carefully, these rules should be easy to remember.
6. Lectures and Homework (HW) from Week 6.
   • Lectures of week 6 (the latter half of Lecture5 from last week).
   HW for Wednesday, October 14. :
   • Chapter-2:
     • Sec 2.4: 1, 2, 3, 24, 28, 29, 35, 37, 41, 42.
     • Sec 2.5: 1, 7, 8, 13, 26.
   The Chain Rule (Section 2.4) is the last but not least important rule for differentiation. We discussed some examples regarding the Chain Rule, namely start with the "outer most function" differentiate that, and then work your way in. -- Using the product rule, quotient rule and chain rule for computing derivatives of a function f(x) enables us to take the derivative of pretty much any "nice" function. Be sure to know how to use these rules inside out! -- In economics we can then use derivatives to compute the marginal cost, revenue or profit, and then use them to predict or estimate the cost, revenue or the profit of producing the x-th unit. This shows us that using derivatives give us a way to make educated guesses of how a given function is is going to behave in the very near future. Also, the change in x will result in a change in f which is going to be approximately the change in x times the derivative f'(x). This can be practical when you want to know the accuracy of quantity that is a function of x when you know the accuracy of x.
7. Lectures and Homework (HW) from Week 7.
   • Lectures of week 7.
   HW for Wednesday, October 21. :
   • Chapter-2:
     • Sec 2.5: 28.
     • Sec 2.6: 1, 2, 3, 4, 8, 9, 10, 25, 28, 29, 47, 52.
   Implicit differentiation is very useful when we need to find the rate of change or the tangent line of a curve where we cannot (or it is really complicated to) solve for y and obtain y = y(x) directly as a function of x. By differentiate the equation implicitly we will obtain an equation involving x, y and dy/dx, which then we can use to solve for dy/dx. In this way we obtain the slope at a given point in terms of both x and y, instead of just x as we do when differentiate a usual function. -- Higher derivatives can be obtaind by differentiating the function more than once, and the second derivative has nice interpretations both physically as the acceleration (when we differentiate twice w.r.t. time) and also for curve sketching.
8. Lectures and Homework (HW) from Week 8.
   • Lectures of week 8.
   HW for Wednesday, October 28. :
   • Chapter-3:
     • Sec 3.1: 1, 2, 9, 10, 11, 12, 23, 24, 25, 35, 36, 45, 68, 69, 77.
     • Sec 3.2: 5, 6, 7, 13, 14, 39, 40, 70.
   One of the most important things about the derivative of a function is that it tells you exactly where the slope of the tangent line is zero. So, if x = cis a relative extremum point, then f'(c) = 0. Hence, when seeking where exactly the extrema values of a function is (where it is maximum and where minimum) it suffices to look where the derivative is negative. If the derivative is positive, then the function is increasing. If the derivative is negative, then the function is decreasing. Knowing the signs of the derivative everywhere tells us what the graph looks like, so we can use this to sketch the graph. But the bottom line is that derivatives can be used to determine the largest value (absolute maximum) and the smallest value (absolute minimum) of a function on a given real interval. All one has to do is to (i) check the endpoints of the given interval and (ii) check the critical points x = c where f'(c) = 0. The maximum and the minimum value of the function is among these. This has many practical applications in optimizations (something we will discuss better next time.) -- The second derivative tells us about the concavity of the graph of the function. If the second derivative is positive, then the graph is concave up and has a the shape of a "smile". If the second derivative is negative then the graph is concave down and has the shape of a "frown". The point at which the graph changes from being concave up to concave down, or vice versa, is called a point of inflection.
9. Lectures and Homework (HW) from Week 9.
   • Lectures of week 9.
   Suggested HW for Wednesday, November 4. (You don't need to turn this in!):
   • Chapter-3:
     • Sec 3.3: 9, 14, 17, 22, 33, 34.
     • Sec 3.4: 3, 4, 35, 44.
     • Sec 3.5: 1, 4, 15, 16, 24.
   Knowing what the first and the second derivative of a function tells us, can give us enough info to sketch the graph of the function in quite some detail, and moreover, understand the nature of the behaviour of the function. The main thing, however, is to know how to compute the critical points of a function. The critical points are the points where the derivative is zero (or where the derivative is not defined, but that very rarely happens.) To compute the largest or the smallest value of a function on a given interval, one must compute the values of the function at all the points where the derivative is zero and also at the endpoints. -- This is a very useful approach for many practical optimization problems. Namely, we reduce the problem of finding where exactly the function takes its minimum or maximum value (depending on the problem at hand) and to do that we find the critical points. -- The exponetial function is a new kind of funtion we have not discussed yet, but is of great importance in economics, physics and any kind of natural situation of progression over time. We will discuss this in detail in the next couple of weeks.
10. Lectures and Homework (HW) from Week 10.
    • Lectures of week 10.
    HW for Wednesday, November 11. :
    • Chapter-4:
      • Sec 4.1: 13, 14, 17, 21, 35, 47.
      • Sec 4.2: 3, 4, 6, 13, 16, 21, 24, 32, 39, 41, 47.
      • Sec 4.3: 3, 6, 10, 24, 33, 35, 40.
    We defined the number e as the base number for the natural logarithm. The exponential function exp which takes this number e and puts it to the x-th power, is an important function in economics and science. We discuss how this comes up in a natural manner better in compound interest. The exp function is esentially the limit function obtained when we compound the interest every split second (in a continuous manner..!) Another important function is the natural logarithm function, which is the inverse of the exp function. The nice thing about the exp function is that it is its own derivative. This makes it easy to differentiate various functions involving exp by using the Chain Rule. Similarly the derivative of the natural logarithm is 1/x, which is also a very nice form. The exp function appears naturally in continuously compounded interest, and logarithm is very practial for solving equations where the unknown is in the power. The thing is that logarithm brings the power down, so we can solve for x.
11. Lectures and Homework (HW) from Week 11.
    • Lectures of week 11.
    HW for Wednesday, November 18. :
    • Chapter-4:
      • Sec 4.3: 45, 49, 75.
    • Chapter-5:
      • Sec 5.1: 1, 2, 3, 4, 5, 11, 12, 13.
    Logarithms are also used by many scientist when they want to estimate the age of certain fossils, since they are used to solve equations that involve exponential functions. -- Our final main topic is that of the antiderivative or integration. This is the opposite of taking the derivative, so when we have a function f(x) and we want/need to integrate it, we must find/compute a function F(x) whose derivative is the given function f(x). This turns out to be a considerably harder question than differentiation. The main reason is that there is, in general, no mechanical way to integrate every function. But many functions we work with can be integrated and we will discuss them and applications of integration.
12. Lectures and Homework (HW) from Week 12.
    • Lectures of week 12.
    HW for Wednesday, December 2. :
    • Chapter-5:
      • Sec 5.1: 15, 17, 20, 25, 26, 30, 49, 65.
      • Sec 5.2: 3, 4, 5, 6, 11, 24.
      • Sec 5.3: 1, 3, 8, 10, 14, 31, 39.
      • Sec 5.4: 1, 2, 3, 4, 7, 8.
    Integration (i.e. finding antiderivatives) has many practical applications. For example, we can find the distance from the starting point at any given time if we know the speed as a function of time. Also, we can compute the area between curves of functions by using the definite integral. Note that the Fundamental Theorem of Calculus states that the definite integral of f(x) from x = a to x = b is precisely F(b) - F(a) where F(x) is an antiderivative of f(x). This is a very powerful and practical way to find complicated areas. Remember, when computing a definitie integral, always first compute the indefinite one, and then plug in the upper and lower bounds to compute the definite one. -- To compute the area between two curves, we first must know where they intersect, that is, we must first find out what the bounding numbers are, then we use these boundary numbers to plug into the corresponding anti-derivative (integration) and compute the area between the curve in that way. -- There are many known methods to compute indefinite integrals, but the two main ones are substitution, which is based on the chain rule and integration by parts, which is based on the product rule. -- That is it folks! :)

• Review sheet for the 1st midterm.

Here below is a hand written draft of solutions to the review sheet above. Please attempt the problems first on your own before you look at the solutions.

• Draft of solutions for the Review sheet above.

Dear folks! The grading of the first midterm exams is currently done and I will give them back to you, Wednesday, October 14th. I am glad to see how many of you understand well what I am discussing in class, but we have a long way to go so please keep on working. The average score was 61.4/100 which is quite good (compared to many other times I have taught this 108 class.) On the other hand, there are quite a few of you who must change the way you work in this class. The sum is the score I will record as your score for this midterm exam.

Below are handwritten solutions to the midterm. Please print out a hard copy for your records. Make sure you understand the solutions. -- If you have any concerns, you are welcome to come and discuss them with me, but read the solutions first.

• Solutions of the first midterm.

Below is a review sheet for the second midterm. The midterm is to be held in the classroom, Wednesday, November 4th, from 9:00 pm to 10:00 pm. Also note that the review sheet is not a recipe for the midterm, but merely a collection of problems you should attempt to practice your skills.

This 2nd midterm will have the same format as the first one: No multiple choice questions, no calculators, no cheat-sheets. Note that by my policy, the midterm can only help your overall grade of this class and cannot hurt you at all! Roughly what will be covered on this midterm is what we will have covered in class Wednesday, October 28th, that is all of Chapters 2 and 3.

• Review sheet for the 2nd midterm.

Here below is a hand written DRAFT of solutions to the review sheet nr.2 here above. Please attempt the problems first on your own before you look at the solutions.

• DRAFT of solutions for the Review sheet for 2nd midterm.

Below are handwritten solutions to the second midterm. Please print out a hard copy for your records. Please look over these solutions to get a good grasp of what was expected in this second miterm.

• Solutions to the second midterm.

Good people! The grading of the second midterm MT2 exams is done. Some of you are showing signs continued good work and that makes me glad :) However, I was slightly disappointed with the outcome of the second problem on the midterm. Differentiation is something you should be able to do in your sleep. Please make sure you know how to take derivatives using the product rule (PR), quotient rule (QR) and the chain rule (CR). The raw average score was 49.1/100. Please compare your own work with the solutions posted here above, so you understand what was expected of you.

Here below is a review sheet for the final exam. Please print out a copy for yourselves. I will discuss these problems the two last weeks of class, both Wednesdays December 2nd and 9th.

• Review sheet for the final exam.

Please note that this is not a recipe for the exam, but merely a collection of problems you should take a look at before I go over them in class. Some of them are easy (and you should be able to do them with ease!) others are slightly more involved.

The final exam is to be held in the classroom (ST -- II room 9) Wednesday, December 16, 7:30 -- 9:30 pm. The final exam will be cumulative and cover everything that we have discussed this semester. Needless to say, I will stress the material we have done since the last 2nd midterm. The exam will consist of 10 problems. Roughly half of the final will be from the stuff we have covered since the second midterm. There are no multiple choice questions and no calculaters or cheat-sheets are allowed either. You should have plenty of time to do the final. Unless you all do exceptionally well, I will base my grade on a curve and I will give partial credit. Remember to show your work in a clear fashion. -- The material covered in this course and hence this final is roughly as follows: Chapter-1: Sections 1,2,3,4,5,6, Chapter-2: Sections 1,2,3,4,5,6, Chapter-3: Sections 1,2,3,4,5, Chapter-4: Sections 1,2,3, Chapter-5: Sections 1,2,3,4.

Some words of wisdom: The question is always, "How should you study for the final?" Well, a good rule of thumb is to drill the problems you have done in homework and midterms. Don't stay up too late the night before if you possible can manage that. Read over the solutions of the problems on the two midterms so you are sure on how to answer the questions, and so you have a good feeling for what I expect as a perfect solution on the final.

Remember, if you feel completely overwhelmed, please make sure you know how to differentiate a function, using the product rule, quotient rule and the chain rule. There are plenty of problems to drill just this in the book. This is a mechanical procedure and should be within reach of anybody who have practiced a bit. -- Also, recall that integration is reversed differentiation. To integrate f(x) is to find all functions F(x) such that F'(x) = f(x).

Although regular office hours will not be valid the final week, I will be in my office Wednesday, December 16. (the final day) more or less until the final is. If you have some questions you can stop by my office during that day. Also, for quick questions, you can always drop me an email.

EOF/EOS