Last Topic: Applications of Calculus to Your Major

From this Calculus 1 class, you have learned about limits, derivatives, differentiation rules, applications of differentiation, integrals and applications of integration. Please share with me and your classmates how you might use the material of this class for your major(s). Do you plan to take Calculus 2 at some point?

If you finish your comments by Thursday next week (Dec 6), you will receive 100 extra-credit points. Remember to write your name at the bottom of your comments. I wish you success for the Final Exam!

Topic 5: Exponential Growth and Decay

"In many natural phenomena, quantities grow or decay at a rate PROPORTIONAL to their size. For instance, if y = f(t) is the number of individuals in a population of animals or bacteria at time t, then it seems reasonable to expect that the rate of growth f '(t) is proportional to the population f(t); that is, f '(t) = k f(t) for some constant k...Another example occurs in nuclear physics where the mass of a radioactive substance decays at a rate proportional to the mass. In chemistry, the rate of a unimolecular first-order reaction is proportional to the concentration of the substance. In finance, the value of a savings account with continuously compounded interest increases at a rate proportional to that value." (Section 3.8, Calculus, Early Transcendentals, 6th edition, by James Stewart)

We can model these phenomena by the equation dy/dt = ky where:

y(t) is the value of the quantity at time t,
k is a constant,
dy/dt is a rate of change of y with respect to t.

If k > 0, then dy/dt > 0 is the GROWTH rate.
If k < 0, then dy/dt < 0 is the DECAY rate.

The above equation is a differential equation because it involves an unknown function y and its derivative dy/dt. The solutions of the differential equation dy/dt = ky are y(t) = y(0).exp(kt) where y(0) is the initial value.



In exponential growth (k > 0), the rate of change increases over time - the rate of the growth becomes faster as time passes. In exponential decay (k < 0), the rate of change decreases over time - the rate of the decay becomes slower as time passes.

Depending on your major or area of interest, please choose ONE example in Section 3.8 of the textbook and answer the following questions:

1) Specify the number of the example you chose (example 1, example 2, etc).

2) In this example, please analyze how the above model is used.

3) Is your chosen example exponential growth or decay?

If you do not have the textbook, you can look at the examples of Section 3.8 on the class website.

The deadline to submit your comments is Thursday, 11/15. Late submission will be graded zero.

PLEASE WRITE YOUR NAME AT THE BOTTOM OF YOUR COMMENTS FOR GRADING. IF YOU HAPPEN TO FORGET YOUR ACCOUNT ID OR PSW, GO TO GOOGLE TO CREATE A NEW ONE. DO NOT SEND YOUR COMMENTS TO ME BY EMAIL!

Topic 4: Rates of Change in the Natural and Social Sciences

Given y = f(x), the derivative dy/dx can be interpreted as the rate of change of y with respect to x in many areas such as physics, chemistry, biology, economics, and other sciences. Depending on your major or area of interest, please choose ONE example in Section 3.7 of the textbook (Calculus, Early Transcendentals, 6th edition, by James Stewart) and answer the following questions:

1) Specify the number of the example you chose (example 1, example 2, etc).

2) In this example, please analyze how the concept of the derivative is used.

3) In the context of your chosen area, please make up an example where a derivative needs to be computed to describe some rate of change.

If you do not have the textbook, you can look at the examples of Section 3.7 on the class website. Notice that the examples are classified under the associated areas (i.e., physics, chemistry, biology, and economics).

The deadline to submit your comments is a week from now (i.e., Thursday, 10/25). Late submission will be graded zero. Besides, there will be NO extension for this assignment since the Review Set for Test 3 will be posted next Thursday.

PLEASE WRITE YOUR NAME AT THE BOTTOM OF YOUR COMMENTS FOR GRADING. IF YOU HAPPEN TO FORGET YOUR ACCOUNT ID OR PSW, GO TO GOOGLE TO CREATE A NEW ONE. DO NOT SEND YOUR COMMENTS TO ME BY EMAIL!

Topic 3: Early Methods for Finding Tangents

The history of calculus is fascinating! Do you know that the first person who formulated explicitly the ideas of limits and derivatives was Sir Isaac Newton in the 1660s? However, Newton acknowledged that "If I have seen further than other men, it is because I have stood on the shoulders of giants."

These giants were Pierre Fermat (1601 - 1665) and Newton's teacher at Cambridge, Isaac Barrow (1630 - 1677). Newton was familiar with the methods that these men used to find tangent lines, and their methods played a role in Newton's eventual formulation of calculus (quoted from Stewart's Calculus book).

In this topic, I want you to do some research online about:
1) Fermat's life and career (25 points)
2) Fermat's method in finding tangent lines (25 points)
3) Barrow's life and career (25 points)

Barrow's method in finding tangent lines is more complicated. However, you are welcome to try to understand this method and describe it here for 25 extra-credit points.

Please post your comments to briefly:
- describe each of the above items. It is important to give credit to the websites that your comments refer to. Please copy their links and paste them in your comments! Otherwise, half of your grade will be deducted.
- compare Fermat's method for finding tangents and the method we have learned in class (25 points).

The deadline to submit your comments is a week from now (i.e., Thursday, 09/27). Late submission will be graded zero.

THE DEADLINE IS EXTENDED TO THURSDAY (10/04) (i.e., a week from now).

PLEASE WRITE YOUR NAME AT THE BOTTOM OF YOUR COMMENTS FOR GRADING.

Topic 2: Limits and Continuity

Since we are learning about limits and continuity, I want to see how much you have understood these concepts. Please post your comments to answer the following questions:

1) In the 1st problem of the homework assignment of Sections 2.1 and 2.2, why are the average velocities and the instantaneous velocity at time t = 2s NEGATIVE?

2) Geometrically, what is "the line tangent to a curve" at a particular point?

3) What is the difference between the statements "f(a) = L" and "lim_{x->a} f(x) = L"?

4) Assume that f(1) = -5 and f(3) = 5. Does there have to be a value of x, between 1 and 3, such that f(x) = 0? (Think about the Intermediate Value Theorem).

5) Using the point-slope form, derive the equation of the secant line going through two points A(1,0) and B(2,1) and having the slope m = 1. Show me two equations:
- one derived from choosing A as the point in the point-slope form,
- the other derived from choosing B as the point in the point-slope form.
Are they different?

The deadline to submit your comments is a week from now (i.e., Thursday, 09/13). Each question weights 20 points. Late submission will be graded zero.

PLEASE WRITE YOUR NAME AT THE BOTTOM OF YOUR COMMENTS FOR GRADING.

Topic 1: Welcome to Calculus I

As a starting point for learning Calculus 1, please write a comment in which you address three issues:

1) Did you take Calculus in high school?

2) Why did you register for this class?

3) By the end of the semester, what do you expect to achieve in this class?

Please post your comment by Monday (09/03/2007). Otherwise, I will assume that you cannot access the blogging site. In this case, you will need to talk to me by the end of Tuesday's lecture.

DO NOT FORGET TO WRITE YOUR NAME AT THE END OF YOUR COMMENT!