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!