3.3: The Chain Rule.

Main Points
For a function y = f(z) and z = g(t), { y = f(g(t)) }, a change in 't' affects z in turn affecting y and the derivative can thus be computed with the formula: dy/dt = dy/dz * dz/dt. In other words: d/dt(f(g(t))) = f '(g(t))*g '(t).

Challenges
I do not understand how we move from this formula: dy/dt = dy/dz * dz/dt, to this one: d/dt(f(g(t))) = f '(g(t))*g '(t). What exactly is the connection between the two?

Reflections
Being able to differentiate compositre functions allows one to take into consideration the further effects of an event rather than only the immediate one. Again facilitating the possiblity of having future plans. Applied calc seems so far to simply be facilitating predictions.

3.2: Exponential and Logarithmic Functions.

Main Points
The derivative graph of an exponential function looks exactly like the original graph. For any function in the form a^x, their derivative graphs are proportional to the original, but for the function e^x the derivative graph is exactly the same. Thus, the exponential rule is d/dx(a^x) = (ln a)a^x. Finally why is e is useful base is explained: the fact that the constant of proportionality is 1 when a = e. The derivative of ln x is 1/x.

Challenges
Why is the derivative at the point x=0 of the 2^x function approx. 0.693 rather than 0? How exactly are functions where x is the power differentiated?

Reflections
With the examples that we have been doing in class it is becoming more and more obvious how differentiation fits seamlessly into everyday life though it may of course be very theoretical with hypothesized figures. I never really thought differentiation was what was used to compute the demand and supply functions and thus interpretations in economics.

3.1: Derivative Formulas for Powers and Polynomials.

Main Points
The derivative of a constant is always 0 as it is simply a horizontal line and the slope of a linear function (a non-horizontal line) is always constant. When a constant, c, multiples a function, the derivative is: cf '(x). The derivative of a sum or difference is the sum or difference of each functions derivative respectively. The power rule of derivatives is: d/dx(x^n) = nx^n-1. To differentiate polynomial, the same rules for the sum, constant and powers apply. The derivative of a function using formulas would always give the function of the tangent at a point.

Challenges
The differences between the two methods used in obtaining a derivative are large. how was the connection between the two discovered? Because the honestly the derivative using the formula is very simple.

Reflections
Starting with lessons with learning to get derivatives by calculating the slope of the point in questions over smaller and smaller intervals does a lot with explaing just how differentiation works and what exactly it is and what it is meant for.

2.3: Interpretations of the derivative

Main Points
The derivative can also be denoted by f '(x) = dy/dx, where the 'd's is useful in reminding the user that the derivative is a difference in y-values over a difference in x-values. A derivative tells us how fast a function is changins thus enabling us to estimate the values of that function at points close to the derivative point.

Challenges
Under the "Using Units to interpret the Derivative," I do not understand the second point in the box.

Reflections
The derivative serves as an investigative tool that helps the user to compute certain figures pertinent to whatever situation they are in. I especially like the cost of building a house example (Example 2, p. 113)

2.2 The Derivative Function

Main Points
The derivative is the slope of a tangent line to a point in a curve. This is derived from a secant where the point are so close together they are almost one. When a derivative (tangent) is positive (sloping up) the graph is increasing, when 0 (horizontal) the graph is constant and when negative (sloping down) the graph is decreasing. To estimate a derivative numerically we simply use a small difference between two points for the numerator of a slope equation.

Challenges
Considering I have dealt with differentiation before without the explanations on how to get to it by taking the differences of two points over a smaller and smaller range, it makes it a little difficult and frustrating to follow. It feels as though what I learnt before is conflicting with what I'm learnign now even thought they are the same thing.

Reflections
Considering derivatives are instantaeous rates of change for different point over a curve, it would be possible to calculate at which points over a certain car's journey for example that it stopped, (rest stop) sped up (high way) or slowed down (traffic jam.)

2.1: Instantaneous Rate of Change

Main Points
This chapter considers the rate of change of ONE POINT of a function rather than a range of points. Instantanoues velocity is the velocity of an object at a specific point in time rather than an average over a period of time. The derivative of a function at a point is the instantaneous rate of change which in turn is the limit of average rates of change of short intervals about the point.

Challenges
Summarizing the entire chapter was difficult considering I did not understand most of it especially the bit about the limits. And thus reflecting also is difficult.

Reflections
I feel as though finding out what, say, the speed of a car was at a specific second in its journey would be something that pertains to investigations and the like where specific detail like that is needed. It might also be useful for projectiles, so that the necessary thrust is enough to get an object to its desired destination; and this could be with the instantaneous velocity for example.

1.3 Rates of Change

Main Points
Rates of change could also be calculated when changes in a certain value are not constant. and the formula used is: dy/dx = f(b) -f (a)/b -a. This formula thus calculate the average rates of change. When the average rate of change is the same at all intervals the functions is linear. A positive rate of change suggests an increase whilst a negative one suggests a decrease. The secant line is that with draws out the average rate of change of two points on a curve. This is also their slope. A concave up graph indicates that a function is increasing whilst a concave down graph has a decreasing function. In effect a line is constant.
The average rate of change of height per time is velocity this is different from speed, as speed is a magnitude that is either positive of zero and velocity is a vector. (ie. it can be negative as well.)

Challenges
Example 5 (a): between -1 and 2, I have no idea whether to say the function is increasing because as x increases and so does y, or to say it is decreasing because it is concave down.
And for the graph between x= 2 and 6, how would you describe wrt to (b) the first half is concave downa nd teh second is concave up...

Reflections
Again, being able to compute certain values using applied calculus and in this case rates of change one is able to predict future values for certain situations and thus act accordingly as they would be theorically prepared. But nothing that happens in theory ever happens exactly the same in real life.

9.1: Understanding Functions of two variables/ 9.2: Contour Diagrams

Main Points
A function with 2 variables can be represented using a table of values, with a formula done algebraically or pictorially with a contour diagram. The best way to learn something about a function with 2 variable is to vary one at a time.
Contour diagrams connect points with similar properties, such as isotherms over a certains geographical area. These diagrams could be used to compare different values and also help in the decision making process for a number of situations.

Challenges
How exactly are contour diagrams obtained? Is there a fomula that is used?
What is the significance of the Cobb-Douglas Production Model other than with the US economy? It seems to be pretty inportant since it has its own title and all.

Reflections
ALMOST EVERY PROBLEM IT SEEMS NOW CAN BE SOLVED BY MATH. Of course the nature of the problem has to be considered. And further more it helps with making predictions about certain situations.

1.10: Periodic Functions.

Main Points
Periodic functions have graphs that look like waves as their values repeat after fixed time periods and this goes on forever. The sine and cosine graphs are the most common periodic functions there are and the graph of the cosine function is the same as that of the sine graph only shifted "pi"/2 to the left. Both have an amplitude of 1 and a period of 2*"pi". Periodic functions come with a family with the general term: y = A sin(Bt) where A and B are parameters. "A" affects the amplitude of the graph and "B" affects the period. (the smaller B is the bigger the period.)

Challenges
Are the functions called periodic simply because they have a constant period? Does that mean that if the period were constant but the amplitude of the graph changed it would still be a perfectly good periodic function? Period is equivalent to wavelength right? Therefore light can be considered periodic until it moves from one medium to another and the wavelength changes...
Is this right?

Reflections
Considering that periodic functions are supposed to repeat basically forever applying the functions to real life situations allows the user to predict and therefore prepare for the future. Granted the data would not fit perfectly on a sin curve but it would suggest such as for the housing project that the builders would be able to project would rough amounts of concrete they would be needing.

1.7: Exponential Growth and Decay

Main Points
Exponential growth and decay occur often in nature such as with radioactive decay and compound interest. Every growth has a time when it has doubled in value (doubling time) and a decay has a time when it is half its value (half-life). Interest can be compounded in two ways: annually and continuously, where interest is compounded a number of times in a year (known as compounding). Up to a certain point compounding can be more beneficial had doing it annually as one collects interest on an interest already paid. Annual interest is calculated using the formula: P = Po(1 + r)^t and compounding interest has: P = Poe^(rt). Most products have a present a future value.

Challenges
I do not understand why the two different compound interest formulas are the way they are. I do not understand how they were derived. Thus the same goes for present and future value.

Reflections
Its cool to think that because we know the effects of radioactivity and the half-lifes of the various materials that give it off, the time until which the area of the iodine spill could be occupied again can be calculated. This has then led to higher and better security and safety methods.

1.5: Exponential functions.

Main Points
The exponential function was defined as being in the form f(x) = a^x, with a being a positive constant. With exponential functions the dependent value increases faster per the independent values and have a constant ratio. A typical example of a exponential function is a population growth.
Another form for the exponential function is P = Poa^t where Po is the initial amount at t = 0. Exponential growth occurs when a > 1 and decay occurs when 0 < a < 1. Also considering "a" is a factor of P, the bigger "a" is the faster P grows or decays.
For both a growth (increase) and decay (decrease) the graphs of the functions both are concave up. The difference between the linear and exponential functions is that the linear functions have an absolute rate of change whilst the exponential function has a percentage rate of change.
e is the most commonly used base.

Challenges
A decay always has the x-axis as its asymptote because it never gets to zero and even if it could it could not go beyond it because it cannot go into the negatives. But this is not the same for a growth, it has no asymptote because there are an infinite number of numbers and therefore P can increase forever. This still does not make sense to me though...

Reflections
Whatever medication you take technically never leaves your bloodstream because it never decays to a value of 0! The fact it never gets to zero is possibly the theory behind immunizations. This also explains why there is always a certain amount of time that must pass between two doses of a certain medication. If this time frame was not observed an overdose is more likely to occur. This is good to know, I nearly took my anti-malaria pill 24hrs too early.