3.3 & 3.4 The Chain, Product and Quotient Rules

Main points:
• Chain rule
○ d/dt(f(g(t))=f'(g(t))*g'(t)
○ In words, the derivative of a composite function is the derivative of the outside function times the derivative of the inside function
○ For functions given by formulas the function is first rewritten using a new variable z to represent the inside function
§ y=(t+1)^4 is the same as y=z^4 where z=t+1
○ If z is a differentiable function of t, then
§ d/dt(z^n)=(nz^n-1)(dz/dt)
§ d/dt(e^z)=(e^z)(dz/dt)
§ d/dt(lnz)=(1/z)(dz/dt)
• d/dt(e^kt)=ke^kt
• Product rule
○ d(uv)/dx=(du/dx)*v+u*(dv/dx)
○ In words, the derivative of a product is the derivative of the first times the second, plus the first time the derivative of the second
• Quotient rule
○ (d/dx)(u/v)=[(du/dx)(v) - (u)(dv/dx)] / v^2
○ In words, the derivative of a quotient is the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator, all over the denominator squared

Challenges:
I have studied the chain, product and quotient rule before so I am quite confident with them. In the past I have had trouble with the questions containing lna, a^x and other more complex identities. Therefore I should spend more time on the more challenging problems in order to become comfortable with these mathematical identities.

Reflection:
I was once again looking for examples on economics, and the section in product and quotient rule did have those whereas the section in chain rule did not. I was happy to find several related to the topics such as total revenue and price of a product. I hope doing these examples will prove to be helpful as I am going to take economics on next semester.

3.1 & 3.2 Derivative formulas for powers and polynomials & exponential and logarithmic functions

Main points:
• Derivative of
○ Constant function is 0
○ Linear function is slope=m (when f(x)=b+mx)
○ Constant times a function is (d/dx)[cf(x)]=cf'(x) (when c is a constant)
○ Sum is (d/dx)[f(x)+g(x)]=f'(x)+g'(x)
○ Difference is (d/dx)[f(x)-g(x)]=f '(x)+g'(x)
• The power rule: (d/dx)(x^n)=nx^(n-1)
• (d/dx)(e^x)=e^x
• Exponential rule: (d/dx)(a^x)=(lna)a^x
• (d/dx)(lnx)=1/x

Challenges:
As going through the examples I found myself making careless mistakes such as putting unnecessary negative sign when differentiating √x. I definitely need to be careful as I am finding myself pretty comfortable with the differentiation in order to avoid simple mistakes. I am also from time to time getting confused with the graphs of function and their derivatives, especially when they are on the same graph. So spending more time with graphing would not be bad idea.

Reflection:
There was no examples related to economics which was something that I did not like, but luckily there are questions related to economics in the problem section. I was surprised how easily I remembered differentiation, but regardless that I was capable of using the rules I could not provide the correct notations for my work, just the answer to the differentiation. I know that differentiation is used many ways in social sciences and I am hoping to see more of its practical applications to the real world problems.

2.2 & 2.3 The derivative function and its interpretation

Main points:
• Sign of derivative indicates whether the function is increasing or decreasing
○ f'>0 ,then f is increasing over that interval
○ f'<0 ,then f is decreasing over that interval
○ f'=0 ,then f is constant over that interval
• Leibniz's notation for derivative is f'(x)=dy/dx
○ Disadvantage: awkward to specify the x-value at which a derivative is evaluated
• Units of derivative: units of dependent variable / units of independent variable
• Velocity= ds/dt and derivative of velocity (dv/dt) is acceleration
• Local linear approximation: dy≈f'(x)dx (for dx near 0)

Challenges:
I am finding it frustrating in a way to find estimations of the instantaneous rate of change, as I would prefer an exact value. I am also having hard time remembering that the derivative is the instantaneous rate of change so it actually implies how much y changes in respect to x. For example with the copper mine example (example 3), it shows the cost of extracting the next ton of copper. I didn’t understand the concept of local linear approximation at all, and it is something what I would like to hear tomorrow in the class.

Reflection:

I found the examples interesting, especially business related such as the copper mine, sugar production and cost of building extra square feet of house. I feel that they are real world examples that I might be pondering myself too at some point. I am always been also interested in the great mathematicians, so I found it interesting to be reading about Leibniz. It's the fact that they often have very unusual backgrounds and maybe I'm hoping to kind of have a chance to identify their thought processes as they made their major discoveries.

1.3 & 2.1 Rates of change and the derivative

Main points:
• Average rate of change of y between t=a and t=b: delta(y)/delta(t)=[f(b)-f(a)]/(b-a)
• The units of average rate of change of a function are units of y per unit of t
• Increasing function: the values of f(x) increase as x increases
• Decreasing function: the values of f(x) decrease as x increases
• Concavity
○ Concave up=bends upwards as we move left to right
○ Concave down=bends downwards as we move left to right
• Average velocity=change in distance/change in time
• Instantaneous velocity of an object at time t is defined to be the limit of the average velocity of the object over shorter and shorter time intervals containing t
• Instantaneous rate of change of f at a (also known as rate of change of f at a) is defined to be the limit of the average rates of change of f over shorter and shorter intervals around a
• Derivative of f at a, f'(a), is defined to be the instantaneous rate of change of f at the point a

Challenges:
It is a long time since I have last done physics so problems with velocity are going to be good practice for me. Other ways derivatives are something what I did study in the past but don’t really remember anymore, so I am going to need some practice with them too. I am also finding myself in need to recap the basic precalculus things, in order to work with the current problems. So I need to set more time for revision of things I have learnt in the past.

Reflection:

The derivatives totally took my attention, as they are needed in so many different sciences. I also didn’t understand them too well in the past so I am looking forward to become more familiar with them. Most of the problems were related to hard sciences and I was hoping to see more problems from economics (especially in the derivatives section). I hope there are going to be some economics related problems in tomorrow's class.

9.1 & 9.2 Functions of two variables and contour diagrams

Main points:
• R=f(x,y) (R is function of x and y)
• R is the dependent variable and (x,y) is the independent variable
• Domain is the collection of all possible inputs (x,y)
• Can be presented numerically by table of values, algebraically by a formula or pictorially by a contour diagram (a graph showing selected contours of a function; also called 'level curves' or 'level sets')

Challenges:
This was the most challenging topic so far as I have not done this before. I was trying to do the example problems but had troubles with some of them, so I am hoping to understand the relevant concepts of this topic more in depth tomorrow.

Reflection:
I found the topic interesting to read as it was something that I had not encountered before. I was also fascinated by the contour diagrams and their use in topographical map outlines. As I am interested in economics I am hoping to set some time aside during the weekend and search for Cobb-Douglas Production Function and learn more about it.

Periodic functions 1.10

Main points:
• Periodic functions are functions whose values repeat at regular intervals (E.g. sine and cosine functions)
• They can be presented in the form of: y=Asin(Bt)+C (A,B,C are called parameters)
○ A is the amplitude (half the difference between the functions maximum and minimum values)
○ B is the period (calculated by 2*pi/IBI
○ C is vertical shift

Challenges:
I might find it difficult in the beginning to work with radians as in the past I have used degrees. I still didn't understand where the sign (positive or negative) comes for amplitude. I am also looking forward to learn more about the nature of sine and cosine, because I came to wonder that aren't they just the same curve? y=sin(x) is the same as curve y=cos(x-(pi/2)) ? So I would like to know where they originate from and what are their major usages in mathematics.

Reflection:

I found the topic interesting to read, as it was revision from my IB mathematics. But I still had forgot the things completely… I'm happy that I did it now because at least one of the questions in lab 1 require discussion about amplitudes and periods. Periodic functions seem to be particularly useful for natural sciences, and even I am not taking them at the moment, I do enjoy biology problems the books has in its homework for this section.

Exponential functions 1.5 & 1.7

Main points:
• Exponential functions are expressed as f(x)=a^x. a is called the base or growth factor.
• P is exponential function of t with base a if P=Poa^t.
• Exponential growth/decay occurs when there is constant percentage change. When a>1 there is exponential growth and when 0 • e equals 2.71828... and is called natural base. Many quantities in nature change according to P=Poe^kt where P0 is the initial quantity and kt is the continuous growth/decay rate.
Challenges:I find it hard to put the mathematical concepts to my own words as English is not my first language. At the moment I do not have confidence with logarithms so I have to revise them too before I can fully approach exponential functions. Especially as e is often used with logarithms (or at least that is what I remember). I still did not fully understand why the factor a=1+r and I am looking forward to learn it in tomorrow's class. I am also looking forward to learn more about the natural base e and its importance, because in the past I have been bit intimidated by it.

Reflection: I found the topic useful through the examples of population growth and compound interest examples. The financial applications are something what I can use to my advantage in my own life.  I was also reminded that the functions can be solved in several ways, as in the past I was not so confident with formulas I used graphs instead. I have studied this in the past but found it too be more difficult than some other topics that I have managed to revise so far. I am looking forward to learn more about the natural base e and financial applications tomorrow.