LA 4.3 Least Squares

Main points:
• Multiple m2 x of x that reaches closest to y
○ Multiple is called the projection of y on the line spanned by x
○ Step from the tip of the projection m2 x to y is the residual vector r
○ m can only be estimated, the quality of estimation is indicated by the length of the residual vector r
• The fundamental problem of Linear Modeling
○ Least-squares curve-fitting with J parameters leads to projection onto subspaces spanned by J vectors.
§ The line y = m2 x is called the least squares fit to the data
○ The optimal linear combination is called the projection of the vector b on the subspace spanned by the vectors u1, . . ., uJ

Challenges:
I understood what the topic discussed in terms what is being solved but I didn’t understand the methods applied in the solving. I also had problems understanding the examples. In the first example I didn't understand how the m2 value (170/89) was obtained.

Reflection:
This topic area seems highly relevant because many social sciences have optimization problems. In my area of interest, economics, this section had even an example of the effect of advertising dollars on sales. This made the whole topic a lot more interesting to study, even I didn’t completely understand the examples.

LA 4.1 - 4.2 Vector Projection

Main points:
• Many of the equations we would like to solve, don’t have solutions
○ Therefore we look for approximate solutions
○ Finding an approximate solution is often easier in vector format
• Dot product (=scalar product)
○ The dot product of two vectors is zero if the two vectors are perpendicular (or orthogonal)
• Residual vector

Challenges:
I had a bit of troubles understanding residual vectors, mainly just the whole logic behind it. Partly it was due because I still have problems understanding what the terminology means.

Reflection:
The topics seemed ok, but I really need to spend more time with them and also be really alert on tomorrow's class. I am mainly looking forward to go through problems in tomorrow's class as I think the new things are otherwise understandable.

LA 1.3, 2.0 - 2.3, 3.0 - 3.3 Linear combinations, linear independence, span

Main points:

• Linear combination=a sum of multiples
• Linear equations: two interpretations
○ Following kind of a question: what multiplies of u and v can you add to get to w
○ Solving
§ Multiples are denoted as x and y
§ Solution is a pair (x,y) which satisfies the simultaneous system of two equations and two unknowns
• The geometry of linear equations
○ Points interpretation
○ Vector interpretation
• System of equations can have
○ No solution (two lines are parallel and not the same; and the two vectors are parallel and the vector we are trying to get to is not on that line)
○ One solution (two lines intersect in a point; and the two vectors can be combined in exactly one way to get to that vector)
○ Infinitely many solutions (two lines are the same and any point on this line works. The corresponding vectors are parallel and so is the vector we are trying to get to)
• Higher dimensions
• Matrix notation
○ Ax=b
○ An m * n matrix is a rectangular array of numbers, arranged in m rows and n columns
• Spanning
○ The span of a list of vectors is the set of all linear combinations that can be made
with those vectors
• Linear independence
○ No vector on the list is a linear combination of the other vectors on the list.
• Linear dependency
○ At least one of the vectors on the list is a linear combination of the other vectors on the list
• Subspaces and dimension
○ Subspace is a set of vectors that is the span of some list of vector
○ Dimension of a subspace is the minimum number of vectors required to span the subspace

Challenges:

I didn't understand what linear combinations are and what is their importance. I didn't understand what are the benefits and disadvantages of using vectors in contrast to system of equations. Not understanding linear combinations severely limited my understanding of spanning and rest of the topics. I am hoping to understand the things more clearly after tomorrow's class.

Reflection:

I was reminded how in math everything's builds on previously learnt things. As I didn't understand linear combinations, I had problems understanding spanning and the rest of the topics. I also came to realize how often I have had problems understanding the topics when I read them from the book, but how much easier it is for me to learn them in the classroom. For tomorrow, I'm especially interested to see how higher dimensions can be tried to "graph".

10.7 Modeling the spread of a disease

Main points:

• The S-I-R Model
○ S=the number of susceptibles, the people who are not yet sick but who could become sick
○ I=the number of infecteds, the people who are currently sick
○ R=the number of recovered, or removed, the people who have been sick and can no longer infect others or be reinfected
○ dS/dt=-aSI
○ dI/dt=aSI-bI
○ dR/dt=bI
○ Threshold population = b/a
§ If S0 is above --> epidemic
§ If S0 is below --> no epidemic

Challenges:

I understood the basics of the S-I-R model, but nevertheless had troubles fully understanding it. Part where trajectories in the phase plane were introduced made me confused, because I didn’t completely understand what the trajectories are. I think I will understand the topic better after tomorrow's class.

Reflection:

Even I am not doing biology, I have done it in the past and I do found it interesting to study the patterns of diseases spreading. It is going to be useful for just my general knowledge and also possibly useful as I am interested in international studies. Today's studying was spred over 2 hours which proved to be very unproductive in the end (due to several distractions) so I need to make sure that in future I allocate more specific timeframe for my math studying.

10.6 Modeling the interaction of two populations

Main points:
• Predator-Prey model:
○ dw/dt=aw-cwr and dr/dt= -br+kwr
• The Phase Plane:
○ Graph of r and w against to see the growth of the populations
○ Phase plane = wr -plane on which the point moves
○ Phase trajectory = the path of the point
○ dr/dw=(-r+wr)/(w-wr)
• The slope field and equilibrium points
• Trajectories in the wr-phase plane

Challenges:
In the Robins and Worms example I didn't understand the meaning of the different constants. Later on the slope field and equilibrium points and their significance went just completely over my head. Similarly trajectories in the wr-phase plane proved to be something that I cannot grasp. Tomorrow's class is going to be difficult as I did not understand the topic at all.

Reflection:
I'm finding that for me spending an hour daily on mathematics is not enough. On the other hand I don’t think I can spend more time on mathematics, so I need to become somehow more efficient with my learning. I am thinking of getting some help at Max Center because preceptors are helpful but they don't seem to have the topics fresh in their mind so it always takes some time for them to remind themselves about the topic and then trying to explain it to me.

10.4 & 10.5 Exponential growth and decay & Applications and modeling

Main points:
• General solution to dy/dt=ky is y=Ce^kt for any constant C
○ Exponential growth for k>0
○ Exponential decay for k<0
○ The constant C is the value of y when t is 0
• General solution to dy/dt=k(y-A) is y=A+Ce^kt for any constant C
○ Arbitrary constant C is the initial value of y-A
• An equilibrium solution
○ Constant for all values of the independent variable. The graph is a horizontal line.
○ Stable if a small change in the initial conditions gives a solution which tends toward the equilibrium as the independent variable tends to positive infinity
○ Unstable if a small change in the initial conditions gives a solution a curve which veers away from the equilibrium as the independent variable tends to positive infinity

Challenges:
I did understand the general concepts, but I still have problems applying them to real problems. Especially translating the problems to mathematics was something that I found difficult. I had problems understanding the example of Pollution in Great Lakes, so I would appreciate if we could go through it in class.

Reflection:
Today was difficult because I was really tired so I kept reading the sentences several times in order to understand them. Shows that I really need to consider whether it is worth trying to study if I am just very exhausted. I liked the examples on continuously compounded interest and company's revenue as the relate to economics.

10.1 & 10.2 Intro. to differential equations, solutions to differential equations

Main points:

• Differential equation
○ Equation that is written when information about functions rate of change or its derivative is known
• Logistic differential equation
○ E.g. dP/dt = kP(L-P)
• Solution to differential equation
○ Any function that satisfies the differential equation
○ Solving numerically
○ Substituting the differential equation separately into the left and right sides of the differential equation checks the solution
○ General solution=solution that satisfies the differential equation (family of functions when C is unknown)
○ Particular solution=solution that satisfies the differential equation together with the initial condition

Challenges:

The differential equation wasn't that difficult to understand, but I am more concerned that I can myself turn worded problems into a equations. I am also not sure when the equations has a constant k. I though it would be used when the rate of change of Q (or other unit) is proportional to Q. I would like to be clarified about this point tomorrow. I am also not sure of the type of problems that require the use of a logistic differential equation, so I would like to see and example of it.

Reflection:

I found the example of net worth of a company interesting because I am planning to take accounting next semester and I am hoping there would be similar problems in the homework more. I am not sure what are all the possible application of these skills but I can imagine that for example in investing I could use this to determine the returns for different initial investments.

9.6 Constrained optimization and Lagrange multipliers

Main points:

• Constrained optimization
○ External circumstances constraining optimization
○ Graphical approach
§ Global max or min occurs where the graph of the constraint is tangent to a contour or at an endpoint of the constraint
○ Analytical approach (Lagrange multipliers)
§ F has a local maximum at P0 subject to the constraint if f(P0)≥f(P) for all point P near P0 satisfying the constraint
§ F has a global maximum at P0 subject to the constraint if f(P0)≥f(P) for all point P near P0 satisfying the constraint
§ Method of Lagrange Multipliers
fx(x,y) = λgx(x,y)
fy(x,y) = λgy(x,y)
g(x,y) = c
§ Lagrange multiplier ≈ delta f / delta g = change in optimum value of f / change in g
□ The value of Lagrange multiplier is approximately the change in the optimum value of when the value of the constraint is increased by 1 unit
□ The value of Lagrange multiplier represents the rate of change of the optimum value of f as the constraint increases
§ The Lagrangian fuction
□ L(x,y,λ) = f(x,y) - λ(g(x,y) - c)

Challenges:

I found the concept of Lagrange multiplier difficult and couldn’t really follow the explanation for it. It was also frustrating because there were several problems relating to economics. Similarly I had troubles understanding the Lagrangian function.

Reflection:

Constrained optimizations was interesting because both examples used in the book related to economics. Regardless that I haven't done either of the topics in the past the constrained optimization was a lot easier to understand compared to Lagrange multipliers. I'm hoping to really understand the Lagrange multiplier after tomorrow's class.

4.3 Global maxima and minima

Main points:
○ f has a global minimum at p if f(p) is less than or equal to all values f
○ f has a global maximum at p if f(p) is greater than or equal to all values f
○ Finding global max and min (including endpoints)
○ Compare values of the function at all the critical points in the interval and at the endpoints
○ Finding global max and min (excluding endpoints or on the entire real line)
○ Find the values of the function at all the critical points and sketch a graph

Challenges:

I ran into troubles as I tried to differentiate sin2x, so I still have weaknesses in the basic skills and I feel that they keep bugging me as we learn new topics. I also found word problems to be lot more difficult compared to problems with a set of given values. So translating word problems to mathematical language and understanding what is asked is still difficult for me.

Reflection:


I understood quickly how to find the global maximum and minimum but applying it to the problems is still difficult. This is mainly due to problems with basic differentiation and understanding written questions. The examples were interesting as I have done biology in the past and the example relating to minimizing gas consumption was good as I found it to relate to an everyday topic I have discussed with my friends.

1.3, 2.4, 4.1, 4.2 Rates of change, second derivative, the local maxima and minima and inflection point

Main points:
• Rates of change
○ 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
• Second derivative
○ f''
○ If y=f(x) then second derivative can be written as (d2y)/(dx2)
○ f''>0 on an interval --> f' is increasing --> graph of f is concave up
○ f''<0 on an interval --> f' is decreasing --> graph of f is concave down
• Local maxima and minima
○ f has a local minimum at p if f(p) is less than or equal to the values of f for points near p
○ f has a local maximum at p if f(p) is greater than or equal to the value of f for points near p
○ "local" because it is only near p
○ Critical point
§ Point p in the domain of f where f'(p)=0 or f'(p) is undefined
§ Point (p,f(p)) on the graph of f
○ Critical value
§ Value, f(p), of the function at a critical point
○ First and second derivative test
• Inflection point
○ A point at which the graph of a function f changes concavity
○ At the inflection point, f'' is zero or undefined (not always!)

Challenges:

Second derivative was pretty straightforward, the ones I had more problems was local maxima and minima and inflection point. Nevertheless I found myself still struggling with more complex forms of equations that contained e, ln, chain rule, product rule etc. I think it would be lot of help to go at least through one example of finding the local maximum, minimum and point of inflection.

Reflection:


Example about investment was interesting and I would like to see more problems like it. Otherwise I still felt that there was a lack of examples and problems related to my are of interest which is economics. This time I spent a bit more time studying the chapter and went through all of the problems, some for the second time too and it really helped to understand the topics. I don't know why I don’t find the time to do so every single time.

Gradient and directional derivatives

Main points:
• The gradient
○ Deltaf(x.y)=[fx(x,y),fy(x,y)]
• Properties of the gradient
○ The gradient vector always points in the direction of the greatest increase. The length of the
gradient corresponds to the steepness of the slope in that direction: the steeper the slope, the longer the gradient vector.
○ The gradient vector is always perpendicular to the level curve at which it
is rooted. All of the gradient vectors on the graph correspond to points
on the level curve f(x, y) = 1. The gradient direction is the direction
of greatest increase. The tangent to the level curve is the direction of no change.
○ The opposite direction to that of the gradient is the direction of greatest
decrease.
• Directional derivatives
○ Allows to find the rate of change in any direction in the (x,y) -plane
○ Directional derivative of f(x,y) at the point (x0,y0) in the direction of a unit vector u=(u1,u2) is
§ Duf(x0,y0)=fx(x0,y0)u1+fy(x0,y0)u2

Challenges:
I understood the method how to find gradients and directional derivatives. I am still lacking the full understanding of the importance of gradient and directional derivatives in real world problems. I also had problems understanding the properties of the gradient. I am hoping that going through the topic tomorrow in class will help me to understand these areas more.

Reflection:
The lack of real world examples continued and I hope to come across more practical examples with the after class problems. I think the current topic is very relevant in the area of economics, but I am hoping still to see the related examples.

Vectors, dot product, and vector components

Main points:
• Scalar multiplication
○ multiply a vector by a constant to get another vector whose length is
rescaled by that constant

• Vector addition
○ add two vectors to get a vector

• Dot products
○ The dot product u·v of two vectors u and v is the real number obtained
by multiplying corresponding coordinates of the vectors and adding

• The length of a vector u
○ IuI=√u*u

• Angle 0 between two nonzero vectors u and u
○ u*v=IuIIuI cos0
• Projection of a vector on a line
• The vector xu is the projection of a on the line spanned by u

Challenges:
I have not done vectors for a while and even when I studied the topic year ago, it was the one area where I had difficulties. I can somehow understand the basic principles but applying my knowledge is where it becomes difficult. So finding whether vectors are parallel is something I have troubles with.

Reflection:

I am still wondering how to apply vectors in the real world, as the examples were just mathematical ones. I would like to see examples how to apply vectors to problems in economics or sciences. This would provide me more motivation and personal interest to master this topic.

3.5, 9.3 & 9.4 Derivatives of periodic functions / Partial derivatives

Main points:

• Sine and cosine function are periodic-->their derivatives must be periodic also
• For x in radians
○ (d/dx)(sin x)=cos x
○ (d/dx)(cos x)=--sin x
• If z is a differentiable functions of t, then
○ (d/dt)(sin z)=cos z(dz/dt)
○ (d/dt)(cos z)=-sin z(dz/dt)
• If k is a constant, then
○ (d/dt)(sin kt)=k cos kt
○ (d/dt)(cos kt)=-k sin kt
• Partial derivative of R with respect to x at (a,b) is the derivative of f with y constant
○ Fx(a,b)=rate of change of f with y fixed at b, at the point (a,b)
• Partial derivative of f with respect to y at (a,b) is the derivative of f with x constant
○ Fy(a,b)=rate fo change of f with z fixed at a, at the point (a,b)
• Local linearity
○ Change in f ≈ rate of change in x-direction * delta x+ rate of change in y-direction * delta y
○ So, delta f ≈ fx*delta x + fy*delta y
• The mixed partial derivatives are equal
○ If fxy and fyx are continuous at (a,b)
§ Fxy(a,b)=fyx(a,b)

Challenges:

The derivatives of periodic functions where ok, as I have done them in the past. I still need to revise them in order to be comfortable with them. Partial derivatives are something what I haven't done in the past, and my first impression was just general confusion. Also the notation of partial derivative of R was something completely new to me and I would like to know where the notation comes from. I would like to see more emphasis put on the partial derivatives in tomorrow's class as they were just lot more complex compared to sine and cosine functions and their derivatives.

Reflection:

I found the airline example interesting in partial derivatives section as it did relate to my field of interest. The problem whether to increase labor or machinery is something what I am very likely going to encounter in my principles of economics class next semester. The topics seemed very relevant to my area of interest (economics) so I am looking forward to really gain deep understanding of partial derivatives.

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.

Introduction to Tapio

My name is Tapio Riihimaki.

I am a first year.

I have not decided my major yet, but most likely economics (possibly Chinese minor).

I previously did International Baccalaureate, and Standard Level Mathematics in it. It included functions and algerbra, trigonometry, matrices, vectors, calculus, statistics and probability.

My weakest part in high school was statistics and it is sad because it is the one are that I want to use later on in my life in social sciences.

The strongest are in my math bacground is most likely functions and algebra.

The course is required by the economics department and I believe that it will prove to be useful as I am interested in social sciences.

I want to build up my confidence in my mathematical skills once again (they are bit rusty at the moment because I had a gap year). I am also interested to learn applying calculus to problems faced in social sciences.

I am interested in economics, business, organizational studies and human resource management. Also nutrition and exercising are very close to my heart.

My worst math teacher did not have self confidence and was not able to keep the class under her control.

My best math teacher showed genuine interest towards math and gave us most of the time homework related to real world mathmetical problems.

I would like to hear Lose Yourself by Eminem.

By the way, I am positively surprised and impressed how much you are trying to engage us in the classroom. Have a great weekend