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.