3.1 and 3.2: Derivative Formulas for Powers and Polynomials and Exponential and Logarithmic Functions

Main Points

3.1 lists several formulas for finding derivatives that i had to memorize last year: the derivative of a constant=0, ect... These rules allow us to find short cuts when determining a function's derivative. The derivative of a^x is closely related to the original function. As values of a change, the graph of the derivative varies in its similarity to the original. When a is a certain number e, it naturally has a derivative that equals the function.

Challenges

why is f'(x) of f(x), m in the equation? I thought the derivative was the slope of the tangent, not the slope of the actual line. I do not understand logarithms in general, their relationship to derivatives,and to e. 

Reflections

I memorized all these rules last year so they are familiar to me. However now that we've done some graphing first i understand why the derivative of a constant is zero. The deriv of a line is a constant, so it makes sense that if u have a constant, it would have percentage change of zero and a deriv. of zero. I was wondering if e had special implications for nature as well as logs and ln. Is it a number that is found in the ration of plants, animals, stars, ect...?

2.2 and 2.3: the derivative of a function and interpretations of a derivative

 Main Points

Because the slopes of the points of a function have numerical values, these numbers can be plotted to create both another graph and another function. The derivative function is the derivative of a y value at an x value.  Because the derivative is the instantaneous change which merely looks at the average rate of change for 2 points infinitely close, the f' can also be written as change in y over the change in x. The equation for this is delta y is approximately f' times delta x.

Challenges.

How can a a bunch of tangent lines (the derivative) form a graph? Why can f'=delta y/delta x. F' is the slope of the tangent line, not f(x). Why does the derivative show how quickly the function is chaning?

Reflections

When first reading the section about derivatives having derivatives, it made no sense to me. I just remembered though that in calc last year we found f' and f'' and f'''. It makes sense because you can graph velocity and acceleration and jerk.

1.3 and 2.1: rates of change and instantaneous rate of change

main points

When giving a a list of data, their average can always be determined. Therefore even when there is no constant rate of change there always an average rate of change. A function is said to be increasing if y increases proportionally to x. The function is decreasing if y increases disproportionally to x. Because y changes with respect to x, as the x values increase the change in y is the change between the y coordinates those two x values. Periodic functions go through increases and decreases and there for concave up and down, respectively. 

The instantaneous rate of change is the the velocity at a specific point. Its value must be calculated, because this value is based on average velocity, which gets more precise for a given point as the intervals decrease. Instantaneous velocity is also the smallest interval of the average velocity of the surrounding points, because two points are required to find an average.

As a tangent line becomes more horizontal, the interval between the two points it hits on the function line decreases, therefore indicating the derivative.

challenges

If values increase proportionally to each other, does that mean that they both increase or does is mean that they both increase by the same factor?

I know they are related because the points for corners of a triangle, but I don't really understand why the change in the function between two x values is delta y.

Why can't the actual velocity of the grapefruit be calculates, as opposed to estimated?

How exactly is the tangent the derivative. How can another line represent the velocity?

reflections

I think that the concept of average rate of change is very important because in real world situations, I assume that data actually literally increases/decreases with a constant rate of change. However a pattern an be gauged and assumptions can still be drawn about the function by using the average rate of change.

The visualizing rate of change section looks very similar to a diagram my old math teacher showed us when explaining derivatives, so I'm guessing they are closely related.

9.1 + 9.2

Main Points

Certain values (variables) can dependent on multiple factors. Therefore the dependent variable R can have both x any y as its independent variable. Increasing functions are those which have one value increasing while the independent variables remain the same. Contour diagrams can be used to visually represent 2 variable functions. Taking a map of the U.S. for example, the map can serve as a graph because the horizontal and vertical increases represent latitude and longitude, the independent variables, while the third is represented by written numerical values  and shown pictorially through a diagram. Contour diagrams can be thought of as a birds-eye-view of a certain geographical feature, when appropriate. The Cobb-Douglas Production Function has important real world implications. This function allows one to determine the output of two different methods, or both combined. Contour tables, and diagrams (which utilize 3 axes). The algebraic formula for contours is f(x,y)=c.

Challenges

Is the definition of an increasing function that R increases while either x or y increases and the other stays constant?

How is dividing a country into zones mean that there is a graph of 2 variables. I am used to thinking of graphs as a line or curve and do not comprehend how a bunch of diving lines constitutes a graph. I don't understand the graphs of the Cobb-Douglas function, or how it can be used to determine how much of both, for example, the number of workers and value of the machines, would increase out-put. I also don't understand what cross contouring is.

Reflections

By looking at one variable at a time, aren't you just essentially making it a normal problem with one dependent and one independent. I therefore don't see the point in it. 

I have used topographical maps for earth science classes but it never occurred to me that they are actually graphs of 3 variable functions. Its pretty crazy that contour diagrams are both diagrams and graphs at the same time. The plugged in formula for contour function seems very similar to that of a circle. 

exponential functions, growth and decay (1.5+1.7)

Back round

Hazel Schaeffer, Freshman

I intend to major in Political Science, and perhaps minor in English or International Studies.

In high school I took trig, pre-calc, and calc.

Trig is my least favorite, what I least remember, and is probably my weakest part.

My strongest part is probably calculus because I just took it.

I am taking this class because I do enjoy math when I understand it, because I wanted to take a variety of classes in my first semester, and because I wanted to satisfy my math requirement while my calculus is freshest.

I hope to pacify my fears that I cannot take a college math class. Because I am intending to major is poli sci and because the world is filled with numbers, I hope to learn skills I can apply to my real life.

My worst teacher was boring so paying attention in class was difficult. However I value clarity over excitement because math naturally flusters me.

My favorite math teachers are engaged and are humorous, therefore encouraging me to pay attention and work hard. If I do not understand something (which is often), they are patient and do not make me feel bad. However they always try to be clear.

Personally listening to music makes it harder for me to study. However supposedly, according to some study, listening to Bach or Mozart or Beethoven (or some classical musician that I do not remember) improves studying.  In general I like the  Red Hot Chili Peppers and Fleetwood Mac.

During class I found the clicker system a hindrance. I felt under pressure to click in before you ended the time, there forfeiting 85% of the participation grade, so I felt flustered and was too busy paying attention to the number of people who had answered to properly concentrate. When I noticed a majority of people had answered I merely guessed so as to not lose the points. I know this is silly and that hopefully I will get used to the system, but I would appreciate if for the first few classes you could go slower so I will feel less pressured.