Main Points:
The derivative of a composite function (f (g(t)) can be calculated as the derivative of the outside function multiplied by the derivative of the inside function (f ' (g(t)) * g ' (t)). This rule can also be used on a composite function. Just substitute the function in for g (t). The product rule states that (fg)' = f ' g + f g'. In other words, the derivative of a product is the derivative of the first times the second plus the derivative of the second times the first. The Quotient rule states that (f/g)' = f ' times g minus f times g ' all divided by g^2.
Challenges:
The first explanation of the chain rule where the book defined it in terms of dy/dt was confusing for me, I guess the form of notation that they put it in was the only part I had trouble with because I understand the concept, but it would be helpful if you went over that way of describing the chain rule in class.
Reflections:
The chain rule is really helpfull in solving derivatives, but I wonder what a graph of the function of a function and then the derivative of that would look like. I can't picture it. I wonder if you could give us a graphic sense of the chain rule in class. This class seems to be all about the why aspect of calculus, but it seems like the book glosses over the proofs for these rules.
Monday, September 29, 2008
Wednesday, September 24, 2008
3.1 Derivative Formulas for Powers and Polynomials & 3.2 Exponential and Logarithmic Functions
Main Points
The derivative of a constant function is always the same. The same is true of a Linear Function. If a function is multiplied by a constant, so is the derivative. The derivative of two functions added together is equal to the derivative of each separate function added together. The power rule states that d/dx (x^n) = nx^n-1. Differentiation is used to find the equation of a line tangent to a slope. If you graph the derivatives of 2^x or 3^x etc. you will see that the derivative graph looks like the original function, but stretched upward. This means that the derivative must be simply the function multiplied by a number. e^x = e^x. e approximately equals 2.718. The exponential rule states that d/dx (a^x) = (ln a)a^x. The derivative of ln(x) = 1/x.
Challenges
The exponential rule is a little mysterious, the solution is more complex than what the book gave, but maybe that's because they don't want to confuse us too much. Anyway, e and natural log just overall I don't know much about, so it's a little confusing. My problem is just not knowing enough about log and e and how they originated.
Reflections
Well now we can solve those pesky derivative problems quicker. I'm understanding everything. I've seen all of this material before, so it's not a problem. I really enjoyed going over the manual ways of finding derivatives and other things, because it makes it relavant, but now we forge on to the land of straight up math, but I think we'll still be able to apply it to the real world easily. I just liked to see how the math methods were made in the first place.
The derivative of a constant function is always the same. The same is true of a Linear Function. If a function is multiplied by a constant, so is the derivative. The derivative of two functions added together is equal to the derivative of each separate function added together. The power rule states that d/dx (x^n) = nx^n-1. Differentiation is used to find the equation of a line tangent to a slope. If you graph the derivatives of 2^x or 3^x etc. you will see that the derivative graph looks like the original function, but stretched upward. This means that the derivative must be simply the function multiplied by a number. e^x = e^x. e approximately equals 2.718. The exponential rule states that d/dx (a^x) = (ln a)a^x. The derivative of ln(x) = 1/x.
Challenges
The exponential rule is a little mysterious, the solution is more complex than what the book gave, but maybe that's because they don't want to confuse us too much. Anyway, e and natural log just overall I don't know much about, so it's a little confusing. My problem is just not knowing enough about log and e and how they originated.
Reflections
Well now we can solve those pesky derivative problems quicker. I'm understanding everything. I've seen all of this material before, so it's not a problem. I really enjoyed going over the manual ways of finding derivatives and other things, because it makes it relavant, but now we forge on to the land of straight up math, but I think we'll still be able to apply it to the real world easily. I just liked to see how the math methods were made in the first place.
Wednesday, September 17, 2008
2.2 The Derivative function & 2.3 Interpretations of the derivative
Main Points:
The derivative function is defined as the Instantaneous rate of change of f at x. The derivative function is positive when the slope of the original function is positive. The opposite is true when the original function has a negative slope. The sign of f ' corresponds with the direction of the slope of f (increasing, decreasing, or constant). Finding the slope at a given point from numerical values given is one way of finding the derivative. To get a more accurate result from this method average the slope from the values on either side of the target point. To find the derivative from a formula, you can take two very close values, plug them into the equation, find their difference and divide that by the difference between the two x values. The derivative can also be described as (delta y/delta x). This can be written (dy/dx). The derivative can be written to represent values such as velocity (v=ds(position)/dt). A local linear approximation can be made when the derivative is given. Delta y approximately equals f '(x) * delta x.
Challenges:
I was confused as to why Leibniz's notation is difficult to use to express what f '(x) equals. The whole infitesimally small change in x and y way of describing derivative was also a bit confounding. I just thought it was a poor way of explaining it.
Reflections:
I never got this extensive background of how a derivative is essentially derived, so it is interesting to see all of the very painfull methods of finding something so simple. In my calculus class last year, the teacher didn't bother explaining the how or why part of calculus, we just did it, so I appreciate this........kind of.
The derivative function is defined as the Instantaneous rate of change of f at x. The derivative function is positive when the slope of the original function is positive. The opposite is true when the original function has a negative slope. The sign of f ' corresponds with the direction of the slope of f (increasing, decreasing, or constant). Finding the slope at a given point from numerical values given is one way of finding the derivative. To get a more accurate result from this method average the slope from the values on either side of the target point. To find the derivative from a formula, you can take two very close values, plug them into the equation, find their difference and divide that by the difference between the two x values. The derivative can also be described as (delta y/delta x). This can be written (dy/dx). The derivative can be written to represent values such as velocity (v=ds(position)/dt). A local linear approximation can be made when the derivative is given. Delta y approximately equals f '(x) * delta x.
Challenges:
I was confused as to why Leibniz's notation is difficult to use to express what f '(x) equals. The whole infitesimally small change in x and y way of describing derivative was also a bit confounding. I just thought it was a poor way of explaining it.
Reflections:
I never got this extensive background of how a derivative is essentially derived, so it is interesting to see all of the very painfull methods of finding something so simple. In my calculus class last year, the teacher didn't bother explaining the how or why part of calculus, we just did it, so I appreciate this........kind of.
Monday, September 15, 2008
1.3 Rates of Change and 2.1 instantaneous rate of change
Main points
The average rate of change is determined by the change in y divided by the change in time for a given data set. A function is increasing if f(x) increases with x. The opposite is true for decreasing functions. The average rate of change between two points of a function is represented by a line connecting the two points. This line is called the secant line. concave up and down refer to which way a function "opens up." The average change in position with respect to time is also known as velocity. Speed is the magnitude of a velocity without respect to direction. Instantaneous velocity is the limit of the average velocity of the object over shorter and shorter time intervals containing (t) time. Instantaneous rate of change is similar except that it is basically a measure of the slope of a function over shorter and shorter intervals. The derivative of a function at point a is the instantaneous rate of change.
Challenges
I had a hard time distinguishing between instantaneous velocity and instantaneous rate of change. I realized that they are actually the same. Yeah I feel stupid. Velocity is a rate of change or basically the slope of a function.
Reflections
I'm used to finding the derivative in the good ol bring the exponent out front and subtract one from that method. It's good to see this method again to see how to "manually" come up with a derivative, it connects the process with its roots, at least for me.
The average rate of change is determined by the change in y divided by the change in time for a given data set. A function is increasing if f(x) increases with x. The opposite is true for decreasing functions. The average rate of change between two points of a function is represented by a line connecting the two points. This line is called the secant line. concave up and down refer to which way a function "opens up." The average change in position with respect to time is also known as velocity. Speed is the magnitude of a velocity without respect to direction. Instantaneous velocity is the limit of the average velocity of the object over shorter and shorter time intervals containing (t) time. Instantaneous rate of change is similar except that it is basically a measure of the slope of a function over shorter and shorter intervals. The derivative of a function at point a is the instantaneous rate of change.
Challenges
I had a hard time distinguishing between instantaneous velocity and instantaneous rate of change. I realized that they are actually the same. Yeah I feel stupid. Velocity is a rate of change or basically the slope of a function.
Reflections
I'm used to finding the derivative in the good ol bring the exponent out front and subtract one from that method. It's good to see this method again to see how to "manually" come up with a derivative, it connects the process with its roots, at least for me.
Wednesday, September 10, 2008
section 9.1 & 9.2
Main Points
The functions of two variables described in this section have two independent variables which affect the dependent variable instead of just one independent. A table of values can be used to demonstrate the function or an algebraic function itself can be used. If you want to see how the variables affect the dependent variable independently (though it's not that useful in every situation), you can define one of the variables to see how that affects the function. Contour diagrams are one graphical way to show information from a two variable function. Contour diagrams show equal values of the function connected by lines. This helps to visualize the idea. Cobb and Douglas came up with a two variable function for worker productivity and capital investment. P= f(N,V)=cN^aV^b. Contours on the diagrams are found by setting the dependent variable equal to a value and then plotting the resulting line equation.
Challenges
The whole setting one variable equal to something and then graphing the two variable function is confusing for me. I would like to see/do an example of isolating one variable in class. I'm pretty sure it's easier than I think, but the two graphs the book shows are slightly confusing.
Reflections
I thought the contour mapping section was really interesting. I never considered that those temperature maps were just lines connecting places with a temp on a ten degree interval.
The functions of two variables described in this section have two independent variables which affect the dependent variable instead of just one independent. A table of values can be used to demonstrate the function or an algebraic function itself can be used. If you want to see how the variables affect the dependent variable independently (though it's not that useful in every situation), you can define one of the variables to see how that affects the function. Contour diagrams are one graphical way to show information from a two variable function. Contour diagrams show equal values of the function connected by lines. This helps to visualize the idea. Cobb and Douglas came up with a two variable function for worker productivity and capital investment. P= f(N,V)=cN^aV^b. Contours on the diagrams are found by setting the dependent variable equal to a value and then plotting the resulting line equation.
Challenges
The whole setting one variable equal to something and then graphing the two variable function is confusing for me. I would like to see/do an example of isolating one variable in class. I'm pretty sure it's easier than I think, but the two graphs the book shows are slightly confusing.
Reflections
I thought the contour mapping section was really interesting. I never considered that those temperature maps were just lines connecting places with a temp on a ten degree interval.
Monday, September 1, 2008
Main Points:
Linear functions are straight lines. The slope of that given line is the the rise of the y value over a certain period divided by the change in x value over that same period. The main form for linear functions is y=mx+b. b represents the y intercept. Exponential equations often are used for functions involving growht and decay. They take the form of y=a^x Exponential growth and decay functions are very similar to exponential functions, but the rate of growth is also exponential in these problems. Doubling time, half-life, financial interest compounding, and present and future value functions all use the exponential growth form.
Challenges:
I had trouble with the idea of compounding interest continuously, the whole concept baffled me for a while, but I believe the idea is basically that the bank is just splitting up the interest payment so that they are constantly paying a tiny fraction of the whole, but that whole sum of money is constantly creeping higher.
Reflections:
This reading helped me understand my bank's operations more, which is usefull. Anytime I gain an understanding of why something is or how something works is a good feeling for me, so I appreciated learning that.
Linear functions are straight lines. The slope of that given line is the the rise of the y value over a certain period divided by the change in x value over that same period. The main form for linear functions is y=mx+b. b represents the y intercept. Exponential equations often are used for functions involving growht and decay. They take the form of y=a^x Exponential growth and decay functions are very similar to exponential functions, but the rate of growth is also exponential in these problems. Doubling time, half-life, financial interest compounding, and present and future value functions all use the exponential growth form.
Challenges:
I had trouble with the idea of compounding interest continuously, the whole concept baffled me for a while, but I believe the idea is basically that the bank is just splitting up the interest payment so that they are constantly paying a tiny fraction of the whole, but that whole sum of money is constantly creeping higher.
Reflections:
This reading helped me understand my bank's operations more, which is usefull. Anytime I gain an understanding of why something is or how something works is a good feeling for me, so I appreciated learning that.
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