Derivatives of composite functions / Derivatives of products and quotients, 3.3, 3.4
1)Main Points
Chapter 3.3 introduces a mathematical concept called the chain rule which states that if y=f(z) and z=g(t), then the derivative of the composite function y=f(g(t)) is dy/dx = dy/dz * dz/dt. This in words means that the derivative of a composite function is the derivative of the outside function times the derivative of the inside function. This in function form is y=(t+1)^4, or y=z^4 when z=t+1 (where z is the inside function). This is simply a way of splitting up a function into different parts so you can calculate the derivative of a function like y=(4t^2 + 1)^7. 3.4 introduces how to find the derivative of a product or quotient of functions. We notice that simply multiplying the derivatives does not give us the correct answer. We learn that (fg)'= f'g + fg', which means the derivative of a product is the derivative of the first times the second plus the first times the derivative of the second. We find that through the product rule we can create a rule for quotients as well which is: (f/g)'=f'g-fg'/g^2.
2) Challenges
3.3 introduces some challenges for me. I find it really difficult to keep all the things in the chain rule strait. It is easy to forget that at the end of calculating the derivative for the inside function you must multiply it by the derivative of the outside function. I should practice these a little and be able to use them without getting confused. The product/quotient rules make a little bit more sense to me but could also present a lot of logistical problems. It is easy to get the variables confused or getting the basic derivatives wrong could put off the whole problem. It will be important for me to memorize the derivative rules and make sure that I write clearly.
3)Reflections
Both chapters introduced ways of finding the derivatives of more complicated compound functions. This, I'm sure will be useful when we encounter some real world problems that are more difficult to solve than with just simple derivative rules. I hope that I can master these skills well enough to apply them.
Monday, September 29, 2008
Wednesday, September 24, 2008
3.1, 3.2 Derivatives of powers and polynomials / Derivatives of exponentials and logs
Derivatives of powers and polynomials / Derivatives of exponentials and logs, 3.1, 3.2.
1) Main Points
Instead of finding the derivative by way of limits and points on graphs or tables, we can calculate the derivative by using an equation. For a linear function, we know that the rate of change is constant, and in y = mx + b form m = slope = f'(x). We also notice that for a particular x value, the derivative varies directly with the slope ie: y = 3f(x), slope = 3m, y = f(x)/2, slope = m/2. If a function is multiplied by a constant then the derivative is also multiplied by the same constant. When adding derivatives: f'(x) = {f(x) + g(x)} = f'(x) + f'(y). To find the derivative: {d/dx(x^n)= nx^n-1}. To find the slope of a tangent line at a certain point x on a function, you can find the derivative equation of the function and then substitute in the value of x. In 3.2 the book attempts to find the derivative of a power function. The find a value for f'(0), and use this as a multiplier. They also ask when (because both the derivative and the power function are increasing in the same direction) the derivative will be equal to the function. This happens with the function e^x. This helps us realize that : f'(a^x)= (ln a)a^x. For logarithmic functions: f'(lnx)= 1 / x.
2) Challenges
A challenge for me regarding derivatives in general is graphing them. It is important for me to remember that the derivative will have a very different shape than the function itself. And when the derivative is zero it means the graph is changing directions around a curve. Something I noticed about the nature of derivatives that may help me in the future is that the derivative is always one less power than the function (n-1). So if I am trying to find the derivative of a cubed function I know it will be a quadratic type line, etc. I am confused about why the book used f'(0) as a multiplier for the derivative of a power function. In this chapter I was introduced to a several ways of finding the derivative, so I will have to make sure I remember them and don't mix them up.
3) Reflections
Now we have expanded our knowledge of derivatives to encompass many of the types of functions we know today. This will be very useful for calculating the rates of change of those harder-to-model real life equations such as exponential and power functions, and logarithmic functions. Also inflation rates and equations that use natural logarithms.
1) Main Points
Instead of finding the derivative by way of limits and points on graphs or tables, we can calculate the derivative by using an equation. For a linear function, we know that the rate of change is constant, and in y = mx + b form m = slope = f'(x). We also notice that for a particular x value, the derivative varies directly with the slope ie: y = 3f(x), slope = 3m, y = f(x)/2, slope = m/2. If a function is multiplied by a constant then the derivative is also multiplied by the same constant. When adding derivatives: f'(x) = {f(x) + g(x)} = f'(x) + f'(y). To find the derivative: {d/dx(x^n)= nx^n-1}. To find the slope of a tangent line at a certain point x on a function, you can find the derivative equation of the function and then substitute in the value of x. In 3.2 the book attempts to find the derivative of a power function. The find a value for f'(0), and use this as a multiplier. They also ask when (because both the derivative and the power function are increasing in the same direction) the derivative will be equal to the function. This happens with the function e^x. This helps us realize that : f'(a^x)= (ln a)a^x. For logarithmic functions: f'(lnx)= 1 / x.
2) Challenges
A challenge for me regarding derivatives in general is graphing them. It is important for me to remember that the derivative will have a very different shape than the function itself. And when the derivative is zero it means the graph is changing directions around a curve. Something I noticed about the nature of derivatives that may help me in the future is that the derivative is always one less power than the function (n-1). So if I am trying to find the derivative of a cubed function I know it will be a quadratic type line, etc. I am confused about why the book used f'(0) as a multiplier for the derivative of a power function. In this chapter I was introduced to a several ways of finding the derivative, so I will have to make sure I remember them and don't mix them up.
3) Reflections
Now we have expanded our knowledge of derivatives to encompass many of the types of functions we know today. This will be very useful for calculating the rates of change of those harder-to-model real life equations such as exponential and power functions, and logarithmic functions. Also inflation rates and equations that use natural logarithms.
Wednesday, September 17, 2008
2.2, 2.3
2.2 and 2.3
1) Main Points
We can plot a line tangent to the point to find the derivative. When we plot a graph of derivatives we notice that is is different from the graph of the function. If f'> 0 then f is increasing, if f' < 0 then f is decreasing, if f' = 0 then f is constant over that interval. Derivates can also b expressed in tables. From these table we can "guess" a formula for the derivative. The derivative can also be written as dy/dx, which means the difference in y divided by the difference in x. If this was for a graph of distance over time then the derivative would represent a graph of the velocity because distance/time = velocity. The units of the derivative are the units of dy/dx. The derivative of the derivative, dv/dt would be the acceleration, which would graph the changes in the rate of change. The derivative can be used to calculate other things regarding rates of change.
2) Challenges
With derivatives, units can get confused and are also very important. It is easy to functions and their derivatives mixed up. Sometimes it is hard to determine what the derivative represents in a word problem (I need to continue to remind my self that it quantifies a rate of change and can be used to predict what happens next in the function).
3) Reflections
Derivatives are the start of real calculus and are obviously very applicable to and also very useful in real life. You can use the derivative to measure the speed of someone running at constant velocity or you can predict the length of his next couple strides if he is running at an inconstant speed. Derivatives have financial applications. You can use the derivative to predict an increase in sales if you increase a variable such as advertising. The derivative goes beyond just modeling what nature does into predicting and showing how it changes.
1) Main Points
We can plot a line tangent to the point to find the derivative. When we plot a graph of derivatives we notice that is is different from the graph of the function. If f'> 0 then f is increasing, if f' < 0 then f is decreasing, if f' = 0 then f is constant over that interval. Derivates can also b expressed in tables. From these table we can "guess" a formula for the derivative. The derivative can also be written as dy/dx, which means the difference in y divided by the difference in x. If this was for a graph of distance over time then the derivative would represent a graph of the velocity because distance/time = velocity. The units of the derivative are the units of dy/dx. The derivative of the derivative, dv/dt would be the acceleration, which would graph the changes in the rate of change. The derivative can be used to calculate other things regarding rates of change.
2) Challenges
With derivatives, units can get confused and are also very important. It is easy to functions and their derivatives mixed up. Sometimes it is hard to determine what the derivative represents in a word problem (I need to continue to remind my self that it quantifies a rate of change and can be used to predict what happens next in the function).
3) Reflections
Derivatives are the start of real calculus and are obviously very applicable to and also very useful in real life. You can use the derivative to measure the speed of someone running at constant velocity or you can predict the length of his next couple strides if he is running at an inconstant speed. Derivatives have financial applications. You can use the derivative to predict an increase in sales if you increase a variable such as advertising. The derivative goes beyond just modeling what nature does into predicting and showing how it changes.
Monday, September 15, 2008
1.3 and 2.1.
1.3 and 2.1.
1)Main Points
Chapter 1.3 deals with rates of change and how to calculate them, especially if they are not constant. The average rate of change can be modeled as the Change in the dependent variable / the change in the independent variable. For a function to be increasing, the values of F(x) must increase as x increases. Chapter 2.1 discusses using the derivative to calculate the instantaneous rate of change as opposed to the rate of change over a long period time. The instantaneous velocity of an object can be determined by taking the limit of the average velocity of the object at smaller and smaller time intervals. The instantaneous rate of change of a function f at a is called the "derivative" of f at a, this is written as f'(a). The derivative is the same thing as a) the slope of the graph at that point and b) the slope of a line tangent to that exact point. The derivative can be calculated mathematically by subtracting a very small change in the dependent variable by a the same change in the independent variable. For example: you know that the rate of change is the change in y divided by the change in x. And you know that y=f(x). So with two points, a and (a+.0001) you can in the derivative of f(x) at a by doing {f(a+.0001)- f(a)/ (a+.0001) - a}.
2)Challenges
On a distance over time graph it is easy to forget that the slope between two points is the average velocity. The concept of limits is something that I've only had a little bit of exposure to but it makes sense when calculating something at in instant in time. It will be important for me to keep in mind that it is not an exact number. Also it is sometimes difficult to recognize the tangent line of a curve in an exact spot.
3)Reflections
The use of derivative seems to have lots of practical applications regarding speeds of objects and many inconstant rates of change. The book used the example of a grape fruit thrown up into the air and calculated its average speed. The derivative could have been applied in this situation calculate the instantaneous speed of the grapefruit instead of just the average speed. There are many real world applications for this chapter, probably more so than linear functions because usually things in the world don't change at constant rates.
1)Main Points
Chapter 1.3 deals with rates of change and how to calculate them, especially if they are not constant. The average rate of change can be modeled as the Change in the dependent variable / the change in the independent variable. For a function to be increasing, the values of F(x) must increase as x increases. Chapter 2.1 discusses using the derivative to calculate the instantaneous rate of change as opposed to the rate of change over a long period time. The instantaneous velocity of an object can be determined by taking the limit of the average velocity of the object at smaller and smaller time intervals. The instantaneous rate of change of a function f at a is called the "derivative" of f at a, this is written as f'(a). The derivative is the same thing as a) the slope of the graph at that point and b) the slope of a line tangent to that exact point. The derivative can be calculated mathematically by subtracting a very small change in the dependent variable by a the same change in the independent variable. For example: you know that the rate of change is the change in y divided by the change in x. And you know that y=f(x). So with two points, a and (a+.0001) you can in the derivative of f(x) at a by doing {f(a+.0001)- f(a)/ (a+.0001) - a}.
2)Challenges
On a distance over time graph it is easy to forget that the slope between two points is the average velocity. The concept of limits is something that I've only had a little bit of exposure to but it makes sense when calculating something at in instant in time. It will be important for me to keep in mind that it is not an exact number. Also it is sometimes difficult to recognize the tangent line of a curve in an exact spot.
3)Reflections
The use of derivative seems to have lots of practical applications regarding speeds of objects and many inconstant rates of change. The book used the example of a grape fruit thrown up into the air and calculated its average speed. The derivative could have been applied in this situation calculate the instantaneous speed of the grapefruit instead of just the average speed. There are many real world applications for this chapter, probably more so than linear functions because usually things in the world don't change at constant rates.
Wednesday, September 10, 2008
9.1 and 9.2
9.1 and 9.2.
1)Main points
This chapter has to do with functions that have two variables. All possible inputs ie; (x,y) is called the domain of f. By keeping one variable constant in a two variable function you can interpret useful data. For example, if you have an equation for the time of diffusion of a medicine in your body, you can keep the amount injected at a constant and notice the time it takes to diffuse, or you can keep the time constant and calculate the remaining medicine in the blood stream for a given injection. Two variable functions can also be modeled by a contour diagram (9.2). When depicting altitude on a contour map, the closer together the contour lines, the steeper the terrain. Also the numbers that label the lines usually stand for altitude on a topographical map.
In contour diagrams such as the corn example given in the book, information can also be collected by keep one of the variables constant and noticing how the effects of the other variable changing. The Cobb-Douglas function can be used to model economic output. A contour consists of all the points (x,y) where f(x,y) has a constant value, c.
2) Challenges
I find drawing contour lines very difficult. It is hard for me to find where the line should be from the data. In general thinking about a function with two variables is confusing, but I find it easier when I can use a real world example such as the office example about increasing staff & equipment. I will also have to remember which variable is which in a problem. For example, if there is a difference between the effects of rainfall than temperature on corn and it is easy to mix up those variables which would give some wrong info.
3) Reflections
Contour diagrams and two variable functions are used in all sorts of applications today. From maps to weather charts, and business or economic models. They model very real world situations and are extremely easy and quick to read. A map with contour lines can show you the altitude and shape of a mountain in an incredibly useful way.
1)Main points
This chapter has to do with functions that have two variables. All possible inputs ie; (x,y) is called the domain of f. By keeping one variable constant in a two variable function you can interpret useful data. For example, if you have an equation for the time of diffusion of a medicine in your body, you can keep the amount injected at a constant and notice the time it takes to diffuse, or you can keep the time constant and calculate the remaining medicine in the blood stream for a given injection. Two variable functions can also be modeled by a contour diagram (9.2). When depicting altitude on a contour map, the closer together the contour lines, the steeper the terrain. Also the numbers that label the lines usually stand for altitude on a topographical map.
In contour diagrams such as the corn example given in the book, information can also be collected by keep one of the variables constant and noticing how the effects of the other variable changing. The Cobb-Douglas function can be used to model economic output. A contour consists of all the points (x,y) where f(x,y) has a constant value, c.
2) Challenges
I find drawing contour lines very difficult. It is hard for me to find where the line should be from the data. In general thinking about a function with two variables is confusing, but I find it easier when I can use a real world example such as the office example about increasing staff & equipment. I will also have to remember which variable is which in a problem. For example, if there is a difference between the effects of rainfall than temperature on corn and it is easy to mix up those variables which would give some wrong info.
3) Reflections
Contour diagrams and two variable functions are used in all sorts of applications today. From maps to weather charts, and business or economic models. They model very real world situations and are extremely easy and quick to read. A map with contour lines can show you the altitude and shape of a mountain in an incredibly useful way.
Monday, September 8, 2008
1.10
1.10
1) Main Points
Periodic means repeating at regular intervals. Amplitude is half the distances between the max and min. The period is the time for the function to complete one whole cycle. y=sin(t), the period of a sine function is 2 pi. The number that multiplies the sin(t) part of the function affects the amplitude. If a number is added or subtracted to sin(t) it is shifted up or down accordingly. If a number is added to t then the graph is shifted left, subtracted it is shifted right. Period = 2pi/B and B is the number that will multiply t (y=Asin(Bt).
2) Challenges
1 cycle is when the function goes down then up then back to the starting point. It will be important for me to keep in mind what a whole cycle looks like. It is counter intuitive that when you add a number to t the graph shifts to the negative direction. I will also have to make sure i recognize that this relationship is backward. I will also need to burn into my memory period= 2pi/B. This is a very important for finding the relationship between the parameter B and the period, which are NOT the same.
3) Reflections
Periodic functions are very applicable to the real world. I can imagine using them to describe temperature/whether patterns, tides, or changes in the stock market. So many things occur periodically in the world and a knowledge in periodical functions can help describe them.
1) Main Points
Periodic means repeating at regular intervals. Amplitude is half the distances between the max and min. The period is the time for the function to complete one whole cycle. y=sin(t), the period of a sine function is 2 pi. The number that multiplies the sin(t) part of the function affects the amplitude. If a number is added or subtracted to sin(t) it is shifted up or down accordingly. If a number is added to t then the graph is shifted left, subtracted it is shifted right. Period = 2pi/B and B is the number that will multiply t (y=Asin(Bt).
2) Challenges
1 cycle is when the function goes down then up then back to the starting point. It will be important for me to keep in mind what a whole cycle looks like. It is counter intuitive that when you add a number to t the graph shifts to the negative direction. I will also have to make sure i recognize that this relationship is backward. I will also need to burn into my memory period= 2pi/B. This is a very important for finding the relationship between the parameter B and the period, which are NOT the same.
3) Reflections
Periodic functions are very applicable to the real world. I can imagine using them to describe temperature/whether patterns, tides, or changes in the stock market. So many things occur periodically in the world and a knowledge in periodical functions can help describe them.
Monday, September 1, 2008
1.5 and 1.7
Blog for reading due Tuesday 9/2, on 1.5 and 1.7
1) Main Points:
Whenever you have a constant percent increase it's exponential growth. If the rate of change number is greater than zero and less than one the function is a decreasing exponential function. Growth and decay can be modeled by P=P(e) to the (k x t) power. Every exponentially growing function has a fixed doubling time. For interest: annually P=P(1+r) to the t, continuously P=Pert. B, the future value of a payment is the amount P would have grown if in an interest-bearing bank account.
2) Challenges:
I was challenged by the example given of a decreasing exponential function. For something that is decreased by 40% for every
unit of time, it took me a while to understand that 60% of it would be left each time. Another confusing aspect of exponential functions is that the absolute rate of change is not constant but the relative rate of change is. For solving exponential growth and decay problems I will have to remember to use natural log, Ln, to isolate the variable. It took me a while to understand the two ways of calculating the future values of sums of money (re: interest), you can either calculate what the actual present value of it is or you can calculate what it will be worth in the future.
3) Reflections:
Exponential functions apply very well to population data, because most often populations grow faster as they get bigger. I also notice this mathematical relationship in cells reproduction. Future and Present values were concepts that were new to me but they show many practical uses for determining not only the changing value of money but of other things in nature the exhibit predictable change.
1) Main Points:
Whenever you have a constant percent increase it's exponential growth. If the rate of change number is greater than zero and less than one the function is a decreasing exponential function. Growth and decay can be modeled by P=P(e) to the (k x t) power. Every exponentially growing function has a fixed doubling time. For interest: annually P=P(1+r) to the t, continuously P=Pert. B, the future value of a payment is the amount P would have grown if in an interest-bearing bank account.
2) Challenges:
I was challenged by the example given of a decreasing exponential function. For something that is decreased by 40% for every
unit of time, it took me a while to understand that 60% of it would be left each time. Another confusing aspect of exponential functions is that the absolute rate of change is not constant but the relative rate of change is. For solving exponential growth and decay problems I will have to remember to use natural log, Ln, to isolate the variable. It took me a while to understand the two ways of calculating the future values of sums of money (re: interest), you can either calculate what the actual present value of it is or you can calculate what it will be worth in the future.
3) Reflections:
Exponential functions apply very well to population data, because most often populations grow faster as they get bigger. I also notice this mathematical relationship in cells reproduction. Future and Present values were concepts that were new to me but they show many practical uses for determining not only the changing value of money but of other things in nature the exhibit predictable change.
Subscribe to:
Posts (Atom)