SECTION 3.3: The Chain Rule
If y = f(z) and z = g(t), then the derivative of y = f(g(t)) is given as
dy/dt = dy/dz . dz/dt as long as dz and dt do not equal 0.
In other words, the derivative of a composite function s the derivative of the outside function times the derivative of the inside function:
d/dt f (g (t)) = f ' (g (t)) . g ' (t)
The chain rule can also be found for larger formulae by delaing with the formula as a single variable. For example,
y = (2t+1)^4 can be differntiated first by assuming a variable z = (2t+1)
Therefore, y = z^4
y' = 4(z^3)*z' = 4((2t+1)^3)*2 = 8(2t+1)^3
Some standard differential identities are as follows:
d/dt (t^n) = n*(t^n-1)
d/dt (e^t) = e^t
d/dt (ln t) = 1/t
d/dt (e^kt) = k*(e^kt) where k is an integer constant
Similarly,
d/dt (z^n) = n*(z^n-1)*dz/dt
d/dt (e^z) = (e^z)*dz/dt
d/dt (ln z) = (1/z)*dz/dt
I found it difficult to understand the example of how to determine the derivative of a composite function from a graph. I couldn't understand how it was being calculated. The theory made sense but the visuals didn't.
Again, learning how to determine rates of changes of irregular slopes over smaller and smaller intervals will always be useful in analyzing future market trends in a subject like Economics.
Thursday, February 28, 2008
Tuesday, February 26, 2008
SECTION 3.2: Exponential and Logarithmic Functions
The Exponential Function:
The graph of an exponential function increases slowly for x<0>0. This implies that the derivative of a function f(x) = a^x, is small for x<0>0. Since the function is increasing for all values of x, the graph of the derivative always remains above the x-axis.
At a = 2.718... it is found that d/dx (a^x) = a^x. For this reason a is defined as the number e.
This implies that d/dx (e^x) = e^x.
The exponential rule of differentiation states that
d/dx (a^x) = (ln a)*(a^x)
Since ln a is a constant, it can be said that the derivative of the exponential function (a^x) is proportional to the original function.
The Derivative of Natural Logarithm:
The graph of f(x) = ln x, shows that ln x is always increasing. Therefore its derivative is positive. The graph of f(x) is concave down, therefore its derivative must be decreasing.
ln x is very large near x=0 and very small at x = infinity.
The natural logarithm rule of differentiation states that,
d/dx (ln x) = 1/x
The difficult part about this section was understanding exactly why those derivative formulae are obtained. Beyond seeing the graphical explanations, there isn't enough to really understand how ones arrives at those formulae.
Logarithms are another method used to interpret graphical trends and hence can be useful in Economics.
The Exponential Function:
The graph of an exponential function increases slowly for x<0>0. This implies that the derivative of a function f(x) = a^x, is small for x<0>0. Since the function is increasing for all values of x, the graph of the derivative always remains above the x-axis.
At a = 2.718... it is found that d/dx (a^x) = a^x. For this reason a is defined as the number e.
This implies that d/dx (e^x) = e^x.
The exponential rule of differentiation states that
d/dx (a^x) = (ln a)*(a^x)
Since ln a is a constant, it can be said that the derivative of the exponential function (a^x) is proportional to the original function.
The Derivative of Natural Logarithm:
The graph of f(x) = ln x, shows that ln x is always increasing. Therefore its derivative is positive. The graph of f(x) is concave down, therefore its derivative must be decreasing.
ln x is very large near x=0 and very small at x = infinity.
The natural logarithm rule of differentiation states that,
d/dx (ln x) = 1/x
The difficult part about this section was understanding exactly why those derivative formulae are obtained. Beyond seeing the graphical explanations, there isn't enough to really understand how ones arrives at those formulae.
Logarithms are another method used to interpret graphical trends and hence can be useful in Economics.
Sunday, February 24, 2008
SECTION 3.1: Derivative Formulas for Powers and Polynomials
The derivative of constant is 0 as the slope of the line passing through that point is also 0.
Hence is f(x) = k, it implies that f ' (x) = 0.
The derivative of a linear function refers to the slope of that line, i.e.,
if f(x) = mx +b, f ' (x) = m
A function can be multiplied by a constant. If this is done, the general shape of the graph changes depending on the magnitude of the constant and its sign. The magnitude changes the slope of the curve at each point while the sign determines whether the function is reflected over the x-axis or not (reflected only if negative).
If c is a constant, then d/dx [c*f(x)] = c*f ' (x)
Derivatives of sums and differences of two functions can be found by simply adding or subtracting their magnitudes, i.e.,
d/dx [f(x) + g(x)] = f ' (x) + g ' (x) and,
d/dx [f(x) - g(x)] = f ' (x) - g ' (x)
Theoretically and graphically proven rules of differentiation:
The power rule:
d/dx (x^n) = n*[x^(n-1)]
The constant multiple rule:
d/dx [3*(x^5)] = 3*[d/dx(x^5)] = 3*[5*(t^4)] = 15(t^4)
The sum rule:
d/dx [(p^5)+(p^3)] = d/dx (p^5) + d/dx (p^3)
Similar rules can be used when differentiating to find the second derivative of a function. Finding the second derivative implies finding the derivative of the derivative of a function.
It is confusing to understand exactly what the graph of a second derivative aims to show of a function. It also brings up the question of whether a third or fourth derivative can also exist in a real world situation.
These formulae can be useful in determining general trends of markets. Anything related to graphs and relations of functions such as this can be useful in the field of economics.
The derivative of constant is 0 as the slope of the line passing through that point is also 0.
Hence is f(x) = k, it implies that f ' (x) = 0.
The derivative of a linear function refers to the slope of that line, i.e.,
if f(x) = mx +b, f ' (x) = m
A function can be multiplied by a constant. If this is done, the general shape of the graph changes depending on the magnitude of the constant and its sign. The magnitude changes the slope of the curve at each point while the sign determines whether the function is reflected over the x-axis or not (reflected only if negative).
If c is a constant, then d/dx [c*f(x)] = c*f ' (x)
Derivatives of sums and differences of two functions can be found by simply adding or subtracting their magnitudes, i.e.,
d/dx [f(x) + g(x)] = f ' (x) + g ' (x) and,
d/dx [f(x) - g(x)] = f ' (x) - g ' (x)
Theoretically and graphically proven rules of differentiation:
The power rule:
d/dx (x^n) = n*[x^(n-1)]
The constant multiple rule:
d/dx [3*(x^5)] = 3*[d/dx(x^5)] = 3*[5*(t^4)] = 15(t^4)
The sum rule:
d/dx [(p^5)+(p^3)] = d/dx (p^5) + d/dx (p^3)
Similar rules can be used when differentiating to find the second derivative of a function. Finding the second derivative implies finding the derivative of the derivative of a function.
It is confusing to understand exactly what the graph of a second derivative aims to show of a function. It also brings up the question of whether a third or fourth derivative can also exist in a real world situation.
These formulae can be useful in determining general trends of markets. Anything related to graphs and relations of functions such as this can be useful in the field of economics.
Thursday, February 21, 2008
SECTION 2.3: Interpretations of the Derivative
The derivative can be noted either as f ' (x) or delta y/ delta x
Leibniz took this to mean a small difference in y over a small difference x.
dy/dx can also be regarded as the derivative of y with respect to x. It essentially defines a very very small change in y divided by an equally small change in x.
It is important to note that the units of the derivative of a function are the units of the dependent variable (y) divided by the units of the independent variable (x). If the derivative of a function does not change rapidly near a point, then it is approximately equal to the change in the function when the independent variable increases by one unit.
An example of an everyday use of the derivative:
Derivative of velocity = dv/dt = acceleration, the units of which are meters per second squared.
Derivatives can also be used to estimate values of functions, i.e., the change in value of dependent quantity with every unit increase in independent quantity. In other words, local linear approximation can be found by using delta y = f ' (x) * delta x.
I found it difficult to understand exactly at which point, the value of the derivative would be most accurate. As all values calculated seem to be approximations, what defines their accuracy and to how many decimal places?
This section is useful in Economics because understanding the unit changes if quantities is very important. Marginal costs or benefits are always important when making big decisions. For example, to understand whether hiring one additional person to the labour force increases a firms efficiency/ profits or not.
Sunday, February 17, 2008
SECTION 2.2: The Derivative Function
The derivative of a function can be estimated graphically by finding the slope of the tangent to a point on a graph. This is so because, in general, the derivative function f '(x) is defined as the instantaneous rate of change of f at x.
Graphically, the derivative graph tells us that,
if f ' > 0, then f is increasing
if f ' < 0, then f is decreasing and
if f ' = 0, then f is constant.
Note: If the magnitude of f ' is large then the graph of the derivative is steep. Conversely, if the magnitude if small, the graph is gently sloping.
A derivative can also be obtained/ estimated from a table of data by finding the change in the x values of the data and dividing it by the change in y values of the data. Essentially, this implies finding the slope of the data when graphed.
Derivatives of certain points can also be numerically estimated by studying the average of rates of changes of point to the left and to the right of it.
What I found confusing in this reading, was the suggestion that derivatives of points could also be numerically calculated by just analyzing points to the right of it.
This reading was very useful in economics as learning derivatives in general helps understand market situations relating to recessionary, inflationary or other phases of the business cycle. For example when f ' = 0, it would imply that the market is going through a lull/ no activity phase.
The derivative of a function can be estimated graphically by finding the slope of the tangent to a point on a graph. This is so because, in general, the derivative function f '(x) is defined as the instantaneous rate of change of f at x.
Graphically, the derivative graph tells us that,
if f ' > 0, then f is increasing
if f ' < 0, then f is decreasing and
if f ' = 0, then f is constant.
Note: If the magnitude of f ' is large then the graph of the derivative is steep. Conversely, if the magnitude if small, the graph is gently sloping.
A derivative can also be obtained/ estimated from a table of data by finding the change in the x values of the data and dividing it by the change in y values of the data. Essentially, this implies finding the slope of the data when graphed.
Derivatives of certain points can also be numerically estimated by studying the average of rates of changes of point to the left and to the right of it.
What I found confusing in this reading, was the suggestion that derivatives of points could also be numerically calculated by just analyzing points to the right of it.
This reading was very useful in economics as learning derivatives in general helps understand market situations relating to recessionary, inflationary or other phases of the business cycle. For example when f ' = 0, it would imply that the market is going through a lull/ no activity phase.
Thursday, February 14, 2008
SECTION 1.3: Rates of Change
Rates of Change can be calculated in a similar way to the slope of a linear function, as analyzed earlier. The rate of change of a non-linear function is determined as an average.
If a function y = f(t),
then the average rate of change of y = change in y/ change in t = f(b)-f(a)/ (b-a),
assuming that the rate is being calculates between an interval of t=a and t=b.
Rates can be positive or negative. They help determined whether a function is increasing or decreasing. Increasing functions have positive rates and decreasing functions have negative rates. A function is said to be increasing if the values of f(x) increase as x increases. Similarly, a function f is said to be decreasing if the values of f(x) decrease as x increases.
Rates of change can be easily visualized:
The change in y refers to the vertical change of the graph while the change in x refers to horizontal change of the graph over a period of time. In the diagram below, the change in y between the two demarcated points would be approximately (90-50) while the change in x would be (0-(-10)).
The shape of a graph can also be determined by its concavity. A graphs that bends upwards from left to right is called concave up while a graph that bends downwards is considered to be concave down. The graph above is an example of a concave up graph.
Everyday rates that we encounter include velocity and speed, i.e.,
Average velocity = change in displacement/change in time while,
Average speed = change in distance/ change in time
SECTION 2.1: Instantaneous Rate of Change
To calculate velocity to exact accuracy required, smaller and smaller intervals on either side of the time taken can be determined until the average velocity agrees to the number decimal places needed. This can also be expressed as instantaneous velocity.
Instantaneous velocity is defined for an object at time t as the limit of the average velocity of the object over shorter and shorter time intervals containing t.
To look at a more general idea of the above states reasoning, instantaneous rate of change of a function at say, t, is also called the rate of change of f at t. It is defined as the limit of the average rates of change of f over shorter and shorter intervals around a. This instantaneous rate of change is also referred to as the derivative of f at point a and is written as f '(a).
The derivative of a function is essentially the slope of the function at a or the slope of the line tangent to the curve at a. Similar to the rate of change, a derivative can be positive or negative thereby determining whether the function is increasing or decreasing.
What I found challenging about these sections was deciphering whether or not having a rate of change still indicated a slope between the two points despite the fact that the graph was not linear.
The concept of average rates of change is always useful in everyday situations as not all relationships between values can be of a linear form.
Sunday, February 10, 2008
SECTION 9.1: Understanding Functions of Two variables
To indicate that a certain result is dependent on two variables we say that R = f(x,y) where
R is the dependent variable
x and y are the two independent variables and
f stands for "function".
All combination of values of (x,y) are called the domain of f. It is said that a function is increasing(decreasing) if one of its variables simultaneously increases(decreases) as the other is held constant.
The combination of x and y variables can be represented in the form of a table. From this table an algebraic formula can be derived. This is usually written in the form of R = ax + by, where a is the amount by which each additional quantity of x changes and b is the amount by which each additional quantity of y changes. We can then also say that R = f (x,y) or f (a,b).
A way to investigate functions of two variables/ quantities dependent on two variables, one cab vary one variable over a period of time while keeping the other constant and vice versa. The change in one variable compared to the other can easily be graphed and hence interpreted in a way in which one can understand the effect of a variable more clearly as the other remains constant.
SECTION 9.2: Contour Diagrams
Functions of two variables can also be expressed as diagrams known as contours or level curves/ level sets. Such diagrams are put to use in everyday situations while deciphering weather conditions through isotherms and by studying topographical maps depicting mountains, valleys, ridges etc.
For the latter, closer lines imply mountain regions while the ones further apart usually refer to plains. The elevation numbers on the contours are important because they represent the curves of the mountains themselves. In general, contours marked in constant intervals and spaced closely together usually depict steeper terrain.
The one important point to keep in mind is that contours can never intersect.
Contour diagrams function in exactly the same way as other algebraic two variable functions. The values through the x and y values from contours can also be depicted in the form of a table if required to do so.
Cobb-Douglas Production Functions:
These are often used by businesses to estimate how revenue can be maximized through various production possibility levels (for example, whether they should initiate growth by increase number of laborers or number of machines). The graph usually looks as following:
The function is written as P = f (N,V) = c(N^alpha)(V^beta) where,
P is the total quantity produced,
and c, alpha and beta are positive constants (alpha and beta lie between 0 and 1)
I had difficulty understanding why the production model has the shape it does... in fact I'm still not exactly sure why that is.
It is evident that the Cobb-Douglas Production Function is relevant to Economics Majors because it helps businessmen anticipate the results of their decisions by weighing the opportunity costs of making one decision over another.
To indicate that a certain result is dependent on two variables we say that R = f(x,y) where
R is the dependent variable
x and y are the two independent variables and
f stands for "function".
All combination of values of (x,y) are called the domain of f. It is said that a function is increasing(decreasing) if one of its variables simultaneously increases(decreases) as the other is held constant.
The combination of x and y variables can be represented in the form of a table. From this table an algebraic formula can be derived. This is usually written in the form of R = ax + by, where a is the amount by which each additional quantity of x changes and b is the amount by which each additional quantity of y changes. We can then also say that R = f (x,y) or f (a,b).
A way to investigate functions of two variables/ quantities dependent on two variables, one cab vary one variable over a period of time while keeping the other constant and vice versa. The change in one variable compared to the other can easily be graphed and hence interpreted in a way in which one can understand the effect of a variable more clearly as the other remains constant.
SECTION 9.2: Contour Diagrams
Functions of two variables can also be expressed as diagrams known as contours or level curves/ level sets. Such diagrams are put to use in everyday situations while deciphering weather conditions through isotherms and by studying topographical maps depicting mountains, valleys, ridges etc.
For the latter, closer lines imply mountain regions while the ones further apart usually refer to plains. The elevation numbers on the contours are important because they represent the curves of the mountains themselves. In general, contours marked in constant intervals and spaced closely together usually depict steeper terrain.
The one important point to keep in mind is that contours can never intersect.
Contour diagrams function in exactly the same way as other algebraic two variable functions. The values through the x and y values from contours can also be depicted in the form of a table if required to do so.
Cobb-Douglas Production Functions:
These are often used by businesses to estimate how revenue can be maximized through various production possibility levels (for example, whether they should initiate growth by increase number of laborers or number of machines). The graph usually looks as following:
The function is written as P = f (N,V) = c(N^alpha)(V^beta) where,
P is the total quantity produced,
and c, alpha and beta are positive constants (alpha and beta lie between 0 and 1)
I had difficulty understanding why the production model has the shape it does... in fact I'm still not exactly sure why that is.
It is evident that the Cobb-Douglas Production Function is relevant to Economics Majors because it helps businessmen anticipate the results of their decisions by weighing the opportunity costs of making one decision over another.
Friday, February 8, 2008
SECTION 1.10: Periodic Functions
Functions that have values occurring at regular intervals are known as periodic functions. Many everyday processes can be referred to as periodic for example, the temperature in a day rise all the way till noon, begins to fall as night approach, and repeats the same pattern every single day.
Amplitude of a periodic wave is the average difference between the maximum and minimum values, i.e., A = (max-min/2). The period of the wave is the time it takes to complete one oscillation/cycle.
To important periodic functions to be studied are the sine and cosine functions. Their graphs are equally important. It is to be noted that the graph of the cosine function is the graph of the sign function, shifted pi/2 to the left.
The graph for sin(x) looks as follows:
As can be seen it has an amplitude of 1 and a period of 2(pi).
The graph for cos(x) looks as follows:
As can be seen, it has an amplitude of 1 and a period of 2(pi) as well.
For an expression of a sine or cosine curve where
y = Asin(Bx+C)+D or y = Acos(Bx+C)+D
A affects the amplitude of the curve. Infact the absolute value of A is the amplitude of the curve.
B affects the period, i.e., period = 2(pi)/absolute value of B.
C affects the horizontal shift of the curve, and
D affects the vertical shift.
The part of graphing such sine and cosine curves that I found most challenging was when determining the amount by which and the direction in which curves moved when experiencing a horizontal shift. The positive and negative signs of C were slightly confusing.
Again, as mentioned before, periodic functions are extremely related to Economics because of business cylces are usually recurring phenomenon, i.e., periodic functions of recessions and booms. Hence the ways in which the operate is directly related to the shifts experienced by a regular sine or cosine function.
Functions that have values occurring at regular intervals are known as periodic functions. Many everyday processes can be referred to as periodic for example, the temperature in a day rise all the way till noon, begins to fall as night approach, and repeats the same pattern every single day.
Amplitude of a periodic wave is the average difference between the maximum and minimum values, i.e., A = (max-min/2). The period of the wave is the time it takes to complete one oscillation/cycle.
To important periodic functions to be studied are the sine and cosine functions. Their graphs are equally important. It is to be noted that the graph of the cosine function is the graph of the sign function, shifted pi/2 to the left.
The graph for sin(x) looks as follows:
As can be seen it has an amplitude of 1 and a period of 2(pi).
The graph for cos(x) looks as follows:
As can be seen, it has an amplitude of 1 and a period of 2(pi) as well.
For an expression of a sine or cosine curve where
y = Asin(Bx+C)+D or y = Acos(Bx+C)+D
A affects the amplitude of the curve. Infact the absolute value of A is the amplitude of the curve.
B affects the period, i.e., period = 2(pi)/absolute value of B.
C affects the horizontal shift of the curve, and
D affects the vertical shift.
The part of graphing such sine and cosine curves that I found most challenging was when determining the amount by which and the direction in which curves moved when experiencing a horizontal shift. The positive and negative signs of C were slightly confusing.
Again, as mentioned before, periodic functions are extremely related to Economics because of business cylces are usually recurring phenomenon, i.e., periodic functions of recessions and booms. Hence the ways in which the operate is directly related to the shifts experienced by a regular sine or cosine function.
SECTION 1.9: Proportionality, Power Functions, and Polynomials
When a quantity y is proportional to a quantity x, then it is implied that y = kx where k is described as the constant of proportionality (the number of times by which quantity y changes with change in quantity x).
The quantities are said to be directly proportional when the increase (decrease) in value of one results in the increase (decrease) in value of the other.
The quantities are said to be inversely proportional if one quantity is proportional to the reciprocal of the other. This implies that the increase in one quantity results in a decrease in the other quantity and vice versa.
For example, to say velocity = displacement/ time indicates that velocity is directly proportional to displacement and inversely proportional to the quantity time.
It is said that y is a power function of x if y is proportional to a constant power of x, i.e., y = k(x^p), where k is still the constant of proportionality and p is the power.
Some graphs of basic power functions can be explored: It is seen the (x^2) is consistently concave up. Values of y decrease for -x and increase for +x. The graph of (x^3) is concave down for all values of -x and concave up for all values of +x. The graph of (x^1/2) is increasing and concave down.
Sums of power functions with non-negative integer expoenets are called polynomials. They can be written in the following form:
y = an*(x^n) + an-1*(x^n-1) + ... + a1*x + a0
where,
n is a non-negative integer called the degree of the polynomial
an is a non-zero number called the leading co-efficient, and
an*(x^n) is called the leading term.
If n=2, the polynomial is quadratic.
If n=3 the polynomial is cubic.
If n=4, the polynomial is quartic
If n=5, the polynomial is quintic and so on.
Personally, I find it challenging to graph polynomials especially when they get more complicated (as in fractions) and their asymptotes are not as obvious.
The relevance of such polynomials and the concept of proportionality is extremely important in Economics to understand the fluctions in the demand and supply with changes in price of goods. The price at which maximum revenue can be attained also depends on the kind of relationship between the good and the people's demand for it. Also, demand for complementary goods is dependent on the concept of direct proportionality as is the demand for substitute goods dependent on the concept of inverse proportionality.
When a quantity y is proportional to a quantity x, then it is implied that y = kx where k is described as the constant of proportionality (the number of times by which quantity y changes with change in quantity x).
The quantities are said to be directly proportional when the increase (decrease) in value of one results in the increase (decrease) in value of the other.
The quantities are said to be inversely proportional if one quantity is proportional to the reciprocal of the other. This implies that the increase in one quantity results in a decrease in the other quantity and vice versa.
For example, to say velocity = displacement/ time indicates that velocity is directly proportional to displacement and inversely proportional to the quantity time.
It is said that y is a power function of x if y is proportional to a constant power of x, i.e., y = k(x^p), where k is still the constant of proportionality and p is the power.
Some graphs of basic power functions can be explored: It is seen the (x^2) is consistently concave up. Values of y decrease for -x and increase for +x. The graph of (x^3) is concave down for all values of -x and concave up for all values of +x. The graph of (x^1/2) is increasing and concave down.
Sums of power functions with non-negative integer expoenets are called polynomials. They can be written in the following form:
y = an*(x^n) + an-1*(x^n-1) + ... + a1*x + a0
where,
n is a non-negative integer called the degree of the polynomial
an is a non-zero number called the leading co-efficient, and
an*(x^n) is called the leading term.
If n=2, the polynomial is quadratic.
If n=3 the polynomial is cubic.
If n=4, the polynomial is quartic
If n=5, the polynomial is quintic and so on.
Personally, I find it challenging to graph polynomials especially when they get more complicated (as in fractions) and their asymptotes are not as obvious.
The relevance of such polynomials and the concept of proportionality is extremely important in Economics to understand the fluctions in the demand and supply with changes in price of goods. The price at which maximum revenue can be attained also depends on the kind of relationship between the good and the people's demand for it. Also, demand for complementary goods is dependent on the concept of direct proportionality as is the demand for substitute goods dependent on the concept of inverse proportionality.
Sunday, February 3, 2008
SECTION 1.7: Exponential Growth and Decay
Many functions can be defined in the form of P = P0 (e ^kt) where P0 is still defined as the initial quantity and k is the continuous growth or decay rate. When k is negative it represents a decay. When it is positive it refers to exponential growth. "e", as mentioned in the previous post, is defined as a natural base.
Two new concepts defined in this sections were doubling time and half-life:
Doubling time of an increasing exponential function is the time required for the quantity to double itself.
Half- life of a decaying exponential function is the time required for the quantity to be reduced to half its current amount.
All growing functions have a doubling time, and all decaying functions have a half-life.
Compound interest on an amount saved in a bank can be defined exponentially. If an initial
amount of P0 is deposited in a bank that has an interest rate of r/ year, assuming P is the balance in the bank over t years, it can be said that
- interest compounded annually implies P = P0 (1+r)^t
- interest conpounded continuously implies P= P0(e^rt) where e is again the natural base.
In business terminology, there are two important terms to remember:
The future value of an amount P is the amount to which P will grow in an interest-bearing bank account.
The present value, P, of a future payment is an amount that will have to be deposited in a bank now to produce the amount of future payment at the relevant time.
In this reading, I was confused by the term 'interest compounded continuously' because I could not easily understand how this could happen when the interest rate itself was defined per year.
It is clearly seen that this section of exponential functions is most related to Economics. Interest rates are crucial to bankers, borrowers and lenders. Understanding how one's savings grow in a bank is equally important and necessary in managing one's own finances.
Many functions can be defined in the form of P = P0 (e ^kt) where P0 is still defined as the initial quantity and k is the continuous growth or decay rate. When k is negative it represents a decay. When it is positive it refers to exponential growth. "e", as mentioned in the previous post, is defined as a natural base.
Two new concepts defined in this sections were doubling time and half-life:
Doubling time of an increasing exponential function is the time required for the quantity to double itself.
Half- life of a decaying exponential function is the time required for the quantity to be reduced to half its current amount.
All growing functions have a doubling time, and all decaying functions have a half-life.
Compound interest on an amount saved in a bank can be defined exponentially. If an initial
amount of P0 is deposited in a bank that has an interest rate of r/ year, assuming P is the balance in the bank over t years, it can be said that
- interest compounded annually implies P = P0 (1+r)^t
- interest conpounded continuously implies P= P0(e^rt) where e is again the natural base.
In business terminology, there are two important terms to remember:
The future value of an amount P is the amount to which P will grow in an interest-bearing bank account.
The present value, P, of a future payment is an amount that will have to be deposited in a bank now to produce the amount of future payment at the relevant time.
In this reading, I was confused by the term 'interest compounded continuously' because I could not easily understand how this could happen when the interest rate itself was defined per year.
It is clearly seen that this section of exponential functions is most related to Economics. Interest rates are crucial to bankers, borrowers and lenders. Understanding how one's savings grow in a bank is equally important and necessary in managing one's own finances.
SECTION 1.5: Exponential Functions
If the percentage increase or decrease in a quantity is a constant, then the function is defined as an exponential function. Population growth in certain areas is an example often used for demonstrating exponential growth.
Exponential functions are defined as P = P0 (a^t) where
P represents the vertical axis of a graph
P0 represents the initial quantity (when t=0), and
a represents the the factor by which the function increases or decreases. It is called a base.
If a>1, there is said to be exponential growth, if a<1, there is said to be exponential decay. Larger values of a imply faster growth. and values of a near 0 im
ply faster decay.
It can also be written as a = 1+r, where r is the decimal representation of the percent rate of change. It is positive for exponential growth and negative for exponential decay.
For an exponential decay to occur, as t increases the functions values get closer and closer to zero. Therefore the t-axis represents the horizontal asymptote of the function.
It is also important to note that graphs of exponential functions are always concave up.
To differentiate between an exponential function and a linear function, it is important to understand that the former refers to a constant relative rate of change while the later refers to an absolute value for the rate of change.
In everyday situations the most common base number is defined as e= 2.71828... This is often called the natural base.
When reading this section, the question that arose in my mind was whether or not the base a could be a negative number, and if not then why?
This material was relevant to what had been studied in the earlier section, especially in discussing why certain data when extrapolated beyond certain intervals could still be considered functions and yet not change at a constant absolute rate. The section helped clearly identify the differences between linear and exponential functions.
If the percentage increase or decrease in a quantity is a constant, then the function is defined as an exponential function. Population growth in certain areas is an example often used for demonstrating exponential growth.
Exponential functions are defined as P = P0 (a^t) where
P represents the vertical axis of a graph
P0 represents the initial quantity (when t=0), and
a represents the the factor by which the function increases or decreases. It is called a base.
If a>1, there is said to be exponential growth, if a<1, there is said to be exponential decay. Larger values of a imply faster growth. and values of a near 0 im
ply faster decay.
It can also be written as a = 1+r, where r is the decimal representation of the percent rate of change. It is positive for exponential growth and negative for exponential decay.
For an exponential decay to occur, as t increases the functions values get closer and closer to zero. Therefore the t-axis represents the horizontal asymptote of the function.
It is also important to note that graphs of exponential functions are always concave up.
To differentiate between an exponential function and a linear function, it is important to understand that the former refers to a constant relative rate of change while the later refers to an absolute value for the rate of change.
In everyday situations the most common base number is defined as e= 2.71828... This is often called the natural base.
When reading this section, the question that arose in my mind was whether or not the base a could be a negative number, and if not then why?
This material was relevant to what had been studied in the earlier section, especially in discussing why certain data when extrapolated beyond certain intervals could still be considered functions and yet not change at a constant absolute rate. The section helped clearly identify the differences between linear and exponential functions.
My name is Proma Sen and I am a freshman at Macalester College, St. Paul, Minnesota. My major is yet undeclared but I have developed a keen interest in the subject of Economics.
My experience in math has been limited to whatever I have learnt through my years of schooling. I studied Math in the Indian system of education till year 10, after which I transferred to a UWC and took MATH HL as a part of the IB curriculum. I am not entirely sure what the weakest part of my Math background would be but I do remember struggling with infinite series for the IB. The strongest part of my Math background is probably Algebra and Trigonometry.
I am taking this Applied Calculus course because it is a pre-requisite for someone who wants to major in Economics. Through this course, I want to have a strong hold on the basics of understanding calculus, the math as well as why it can be relevant in today's world.
My interests are varied. I play the piano and the guitar. I sing in an a capella group on campus. I am heavily into music of all sorts but mainly old school rock. I used to play basketball in high school but have stopped since. I love watching football (soccer). I also like to read books outside of coursework when I get the time.
The worst Math teacher I ever had was in the 7th grade. My teacher would simply work out all the problems from the textbook on the board by herself and have us copy them down without any explanations. These were what we were to use as notes for the exam.
The best Math teacher I ever had was someone who always came up with entertaining everyday examples related to math and what she taught. Her quizzes were fairly challenging and it was a very competitive class but none the less enjoyable.
SECTION 1.2: Linear Functions
Linear functions are represented as straight line graphs. They may be increasing (olympic world record) or decreasing (a car decelerating) functions. These functions can be written in the form of y = f(x) = ax+b where
y is the value represented in the vertical axis of a graph (it is a function of x)
x is the value represented in the horizontal axis
b is the y-intercept of the graph (when x=0), and
a is the rate of change of y in relation to the change in x, i.e., it is the slope of the graph (slope= rise/run)
The reading section also identifies problems of extrapolation by which one tends to assume that the linear function demonstrates in the intervals in which the graph is drawn carry on endlessly. This may not always be the case. This depends on whether or not the data of the graph is discrete or continuous. Only a continuous function would continue to change at a constant rate on extrapolation.
The slope of the graph is a very important factor in understanding the function. The symbol delta is used to symbolize a change in values. Essentially, the slope,
which can also be written as 
This is defined as the difference quotient.
The equation of a line of slope m that passed through a point (x1,y1) can also be written as
y-y1= m(x-x1)
Note that a slope, m=0 implies that the function is a horizontal line, i.e., y=b.
Functions that share certain properties are defined as a family of linear functions. m and b are the parameters.
I personally found it challenging to differentiate between intervals of discrete data and continuous data.
The reading was very clearly relatable to Economics because I think functions, linear or otherwise, are important to all businesses when assessing their pros and cons of a financial decision with regard to the rate of change of fluctuating prices in the market.
My experience in math has been limited to whatever I have learnt through my years of schooling. I studied Math in the Indian system of education till year 10, after which I transferred to a UWC and took MATH HL as a part of the IB curriculum. I am not entirely sure what the weakest part of my Math background would be but I do remember struggling with infinite series for the IB. The strongest part of my Math background is probably Algebra and Trigonometry.
I am taking this Applied Calculus course because it is a pre-requisite for someone who wants to major in Economics. Through this course, I want to have a strong hold on the basics of understanding calculus, the math as well as why it can be relevant in today's world.
My interests are varied. I play the piano and the guitar. I sing in an a capella group on campus. I am heavily into music of all sorts but mainly old school rock. I used to play basketball in high school but have stopped since. I love watching football (soccer). I also like to read books outside of coursework when I get the time.
The worst Math teacher I ever had was in the 7th grade. My teacher would simply work out all the problems from the textbook on the board by herself and have us copy them down without any explanations. These were what we were to use as notes for the exam.
The best Math teacher I ever had was someone who always came up with entertaining everyday examples related to math and what she taught. Her quizzes were fairly challenging and it was a very competitive class but none the less enjoyable.
SECTION 1.2: Linear Functions
Linear functions are represented as straight line graphs. They may be increasing (olympic world record) or decreasing (a car decelerating) functions. These functions can be written in the form of y = f(x) = ax+b where
y is the value represented in the vertical axis of a graph (it is a function of x)
x is the value represented in the horizontal axis
b is the y-intercept of the graph (when x=0), and
a is the rate of change of y in relation to the change in x, i.e., it is the slope of the graph (slope= rise/run)
The reading section also identifies problems of extrapolation by which one tends to assume that the linear function demonstrates in the intervals in which the graph is drawn carry on endlessly. This may not always be the case. This depends on whether or not the data of the graph is discrete or continuous. Only a continuous function would continue to change at a constant rate on extrapolation.
The slope of the graph is a very important factor in understanding the function. The symbol delta is used to symbolize a change in values. Essentially, the slope,
which can also be written as This is defined as the difference quotient.The equation of a line of slope m that passed through a point (x1,y1) can also be written as
y-y1= m(x-x1)
Note that a slope, m=0 implies that the function is a horizontal line, i.e., y=b.
Functions that share certain properties are defined as a family of linear functions. m and b are the parameters.
I personally found it challenging to differentiate between intervals of discrete data and continuous data.
The reading was very clearly relatable to Economics because I think functions, linear or otherwise, are important to all businesses when assessing their pros and cons of a financial decision with regard to the rate of change of fluctuating prices in the market.
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