Algebra Vs. Calculus

1. Depending on the situation, there may or may not be a difference between the limit of a function at x=c and plugging in the number x=c. When finding the limit at x=c, one is looking for the number the function outputs as the input approaches c. There may be an hole at x=c, but if the function approaches that point from both sides, it is indeed the limit. When searching for f(c), one is looking for the output at the x value c. The output does not necessarily have to be the same as the limit of x=c. However, if the limit at x=c equals f(c), then the function is continuous.

Only when the function is condtinuous does the limit at x=c equal f(c).


2. When finding the slope of a line, one is finding the (change in y)/(change in x) of the entire line, not just between two pints. To find the slope of a line, one uses the formula (change in y)/(change in x). (y2-y1)/(x2-x1), another variation of this formula, is sometimes used. Finding the slope of a line typically does not require much more than counting rise over run or plugging in numbers into a simple formula.

To find the derivative of a line, one has the choice of using various different formulas, three of which involve limits, and one which does not. The typical, most basic formula is limit as h approaches 0 f(h+a)/h. When finding the derivative, one is basically finding the slope of a line at one specific point (this is why h is approaching 0).

Finding the slope and finding the derivative of given lines are similar in the way that both of them deal with finding the change in x and y.

I've Reached My Limit !!

Limits seem to perplex me a bit. I understand the basic concept of limits, however, i do not understand certain small details.

1. I am confused on how to find the limit of a function as it approaches a vertical asymptote. The way i solve such problems is by plugging in numbers in order to get an idea of how the graph looks like. Then, i can easily see the right- and left- hand limits of a certain point. I do not think this is how to properly solve it, for it is too time consuming. I was wondering if anyone knew how to solve such problems quickly and easily ??
   
2. I am a little unclear on how to find the limit of a sine or cosine function. I am not sure whether the limit is simply positive and negative one, or just positive one or just negative one.
3. I do not completely understand how to find out if a discontinuity is removable or non-removable. An example of such a problem would be # 25 from homework g1. I also do not know how to find values of x for which a function is discontinuous. An example of such a problem would be part e on question 18 of the test on chapters 2.1-2.3.
   
4. I do not understand how to solve #3 from the test on chapters 2.1-2.3.

Thanks for considering my problems ! :]

Colleges, colleges!

1. These are a few of the majors that struck me as utterly fascinating:

• Environmental Science: This major involves using biology, chemistry, and geology skills to find possible solutions to some of the world’s most pressing problems. I think i would be very interested in majoring in environmental science because i am very passionate about the health of our planet and would be willing to dedicate lots of time and effort into discovering new ways to conserve the earth.
  

• Epidemiology: This major deals with disease and disability among and between certain groups. Epidemiologists explore ways of stopping the spread of disease. An epidemiologist must be familiar with statistics, chemistry, biology, and genetics. I think majoring in epidemiology would be rather fascinating. I would spend lots of time analyzing data and trying to find patterns and trends as well as ways to prevent diseases. I would feel accomplished doing such work.
  

• Journalism: This major involves bringing news to the public. It can consist of covering world events for a major newspaper, reporting on sports for a radio station, or even writing about entertainment on the Internet. A journalism major would allow me to learn about important issues that affect today's society. I love writing, and when given an interesting topic, i truly lose myself in the writing task. I believe journalism is something i can truly be passionate about.

• Logic: Students who chose this major focus on the art of the reasoned argument. They study classical logic, logical fallacies, and symbolic logic among other things.I would love to major in logic in order to think more thoroughly and be more persuasive. I believe majoring in logic will actually enhance my thinking skills and show me how other people think. This will prove to be very useful in life.
  

Colleges that offer the majors that interest me:
Boston University

• Private 4-year university
• 80% of 1st year students come from out-of-state
• offers bachelor's, master's, doctoral, and first professional degrees
  
• urban setting
• offers more than 250 fields of study
  
• 54% of applicants are admitted

University of Pennsylvania

• Private 4-year university
• 72% of 1st year students had a high school G.P.A. of 3.75 and higher
• 99% of 1st year students were in the top 10th of their high school graduating class
  
• urban setting
• 83% of 1st year students come from out-of-state
  
• 17% of applicants are admitted

New York University

• private 4-year university
• 21,269 undergraduates
  
• 60% of 1st year students are women
• 71% of 1st year students come from out-of-state
  
• 32% of applicants are admitted
• urban setting
  

Tips & Hints

1.) The way i remember transformations is by doing the opposite of what my first instinct tells me to do. For example, if the equation happens to be y= sin (x + 2), the graph of sin x is shifted 2 units to the left. Although i would think that adding 2 to the input would make the graph shift to the right, it actually shifts the graph to the left. This is what i mean by doing the opposite of my first instinct. On the other hand, if the given equation is y= sin (x-2), the graph of sin x shifts 2 units to the right. The way i remember this transformation is if something is added to the input (the x), the graph moves along to the x-axis, either to the right or left. If the entire function is multiplied, for instance, in the case of y= 3sin x, the graph is stretched vertically. The graph will span over double the number in front of the sin x. For instance, in this case, the graph will range from -3 to 3. This is a range of six units in total, which is 2 times 3. Another way to manipulate the graph of a parent function is by affecting the period. The "do the opposite" hint applies to this. For instance, if the equation is sin2x, the period of the graph changes from 2 pi to pi. The graph "speeds up" and goes the same distance in 3.14 units as it would before in 6.28 units. If the equation is y=sin x + 2, then the graph of sin x is moved 2 units upwards. If the equation is y= sin x -2, the graph of sin x simply shifts 2 units downward. This transformation affects the output (the y), and therefore moves along the y-axis, either up or down. The "do the opposite" hint does not apply to this transformation.

2.) The way i understand trigonometry is by knowing the unit circle measurements, both the radians and points. The only trick to this is memorization. All that really needs to be memorized are the measurements in the first quadrant. With this knowledge, one can simply "fold over" the measurements and find the measurements of the proceeding quadrants. The radians in the 2nd quadrant will always be one unit away from being an entire pi. The radians in the 3rd quadrant will always be one unit more than pi. The radians in the 4th quadrant will always be one unit away from being 2 pi. As far as points, they remain the same as the points in the first quadrant. In the 2nd quadrant, the first point (cos x) becomes negative. In the 2nd quadrant, both points are negative (cos & sin). In the 4th quadrant, the last point is negative (sin x). Of course, in the 1st quadrant both points are positive.

The graphs of sin x, cos x, tan x, csc x, sec x, and tan x, as well as their inverses are important in trigonometry. Again, memorization is a key factor. The graphs can be plotted by actually plugging in points or, more quickly, by memorizing how the graph looks, its domain, and its range.

3.) I believe trigonometry is pretty straightforward and all that is really needed to understand it is time and patience, for memorization is a key factor. What worries me about trigonometry is the graphs of the inverses of sin x, cos x, tan x, csc x, sec x, and tan x. I am a little fuzzy on the graphs of these functions, but i know that i just need to spend some more time memorizing the graphs, their domain, and their range in order fully understand them.

Logs & Inverses

4 Major topics that i have understood:

Logs: I understand how to solve logarithms. I learned that when given a problem, (for instance log base 2 of 32), i am actually looking for the power. In this example, the answer would be 5, since 2^5 = 32. I also learned how to solve more complex logarithms, such as log base 6 of 1/36^1/5. In this case, the answer would be -2/5, considering 6^-2/5 is 1/36^1/5.

Inverses: I understand the term one-to-one. In order for a function to be one-to-one, it must pass the horizontal line test. This means that the graph of the line must not intersect a horizontal line at more than one point. This test is indeed similar to the vertical line test done to check if a graph is a function. However, the horizontal line test deals with the use of a horizontal line, of course.

Inverses: I understand how to find the inverse of a function. To find the inverse graphically, i learned that the graph of f(x) is simply reflected across the line y=x. The line that results is the graph of f^-1(x). In this line, the original values of x and y are simply switched to form the graph of the inverse of f(x). To find the inverse of a function algebraically, i learned that you take the following steps:

• Replace the x with a y.



• Isolate the y variable.



• Divide by the number in front of the y variable (if necessary) in order to get 1y (as opposed to 2y, 5y, etc).



• Change y=something into f^-1(x)= something. This form is the right form for your answer and is much more "telling" as Ms. Hwang says.

Example: find the inverse of f(x) = 2x +3.
f(x) = 2x +3
x = 2y +3
-3 -3
x-3 = 2y
x-3/2 = 2y/2
x-3 = y
f^-1(x) = x-3

Inverses: I understand that if f^-1(x) is really the inverse of f(x), then f(f^-1(x)) =x and f^-1(f(x))=x. This is very useful when using algebra to find the inverse of a function. After solving for the inverse, i learned to check my answers by solving f(f^-1(x)) and f^-1(f(x)). If i get x as an answer both times, then i know i found the correct inverse.

What i did not understand completely:

I did not understand how to graph logarithm functions. For example, i didn't know how to do numbers 39-42 of homework C2. I know you can use a graphing calculator to graph these, but i want to learn how to graph these equations myself.

I also did not know how to solve problems 35 & 36 of homework C2.

I did not understand how f(g(x))=x but yet, g(x) is not the inverse of x. (# 53 of HW C3)

I also did not understand how to solve # 60 from HW C3.

If anyone knows how to solve these, please help me:] I would very much appreciate your effort!

Even and Odd Functions

Even Functions

Hello everyone!

Mathematically, even functions are defined as f(x)=f(-x).

All even functions are symmetrical about the y-axis. This means that all points graphed in Quadrant 1 will be reflected in Quadrant 2 (& vice versa) and all points graphed in Quadrant 3 will be reflected in Quadrant 4 (& vice versa).

Some examples of even functions include f(x)= absolute value of x, f(x)=x^2, and f(x)=cosx.

For those visual people out there, the graphs of even functions look like so:

Example:

Consider the function f(x) = x^2.

Recall: For this function to be even, f(x)must equal f(-x). For this function to be odd, f(-x) must equal-f(x).

f(x)=x^2

f(-x)=(-x)^2= x^2

-f(x)= - (x^2)

In this case, f(x)=f(-x), therefore, function f is even.

Odd Functions

Mathematically, odd functions are defined as f(-x)=-f(x).

All odd functions are symmetrical about the origin. This means that all points graphed in Quadrant 1 will reflect like a mirror image in Quadrant 3. Also, all points graphed in Quadrant 2 will reflect like a mirror image in Quadrant 4.

Some examples of odd functions include: f(x)=1/x, f(x)=x^3, and f(x)=sinx.

Some visual representations of odd functions include:

Example:

Consider the function f(x) =x^3.

Recall: For this function to be even, f(x)must equal f(-x). For this function to be odd, f(-x) must equal-f(x).

f(x)=x^3

f(-x)=(-x)^3= -x^3

-f(x)= - (x^3)=-x^3

In this case, f(-x)=-f(x), therefore, function f is odd.

I hope this explains even and odd functions rather accurately. For further explanation, use the following link to watch a useful video i found on youtube. :]

http://www.youtube.com/watch?v=v2ni9bwO4v4

About Denise:]

Hello fellow bloggers, I'm Denise Acosta! I'm a junior at Polytechnic high school and i'm also very involved in school activities. I'm involved in Cheerleading, Junior council, Medical Exploring, California Scholarship Federation, the HEART program, Project STEPS, National Honors Society, Interact, and UCLA EAOP. I love to read and often find myself enveloped in a whorl of unprecedented thoughts. My all-time favorite book is Gone With the Wind, which i highly recommend. I believe in working instensely in order to achieve what i want in life. I work hard, but i also play hard. I'm anticipating a higher education after high school, but i'm not yet sure of what university i will attend. I'm currently learning how to play the acoustic guitar, and i'm semi-pro:]
I'm looking forward to this AP calculus class because i feel will learn so much and discover an entirely new world. I want to prove to myself that i can do anything and i want to show that i can do it with a smile. :]
"I think I'm different, but if everyone's different, how's anyone different?"