Mean Value Theorem II

f ' (c)= f(b)-f(a)/b-a
1. In the graph the slope of point c is parallel to the slope of chord AB which shows that both have the same slope. When solving for f ' (c) you are basically looking for a point in the graph that has the same slope between two points in the graph.

2. In order to meet the conditions of the Mean Value theorem a function needs to be continous and differentiable. In this example the function in the graph is discontinous. The y- axis is an asymptote.The mean value theorem does not apply here because there is no point in the function that has the same slope of points A and B.

3. The function 7sin(x^2) does not satisfy the conditions of the mean value theorem because the graph is continuous but not differentiable. The slope of points A and B is zero but there is no point in the function with a slope of zero. As seen in the graph the peaks are undifferentiable with an undefined slope which means the mean value theorem does not work in this case.

Mean Value Theorem

f ' (c)= f(b)-f(a)/b-a

1. In the graph the slope of point c is parallel to the slope of chord AB which shows that both have the same slope. When solving for f ' (c) you are basically looking for a point in the graph that has the same slope between two points in the graph.

2. In order to meet the conditions of the Mean Value theorem a function needs to be continous and differentiable. In this example the function in the graph is discontinous. The mean value theorem does not apply here because there is no point in the function that has the same slope of points A and B.

In this example the function is continous but not differentiable at x = 3. The function does not satisfy the conditions of the mean value theorem because the slope of points A and B is zero but the slope of point x=3 is undefined. There is no point in the function with a slope of zero.

The Function f(x) from the graph f '(x)

1. The function f(x) is increasing in the interval (-2,0) U (0,2), since the graph is showing the derivative of f(x) when f(x) has a positive slope f(x) is increasing.

The function f(x) is decreasing in the interval (-infinity,-2) U (2, infinity), because a function is decreasing when the slope is negative.

2. There is local minimum at x=-2. Extremas can be found in critical points when f '(x)=0 or undefined. x=-2 is a local minimum because the slope of f(x) changes from being negative to positive.

There is local maximum at x=2 because at this point the slope of f(x) changes from being positive to negative.

3. The function is concave up in the interval (-infinity, -5/4) U (0,5/4). A function is concave up where f ''(x)>0.
The function is concave down in the interval (-5/4,0) U (5/4, infinity). A function is concave down where f ''(x)<0.
I can tell from the graph f ' (x) where f ''(x) is positive and negative by looking where the slope is positive and where the slope is negative since f ''(x) is derivative of f ' (x).

4. f(x) is a power function of 5 because the graph of f '(x) is to the power of 4 so its anti-derivative has to have a power of five.

Growth vs. Fixed

1. Which mindset do you think you are a part of when it comes to "intelligence"? According to the reading, what tells you that you are of this mindset?
I consider myself to have a growth mindset when it comes to intelligence typically because I always tell myself that the more you know the smarter you get but also have some characteristics of a fixed mindset. In the reading a person who has a growth mindset is described as someone who does not back down from challenges. I take challenges to see how strong I am in a certain area but am also cautious about the challenges I choose. I always try to learn from people who have had success to see if there is anything I can relate to.

2. How has this mindset helped or hurt you in math?
A growth mindset has helped me put an effort into my work such as like in math problems. I always make things as simple as I can make them because the simpler it is the easier it gets.

3. What is your reaction to finding out that the brain is just a big muscle that can be trained?
I was not very surprised because the brain is just like any other muscle the more you put it to the challenge the stronger it gets. Some people train their muscles (brains) more than others which are why some are more successful than others.

4. How do you see this new piece of information affecting your future?
The more a person works at something the better the outcome is. This information will help me in being able to understand others by considering the way they think.

Algebra vs. Calculus

1.The difference between finding the limit of a function as x=c and actually plugging in the number x=c, is that the limit of a function as x approaches c could not be a value of the function if the function is not continuous at c. In a limit you are finding the value as x is approaching the constant c which can be different to what the actual value is. When plugging in x=c to the function you are finding the exact value of the function. The best way to see the difference is through a function that is not continuous for example a piecewise function which can have several different functions with different limits when x=c or a different value when x=c.
The two cases are the same when the function is continuous were the limit when x approaches c is the same as the value of the function when x=c.

2. The similarities when finding the derivative of a line and the slope is that the value is the same in the case of a line because in both functions you are finding the change in y over the change in x. In a derivative when finding the value we use limit as h approaches 0 f(a+h)-f(a)/h meaning that we are assuming that the difference as h approaches a is infinitely small.
Derivatives are mainly used to find the slope of a point in a curve by finding the slope of its tangent line.

Limits

The whole concept of how limits worked took me a while to figure out mainly the part of discontinuity and continuity. I have trouble in trying to figure out whether a piecewise function is continuous or discontinuous.

One of the biggest problems I have is trying to figure out End behavior models for trigonometric functions without using a calculator for example problem number 38 on pg 95 which is f(x)=ln[abs(x)]+sinx. I was able to figure out the right end behavior model knowing that sinx was not really significant because the smallest value is -1 and that the biggest value is 1 so I knew that ln[abs(x)] was the right end behavior model.

I still have trouble in trying to figure out how to solve for horizontal asymptotes for a function. I am not sure whether there is a way to solve for it without having to graph it.

I was really confident about the limits test after seeing the practice test we got primarily because I did not see anything that gave me too much problem. After taking the test I was blindsided by a couple of problems on the test which is still bugging me that I was not able to solve which was one about explaining why a trigonometric function was continuous and to graph a sketch. The other one was the free response which had a piecewise function with a function having a variable on it.

Life after high school

There are many majors that I found a lot of interest in but the ones I found the most interesting were in the region of Biological and Biomedical Sciences and also in the region of engineering. I have not really decided what I want to be but know it is between a surgeon and an engineer.

The first major I am interested in is Biomedical Sciences which is basically biology but more in depth. It also deals with how it relates to the body and health, medicine, and disease. People who study this major research for new ways of treating people with various diseases and also analyze technical procedures to improve human human health.

The second major I am interested in is Biological engineering which deals with knowing how to incorporate engineering to helping solve medical problems. People in this field look for new ways to improve machinery to aid in diagnosing and treating certain diseases. I am very interested in majoring in this area particularly because it deals with engineering and medical issues.

The third major I am interested in is Applied Mathematics because it is a more in depth area of mathematics and requires a lot of thinking and reasoning. People in this field use math and statistics to help solve problems in engineering. I am pretty strong in Math so I think I would be very successful in this field.

The first college I am interested in UC Los Angeles because they offer various majors in the area of biology and health. It is also a top university and has a very good medical school. The university is also highly ranked because it has a lot of resources.

The second college that I am interested in is UC Berkely which offers various majors in the area of engineering. It is a large university in a small setting and is very well known around the nation as a top university. It has a lot of facilities in engineering and is popular in giving degrees in the engineering area.

The third college I am interested in is California Institute of Technology. It is a top university and is extremely competitive but offers many degrees in the area of math. Admission is very low because they only accept a small number of students but the university offers many resources and aid.