MEAN VALUE THEOREM

1. The Mean Value Theorem works :
In the equation f(x)=x^2 works in the Theorem in [0,2]
-*****It is both continuous and differentiable within that interval.
f(0)= 2 and f(2)=4, the MVT guarentees a point c in the interval.

f'(c) = f(2)-f(0)/2-0
2c = 4-0/2 = 2
c=1

2. The Mean Value Theorem only works for continuous and differentiable functions. When there are gaps, or when non-differentiable, as in the case of f(x) = sqareroot(x^2) + 3.
-This graph has a corner at x=0
-Non-differentiable
-There could not be any tangent lines throughout the graph.

Derivative Graph

1. The function is increasing from the interval [-2,2] because f'(x) is positive. The function is decreasing from approximately (-2.2, -2) u (2,2.2) because f'(x) is negative. This can be seen by where the graph is positive or negative on above or below the x-axis.
2. The is an extrema at x= -1.2, 0, 1.2 because that's where the slope is equal to zero, hence there are extrema.
3. The graph is concave up from (-2.2, -1.2) u (o,1.2) and it is concave down from (-1.2,0) u (1.2,2.2) because that's where the f''(x) is positive for the concave up and negative for the concave down.
4. on paper

MINDSETS

1. I believe that I have both a Growth and a Fixed Mindset; growth when it comes to subjects I enjoy and Fixed when it comes to subjects that I struggle with. I used to believe that I was the way I was and tried to do things that were in my "ability", but I now feel that a growth mindset will better suit anything that needs to be accomplished. My Growth Mindset comes into play when I find something interesting, like now in Calculus with Derivatives; they are challenging but I have the need to try to solve them and learn from them more.

2. My mindset has created many problems with math because I would not take challenges "head-on" and would then crumble when it came to tests. Also, with the lack of, or with the prior "fear", I would then create a Domino Effect that harmed the rest of the sections that included the prior knowledge.

3. My reaction to finding out that my brain was a muscle that could be trained, was becoming relieved. I felt like there is always time for change and it is never too late to improve, both in life and in the classroom, with regards to math.

4. I believe that this information will be useful in that I will intake it and make the best of it; I will use it as a self esteem booster and as confidence to do better in anything that I do. I see my future brighter and full of improvement.

P.S: This was a great Article. Thank You!

Limits

I still do not have a conceptual understanding of problems that have to do with the types of discontinuity. The types of discontinuity like Jump discontinuity, removable discontinuity, non-removable discontinuity, etc. I belive that my lack of my knowledge in this concept is my though of mastering the subject before I even grasped it. Questins that I had dificulty with were Q's #25 part e, #29 part c, and #32. I just need more examples to master this concept.

Colleges

3 Majors:

1. Premedicine: Majors under the premedicine branch include Biochemistry, Biophysics, and Molecular Biology, etc. Under these majors one can fulfill all requirements in order to get into medical school as well as grasp a strong foundation to all the other classes they will take in the future. Also, this major is a gateway to get into all other parts of science, in regard to what exact major you choose.

2. Neuroanatomy: This major really intersts me in the aspect that I have always wanted to be a neurologist. Neuroanatomy is the study of the brain and the nervous system with a special focus in the structure and function. It focusesd on the brains signals to the body, and how the body responds. I am really interested in this major and would love to pusue this major.

3. Biomedical science: People in this major usually pursue a degree in medicine and grow a true foundation in biology, that could integrate all sciences together to be in any field. They also explore human health, medicine, and diseases. This major ties to the field that i want to be in so it would really benefit me to pursue this major, also i would really enjoy grasping major concepts.

3 Colleges:
1. Penn State(All): Pensylvania state University is a public institution that has many opportunties to pursue any carreer. 78% of all applicants that meet their criteria are accepted and it is nationally accredited. In my eyes it is the equivalent to a Cal. state in california, and offers B.A.'s as well as A.A's. The major that I would interested in and they offer is Biomedical Sciences. Only about 7% of all students who attend Penn State are out of state students. Only 3,367 students, who are undergrad's, attend Penn State. Also, the football program at Penn State is nationally known and usually attend national competitions.
2. Yale University: Although this university does not offer some of my intended major, I really would love to attend this institution. Yale is a really competitive school, where only 6% of all the students attending their undergraduate program are in state students, while a wapping 94% of all students are out of state. Only about 9% of all aplicants are accepted due to the fact that its is nationally ranked amoung the top schools. There all only about 5,200 students who attend Yale, and it is in a small city. Attention is really crucial at this institution, therefore the ratio of student-to-faculty is 6:1, meaning that for every 6 students there is one faculty member. I would really want to attend next year and am hoping for the Yes this march.
3. Brown University: This univrsity, set in Rrovidence, Rhode Island, is a member of the Ivy Leaugue Schools and is known nationwide as a top institution. Approx. 14% of all applicants are admitted, while 86% are either rejected or waitlisted. It is a 4-year liberal arts college that combines rigorous coarsework with opportunties to excell. There are 6,095 students in Browns undergraduate program and 86% of all the students attending Brown are out of state students. I would love to go to Brown University and am also looking forward to a Yes from them also.

Remembering

Remembering Transformations: I remember transformations in an easy way.
1st. If its in parenthesis, it is going left or right according to the symbol. (when its a -negative, then it is going right but if its a positive+ then it goes to the left.)
2nd. When there is a number added or subtracted to the end of the equation without parenthesis, the graph of the line or equation is gonna go up or down corresponding to its sign. (meaning that if its negative its gonna go down, and if its positive its gonna shift up.)
3rd. Finally, when there is a negative before the equation, the equation is gonna be flipped about the y=0 line and you will get the reflection of the parent function.

Trigonometry: 2nd and 3rd question combined...
Honestly trig. passed me like crazy, i went from trig. A and learning only up to chapter 3, then to Ms. Gapacs class starting on chapter 6.3. (therefore i skipped 3 whole chapters where she introduced the whole unit circle topic. This made learning anything 1000x harder!!!) I don't really remember any short cuts or tips. sorry=(
I know that there are easy ways for short cuts in trig. but i always had to do catching up instead of learning short routes. thanks for undertstanding....!!!!=)

Logs and Inverses

4 Major Concepts learned:
1. Inverses are symmetrical about the the y=x line. -The input and outputs are switched and then the inverses are shown.
2.One to One functions- Is when both f(x) and f^-1(x) are functions, when the are both functions they pass the vertical line test. For every line, there is exactly one point that hits every part of it, not more than one.
3. I understood that "e" in logarithms and inverses equals to 2.71828 and that it is a base of a logarithm (loge).
4. When an equation that includes a loge but has no base, it is always a log with a base of 10, this enables the log to be canceled and used with other like logarithms. (log10)

What I did not understand:
1. I did not understand how to graph any graph that has a complex, well at least for me, form like f(x)=3x=1/x+1, or any thing that has any part under it. Any help???