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!