Ellie's Final Reflection

Alright you guys!!! We've gotten through calculus and we've all made it. When this class first started, I wasn't stressed out by the least. Having a teacher like Mrs. Robinson the year before, I knew what I was getting into. The first week was oddly enough, expected! We got so many formulas to memorize and so many ways of taking a derivative a lot of us were going crazy, but not me, I was kind of like "whatever" with it. Turns out two or three weeks into it, I got really scared, my grades weren't really slipping but I wasn't getting the new, or harder, derivatives. After that, I buckled down and got to work. Here's what I've learned throughout the year...

Easy Concepts For Me:
--How to Take a Derivative:
You take the exponent and times it by the coefficient, what you get is the new coefficient and you're new exponent will be the original exponent minus one!
--The First Derivative Test:
You have to take the derivative of the function and set it equal to zero. Then solve for the critical values (x values). Set those values up into intervals between negative infinity and infinity. Plug in numbers between these intervals into the first derivative to see if there are max or mins or if the graph is increasing or decreasing.
--The Second Derivative Test:
You take the derivative of the function twice and set it equal to zero. Solve for the x values and set them up into intervals between negative infinity and infinity. Plug in numbers between those intervals into the second derivative to see where the graph is concave up, concave down, or where there is a point of inflection.
--Maximums, minimums, critical values, increasing, decreasing:
All of these are related to first derivative test. it's simple. you take the derivative, set equal to 0, solve for x. Set up some intervals using these numbers. Plug in numbers and test your intervals. pos to neg is a min. neg to pos is a max. pos = increasing, neg = decreasing. simple stuff. remember it.
--Point of inflection, concave up, concave down:
It's the second derivative test. set up intervals, if the intervals change signs, it is a point of inflection there. also, if its negative, that interval is concave down, positive is concave up.
--Finding Critical Values:
To find critical values, first take the derivative of the function and set it equal to zero, solve for x. The answers you get for x are your critical values.
--Limits:
If the degree on top is smaller than the degree on the bottom, the limit is zero.
If the degree on top is bigger than the degree on the bottom, the limit is infinity.
If the degree on top is the same as the degree on the bottom, you divide the coefficients to get the limit.



Some Concepts that...after reviewing over and over for weeks/months...it just clicked:
--Slope of a normal line:
Take derivative, plug in x. get a slope. however, make sure you use the negative reciprocal of the slope (normal means perpendicular to). use point-slope formula.
--Absolute Extrema:
If you are given a point, plug those numbers into the original function to get another number. Alos, solve for critical values and plug those into the original function. Once you get your second numbers, you set each pair into new sets of points. The highest point is the absolute max and the smallest point is the absolute min.
--Average Rate of Change:
f(b)-f(a)/(b-a)
--Average Value:
This is just an integral times by 1/(b-a).
--Equation of a Tangent Line:
Take the derivative and plug in the x value.
If you are not given a y value, plug into the original equation to get the y value.
then plug those numbers into point slope form: y − y1 = m(x − x1)


And-of course-the things that I always struggled with, and hope to never encounter again even though I know I will....:
--linearization:
Identify the equation
Use the formula f(x)+f ' (x)dx
Determine your dx in the problem
Then determine your x in the problem
Plug in everything you get
Solve the equation
--Riemanns Sums:
LRAM-Left hand approximation=delta x[f(a)+f(a+delta x)+...f(b-delta x)]
RRAM-Right hand approximation=delta x[f(a+delta x)+...f(b)]
MRAM-Middle approximation=delta x[f(mid)+f(mid)+...]
Trapezoidal-delta x/2[f(a)+2f(a+delta x)+2f(a+2 delta x)+...f(b)]
delta x=b-a/number of subintervals
--Volume by Disks:The formula is pi times the integral of the [function given] squared times dx. so just solve it by taking the integral of it and then pluging in the numbers they give you. just like before you'll have two numbers so whatever the answer is for the top one will be first and then you subtract the answer you get for the bottom one. then graph
--Volume by Washers:
The formla is pie times the integral of the [top function] squared minus the [bottom function] squared times dx. so to do this, if you don't have the in between number you have to set the functions equal, but if you do, then it's worked the same way as above. square the formula's that were given and simplify. then take the integral of it and plug in the numbers they give you or you found by setting the formulas equal to each other and then solve like any other one by subracting them. then graph.
--OPTIMIZATION...
I just can't do it =(


Just a shout out:
To the seniors in the Calculus Class of 2010 - - I'll Miss Yall, we've grown up together basically. I won't lie, I didn't like MANY of yall when I first came here because of different things that happened; however, we've all matured [thank God] and we've become friends. Although some of us aren't best friends I can honestly say that I will miss each and every one of you all because yall are the people I spent all day every day with for the past 6 years.
To the juniors in the Calculus Class of 2010 - - I'll miss yall just as much. There's many reason's why I will..some were my first friend at Riverside Academy because of the Cheerleading Squad, some have always been in my life because we grew up living next door to each other, some have been with me through the years with dancing and just being friends, some HOWEVER have become my friend in this past year and believe me I love yall just as much!

So back to the FARWELLS:
No, I'm not sad I'm graduating. My parents say that it hasn't hit me yet...but most of yall know me already and I'm so happy to finally be just about OUT of Riverside Academy that nothing else matters. Also, I'm not sad that I'm leaving any of yall. Yall have been great friends and I'm glad we've become as close as we have or, not as close as we have in some cases, because LIFE GOES ON..We cannot dwell in the past much less live it. I won't be back at Riverside once I'm out, there's no need....I'm finally getting my life started and I CAN'T WAIT!! =) Yeah, I'll miss yall when certain things come up and I realize what [so and so] would've said and they wont be there..but most of all .. I'll miss US as a class together and how we are with each other! So to everyone - - THANKS! And have a GREAT time either being a senior, or going out into the REAL WORLD and EXPERIENCING everything!

Love Always,
Ellie Kliebert