2005 FR 5

a. fnInt(2+5sin(4(pie)t)/25),t,0,6=31.815 yards^3/hr

b.Y(t)=fnInt(2+5sin(4(pie)t/25)-fnInt(15t/1-3t)+2500

c.Y(4)=fnInt(2+5sin(4(pie)(4)/25-fnInt(15(4)/1-3(4))+2500=2426.1824

d.The minimum is at t=0 because at that point the output is 2 and then the points before 0 were negative makign 0 a minimum. Also all the other times for t thre output are all positive.

The Mean Value Theorum

The mean value theorum helps find certain points is a graph in the intervlas of [negative infinity, positive infinity] from the f(x)=x^3+1

But the mean value theorum can only work for continous functions not for functions like the ones with square roots like f(x)=(x)^1/2+1 in the interval of [negative infinity, positive infinity]

f(x) vs. f ' (x)

1. Function Increasing (-2,-1)U(0,1) here the slope is increasing; Function Decreasing (-1,0)U(1,2) because the slope is decreasing . Also because f '(x) is the derivative of f(x) and the derivative is the slope of the original function.
2. Local Extrema local maximum x=+1 and x=-1 there the slope was positive and then had a flat slope to turn negative. local minimum x=0 there the slope was negative then reach to a slope of 0 and then switch to a positive slope making a critical point.
3. Concave up between (-1,1) where it looks like it can hold water. Concave down is (-2,-1)U(1,2) where it looks like its going to spill water.
4. x^3+x^2 is f(x) on the given slopes and concavity shows it as the two power function are combined.

State of Mind

1. I believe the mindset I have when it come to my intelligence is Growth Mindset for a couple reasons. Growth Mindset is when when they are persistent in obstacles in I have been through many obstacles that I had to overcome and some that I have to overcome now! Also setting efforts show that I have a Growth Mindset because when I set effort to something important to me I really give it my all. I also accept many criticism which I learn my mistakes and improve. Also finding inspiration from others are usually my heroes or if someone beats me that person can inspire me to train harder.
2. The mind set has help me a little to improve because it motivates me to do better in the class and try to get a good grade.
3. I was surprise and I told myself that it will be pretty cool to now workout with my brain to make it stronger.
4. This can help me in the future of becoming a better person not just physically but also mentally and in order to be a healthy person i got to have a strong mind and a strong body. :)

Logs and Inverses

The things on what in understood about logs and inverses were:
1. How to get rid of logs in order to find the variable. Like 7log7^X well the 7 cancels the log7 leaving the x by itself.
2. Also to find the log of a number like log2 32 that is equal to 5. That log is also equal to 2^5=32.
3.Also to extend the log. Like log5(h+d) equals to log5 h + log5 d.
4. Finding the inverse of the log is what is pretty easy most of the times. To find the inverse is just to switch the inputs and outputs.

The things i don't understand:
Mostly i will understand the algebra of the logs and inverses but they are hard when it comes to graphing them. Usually i graphing mess me up when i have to graph the parent function with a new function coming out of the parent function. Then I'm not really sure how to get the new function from the parent function. he the asymptotes to confuse me as well at times.

Even and odd functions

For the even functions in a graph, they are symmetrical. The y-axis cuts the graph function in half. The outputs of the function will stay the same for being symmetrical, but the inputs of the function would actually become the opposite. For example if the input is 2 then the opposite input will be -2. That way the function would look the same in both quadrants either if it is quadrant 1 and 2 or quadrant 3 and 4 .
Odd functions are a little different are symetrical also symetrical in a way. In way it has to be folded twice. The 1st time in half and then flip down and maybe that way you can tell that it is a fuction. Or you can just check the inputs and outputs. Because unlike even functions both the inputs and outputs are the opposite of the numbers fron the original function. Like points (2,3) in quadrant 1 will be the opposite of (-2,-3) in quadrant 3.

Algebra VS Calculus

1. The difference of finding the limits when is when x is approach the output it helps find the input as it approaches c. But when the constant is plug in which is the algebra thta helps find thethe function or the slpoe. There are tiames when they are the same like when x approches 0.
lim f(x) x-> c = limf(x) x->c= = limf(x) x->c-
2. The derivative means that it does have a slope. Finding the slope of the line is looking a the change in y n inthe change in x. Both do mean is to find the slope. But there are different ways to find the slpoe. For the derivative is lim x-> f(a+h)-f(a-h)/2h. And finding the slope is Y2-Y1/X2-X1