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. :)