FR 5

A)R(t)=rate of which sand is removed. The is how much is being removed. so take the integral. So to see how much the tide take away the first 6 hrs set the integral from 0-6 making the equation:
fnInt(2+5sin(4nt/25),T,0,6) = 31.81593137 simplified it is 31.816 cubic yards

B)To fond the amount of sand at a certain time you must subtract S(t)-R(t)since there is no exact time but a starting point the integral will be from 0-x. At t=0 there is already 2500 sand so it must be added so the equation is:
Y(t)=2500+fnInt (S(t)-R(t), t, 0, x)

C)Y(t)=total number of cubic yards of sand on the beach. Y'(t)=rate at which the total number of sand is changing. so the derivative is Y'(t) is just S(t)-R(t). now we just have to find it at t=4 giving the equation Y'(4) = 15(4)/1+3(4) - 2 + 5sin(4(4)π/25). After the simplifying it equals Y'(4) ≈ -1.909 cubic yards/hour.

D)To find the minimum amount at the equation at part A don't you graph it and find the intersection. If so then the intersection is at (5.1178, 4.6943)so at 5.1178 it equals 4.6943 + 2500=2504.6943 cubic yards.

DONE???

MEANIE

The mean value theorum means: F'(c) is the derivative at x=c and the slope of the secant line through the interval [a,b] is F(b)-F(a)/b-a all this basically means this Point of Secant line is equal to the Point of the Tangent line.
For Instance my graph would be x²


lets say the intervals are from [o,3] so then a=0 and b=3 now we got the points now all we need is to use them in the equation and find c so F(3)-F(0)/3-0 =3 so the slope of the secant line is 3 and the equation is the derivative 2x



Ok so skipping all of the equations blah blah blah we find that c lies on x=1 and knowing that we can do the tangent line. Which again blah blah blah we find that it passes through (1,1)and since the tangent line is parallel to the secant use that knowledge and pont slope form and then we get the equation F(x)=2x-1 and bam definition graphically



2. this doesnt work for equations that end up being cusps, corners, etc like and absolute value equation it is not diferrentible at x=0 so there is no tangent point there right? hehehe

F(x) to F'(x)

1) It increases at (-2,0)U(0,2) because durint those intervals it only output positive numbers basically f'(x)>0. It decreases at (-00, -2)U(2, 00) because f'(x)<0

2)at (0, 0) Right? Because an extrema is when F'(x) is zero or undefined

3)Concave up:(-00,-1.5)U(1.5, 00)
Concave down: (-1.5, 1.5)
concave up wher the slope is positive and concave down wher it is negative

4)it is probably x^5 cause it changes direction a couple of times also because the f'(x) looks like x^4.

Mind Boggling

1. I am a growth mindset I always like to learn new things. i belive that anybody can improve themselves.
2. It helped me because it allowed me to stick to math because I love challenges.
3. Makes me want to exercise the hell out of it i could make better than ever before. Funny because even though I am not in shape my brain could be.
4. It allows me to know that I can push myself a lot further than I know.