Calculus study realm

Mehehe I'm back!

I've just checked my e-mail. The teacher sent us secret note about the content of the exam.

1) There will not be any problems involving z-substitutions, volume of objects with known cross-sections, surface area of a volume of revolution, centre of mass, arithmetics sequences and series, alternating series, absolute and conditional convergence.

2) You are to know all the topics on the sheet entitled "Theoretical Questions" I gave out in the last week.

3) Look into the last questions on the exercise sheet entitled "Differential Equations".

4) There will be at least one question on the integrative component – acceleration, velocity, position.

Differential equation. I haven't look to it yet!

That's because you were sleeping and watching MVs.

No, I was memorizing the theorems. I really couldn't remember any integral theorem after the first test.

So, let me try the last questions of differential equation. =3

Oh, man. I should have seen this e-mail before. I studied for nothing! But the test is going to be easy.

7) Solve the following stuff
a) dy/dx = e^(2x)/(4y^3)
∫4y^3 dy = ∫e^(2x) dx
y^4 = e^(2x) /2 + C

b) dy/dx = xy
∫1/y dy = ∫x dx
ln|y| = x^2 /2 + C
y = e^(x^2 /2 + C) = e^(x^2 /2) e^C = Ce^(x^2 /2)

c) dy/dx = xy/(2lny)
∫2/y lny dy = ∫x dx

Consider ∫2/y lny dy = 2ln^2(y) - ∫2/y lny dy
∫2/y lny dy = ln^2(y)

ln^2(y) = x^2 /2 + C

d) dy/dx + e^(y+x) = 0
dy/dx = -e^y e^x
-e^-y dy = e^x dx
e^-y = e^x + C

Finish! That was easy actually. ——

That not the last question. That last is on the second sheet.

There's a second sheet? No wonder I finished it so fast and thought I was genius.

10) Polonium-210 has a half-life of 140 days. If a sample has a mass of 200 mg,
a) what is the D.E. that describes this situation?
b) find a formula for the mass that remains after t days
c) find the mass after 100 days
d) when will the mass be reduced to 10 mg?

Damn chemistry. Why do I have to care that?


I don't understand the question! @__________@

I don't even understand the correction.
Spirit or Dark, SOS!

I don't think the teacher have covered this in class.

What I figured is that: define first the variables
So let M = mass, and t = time
We know that time is always independent. Thus it's dM/dt
let dM (change in mass) -M (the mass) (negative because the slope must be negative --> the function must decrease)
and let dt = 1/k (a constant)

So the equation is dM/dt = -kM

Then just do what you always do to solve differential equation:
dM/M = -k dt
ln|M| = -kt + C

Isolate M for b)
M = e^(-kt + C) = Ce^(-kt)

We know that when t=0, the mass stays unchanged, thus
200 = Ce^(-k0)
C = 200e^(k0) = 200

We know that when t=140, the mass has decreased by a half: 100
100 = 200e^(-k140)
1/2 = e^(-k140)
ln(1/2) = -ln2 = -k140
ln(2)/140 = k

Thus the formula is M = 200e^(-ln(2)t/140)

So then everything becomes easy
c) M = 200e^(-ln(2)100 /140) = 121.9 mg
d) 10 = 200e^(-ln(2)t/140); 605.1 days

You are our God! I was just thinking about going to his office hours. XD
Lemme try the previous question.

The temperature of a roast turkey just taken out of the oven is 185°F. It is placed in a
room where the temperature is 75°F.
a) Using Newton's Law of Cooling, give the D.E. that describes this situation
b) Solve this differential equation
c) If the temperature of the turkey is 150°F after 30 minutes, what is it after 45 min.?
d) When will the turkey have cooled down to 100°?

Newton's Law of Cooling... Whatever it is, I use calculus to do this sh*t. Mehehe

a) dT/dt = -k(T - 75) (always plug in the temperature that is not related to the turkey (variable). At least that's what's the correction suggest.)
dT/(T - 75) = -k dt

b) ln|T - 75| = -kt + C
T = Ce^(-kt) + 75
at t=0; 185 = Ce^(-k0) + 75; C = 110
Thus, T = 110e^(-kt) + 75

c) at t = 30; 150 = 110e^(-k30) + 75; k = -ln(15/22)/30
T = 110e^(ln(15/22)(45)/30) + 75 = 136.93°

d) 100 = 110e^(ln(15/22)t/30) + 75 ; t = 116 minutes

Good, you get it. I'm so pro at teaching

I still don't get it. Why do we have to plug in the 75 in the change in temperature. Can't we just put T like in the polonium question?

Don't ask. You just have to plug in all the given numbers in the question.

Not all. Not 210 in polonium question.

But that's obviously not a useful information to solve the question. Everyone can recognize that.

So, next what do we have?

There will be at least one question on the integrative component – acceleration, velocity, position.

If seems that we did not do any of it in the course. Can we have an example of that?

There was a question like that in the mid term. And I failed it.

We leaned that last semester. How can you miss that?

Because I didn't know it would reappear on that test.

Anyway. I think I still have a copy of that question. Let do it.

You didn't get the answer. I burned my exam. And of the others didn't get it or don't have it, how are we supposed to verify?

Communal verification! Everyone do it and we compare our answer. The one we agree the most wins.

Here's the question: A particle is moving back and forth along a straight line according to the following data. a(t) = 3 cost, v(0) = -3/2. Find the displacement and distance traveled as t varies from t=0 to t=π/2.