"Precalculus Mathematics in a Nutshell" + "Spivak's Calculus"?

Hey, I'm currently a senior in HS attending Elementary Calculus with a decent grade (B+).

I recently had an epiphany that I haven't actually learned anything, I've just temporarily memorized how to derive the answer from the set of information given. That fact disturbs me for whatever reason, seeing as how whatever future career path I choose will be heavily mathematics-based.



Though I skipped Pre-Calc and all that it entails yet somehow am passing Calc easily, I'm wondering if "Precalculus Mathematics in a Nutshell" will suffice in nailing down and/or covering all of the high school math I've learned and will completely prepare me for Calculus? It has amazing reviews. If so, I'd like to move onto Spivak's Calculus right after.



Is this a solid attack plan? Or do you not recommend one of these books?

If you don't recommend one, please provide alternatives. I simply like the simplicity of this plan.

Current in a Circuit with Multiple Voltage Sources

What exactly are the rules for conserving the currents in a circuit? If we have multiple emfs in parallel with each other, how can we figure out the direction and magnitude of currents in each segment of a circuit?



It makes perfect sense to me for one emf, and for the most part with two I can usually figure it out as well. So could we consider three emfs in parallel, and 3 resistors parallel to each other and each in series with the 3 emfs? In a set-up like this, how do we add up the currents? Do we do so arbitrarily?



I tried attaching an image, but am not sure if it worked.



Thanks for any help!

Attached Images

20_82alt.gif (3.0 KB)

Is energy conserved?

If I push a book horizontally across a table I do work. But is energy conserved?

Using calories of food (as a human body), several questions

Hello,

I would like to understand what exactly is meant by "calories" when we read for example that a chocolate bar has 300 kcal. I understand what is a (kilo)calorie.

What I do not know is whether the written values are the raw values in calories or if they are the values our body can "extract" from the food.

In other words, if I eat the chocolate bar of 300 kcal, can my body use 300 kcal energy entirely taken out from the chocolate bar? Or can my body only extract a lesser quantity than that due to some inefficiency?



Another question I have is, say a quarter of a glass filled with oil has also 300 kcal. If I drink it, can I get as many "useful" calories than with the chocolate bar? By useful I mean, can my body use as many calories as with the 300 kcal from the chocolate bar. Or we do "extract/absorb" different food differently?



Thank you.

Showing SHM

1. The problem statement, all variables and given/known data



A cube of mass m is connected to two rubber bands of length L, each under tension T. The cube is displaced by a small distance y perpendicular to the length of the rubber bands. Assume the tension doesn't change. Show that the system exhibits SHM, and find its angular freqency ω.



3. The attempt at a solution



So basically from a FBD of cube, I have vertical forces: -2Tsinθ - mg = m[itex]\frac{d^2y}{dt^2}[/itex] and the horizontal components of tension from each band cancels. Now since the cube is displaced by a small distance y, I assume we can approximate sinθ ≈ θ. But then I'm not sure what to do?

I tried using sinθ = [itex]\frac{y}{(y^2+L^2)^{1/2}}[/itex], but then I get a complicated expression.

I know I need to obtain a -constant*y on the LHS. Any suggestions.

Potential across capacitance

A first capacitor is initially connected to potential source ?. The charged capacitor is then removed

from the source and connected to a second initially uncharged capacitor. Determine the final potential

across the first capacitor long after the switch is closed.









1. The problem statement, all variables and given/known data

Q1 charged by capacitor C1



2. Relevant equations

Q=Cε



3. The attempt at a solution

Set Notation/Theory help

1. The problem statement, all variables and given/known data



Three tests are available to identify six different substances.



Test 1 is true in the presence of substance 1,2,3, and 4

Test 2 is true in the presence of substance 3,4, and 5

Test 3 is true in the presence of substance 2,5, and 6



Using set notation to denote the events that a substance under test is



a) substance 1

b) substance 2

c) substance 6



2. Relevant equations



Let [itex]E_{i}[/itex] denote the event that the i'th test is positive, [itex]i= 1, 2, 3[/itex]



3. The attempt at a solution



How do I properly represent the answers in set notation?



Currently here are my attempts at a,b, and c.



a) [itex] S_{1} = E_{1}[/itex] [itex]\cap[/itex] [itex]E_{2}^{'}[/itex] [itex]\cap[/itex][itex]E_{3}^{'}[/itex]



b) [itex] S_{2}[/itex] cannot be identified



c) [itex] S_{6} = E_{3}[/itex] [itex]\cap[/itex] [itex]E_{2}^{'}[/itex] [itex]\cap[/itex][itex]E_{1}^{'}[/itex]





Is this how you correctly write the answer in set notation?



Also with b) I am also unsure of my answer. Would it be correct to say that E1 is true AND not S6?

test functions for tempered distributions: analytic?

When considering tempered distributions, I am only aware of the definition of test functions of a real variable. However, is it okay to use test functions of a complex variable z that are analytic in a strip that includes the real axis? (of course they still must fall off fast enough as the real part of z goes to +/- infinity). I know of at least one test function of a real variable, [itex]e^{-x^2}[/itex] for whom the analytic continuation to the strip is trivial.



I am asking because (just for fun) I am looking at the Fourier transform of the Heaviside step function, [itex]u(t)[/itex] that is zero for t<0 and one for t>0. If we let [itex]\hat{f} [/itex] denote the Fourier transform of an arbitrary [itex]f[/itex] (test function or distribution), and let [itex]\psi[/itex] be a test function, then by definition of the Fourier transform for distributions

[tex]

\langle \hat{u}, \psi \rangle = \langle u, \hat{\psi} \rangle = \int_0^{\infty} dx\, \int_{-\infty}^\infty dt\, e^{-i x t} \psi(t)

[/tex]

I want to swap the order of integration but cannot since the reverse order integral is not defined. However, the above integral is well behaved so it should be equal to

[tex]

\lim_{\epsilon \rightarrow 0} \int_0^{\infty} dx\, e^{-\epsilon x} \int_{-\infty}^\infty dt\, e^{-i x t} \psi(t).

[/tex]

Now I can swap the order of integration and perform the first integration to obtain,

[tex]

\lim_{\epsilon \rightarrow 0} \frac{1}{i} \int_{-\infty}^\infty dt\, \frac{\psi(t)}{t - i \epsilon}.

[/tex]



This is where I would like [itex]\psi[/itex] to be analytic in a strip including the real axis and extending into the lower half plane (even by just a little bit). In that case the above integral is simple by contour integral techniques - the integral is along the real axis and as [itex]\epsilon \rightarrow 0[/itex] we indent the contour into the lower half plane and I get

[tex]

\langle \hat{u}, \psi \rangle = \pi \psi(0) + PV\int_{-\infty}^\infty dt\, \frac{\psi(t)}{it}

[/tex]

where [itex]PV[/itex] indicates the Cauchy principle value (symmetric limits about t=0). Hence,

[tex]

\hat{u}(x) = \pi \delta(x) + PV \frac{1}{ix}.

[/tex]

It works out so nice it seems like it should be fine to do this, but lots of things can be done formally that make no sense! Everything up to where I want to use contour integration is easy to find in textbooks (right now I am looking at "waves and distributions" by Jonsson and Yngvason), but I don't see authors doing the last step with contour integration and I am wondering if there is a reason - I am guessing there is but I just don't see it.



Thanks,



Jason

calculating G Force

Hey everyone I have a question that I'm wondering if it can be answered, I'm not into physics and don't know how to calculate it out. my question is if 40 lbs fell from a height of 8 feet how many G forces would be created on impact. the reason i'm asking is due to a real life scenario. thanks.

Seeming paradox when squaring distance depending on units

Hello all



This is probably simple and I'm overlooking something



1 mile = 5280 feet



10% of a mile is 528 feet



528 feet squared is 278,784 feet which is 52.8 miles squared



But 0.1 miles squared is .01 miles squared



So depending on if you square it as 0.1 miles, or if you convert it to feet, then square it and convert it back, you get wildly different answers



Why is this.? Not sure why I thought of this.



Thanks in advance

Power

1. The problem statement, all variables and given/known data



In a hydroelectric plant, water falls from a high elevation to a lower elevation,losing PE in the process/ In doing so, it turns turbines which generate electricity. In a particular hydroelectric plant, water flows at a rate of 400 kg/s. If 80% of the PE that the water loses gets converted to electricity, what is the power output of the dam, in watts?



2. Relevant equations



PE=mgy

PE lost=KE gained



3. The attempt at a solution



Known information:



m=400 kg/s (Right?)

g=9.8 m/s^2

y=30 meters



KE= .8(400 kg/s*9.8 m/s^2*30 m)=9408 Joules



That's easy enough, right? But don't a need a time to convert to Watts? Is the time just one second, so that my answer is 9408 Watts?



Thanks!

Connection between cesaro equation and polar coordinates

First, I'd like that you read this littler article (http://ift.tt/1pGnsiw). The solution given by Euler that coonects the system cartesian (x, y) with the curvature κ of the "cesaro system" (s, κ), is that the derivative of the cartesian tangential angle φ* wrt arc length s results the curvature κ.



However, in the 2D plane is definied the polar tangential angle ψ* too. Thus if I want express a curve given s and κ in terms of r and θ (or vice-versa) I need to establish a connection between the curvature κ and the polar system. So, would be correct to say that the derivative of ψ wrt s is equal to κ?



*

http://ift.tt/1ft93TU

http://ift.tt/1gElokS

http://ift.tt/NEf8B8

Thermal Equilibrium of a system

1. The problem statement, all variables and given/known data

A combination of .25kg of H2O at 20C, .4kg of Al at 26C and .1kg of Cu at 100C is mixed in an insulated container and allowed to come to thermal equilibrium. Ignore any energy transfer to or from the container and determine the final T of the mixture.





2. Relevant equations







3. The attempt at a solution



At first this seemed really simple to me I just figured out if the T would be below or above 26 to determine which parts will lose energy and which will gain.

So it turns out that Tf will be less than 26 because the energy required to heat up water to 26 is greater than the copper can supply.



Setting up the equation is where I get lost.



If the energy lost by the Al and Cu must equal the energy gained by the water then shouldn't the equation be:



Q(Cu)+Q(Al)=Q(H2O)



where Q= mCΔT



The book has it set up as



Q(Cu)+Q(Al)+Q(H2O)=0



which is really confusing me.



Ok so if I let the lost energy be negative I get the same thing as the book does but then it seems to me like I'd get the same answer in every case.



The book's way:

-Q(Cu)-Q(Al)=Q(H2O)

Q(Cu)+Q(Al)+Q(H2O)=0



But then let's say Tf > 26 then we get



-Q(Cu)=Q(Al)+Q(H2O)



Which simplifies to



Q(Cu)+Q(Al)+Q(H2O)=0



exactly the same thing! That doesn't make any sense to me.

Static Equilibrium Problem

1. The problem statement, all variables and given/known data



Two supports, made of the same material and initially of equal length, are 2.0m apart. A stiff

board with a length of 4.0m and a mass of 10 kg is placed on the supports, with one support

at the left end and the other at the midpoint. A block is placed on the board a distance of

0.50m from the left end. As a result the board is horizontal. The mass of the block is:



A. zero

B. 2.3kg

C. 6.6kg

D. 10 kg

E. 20 kg



2. Relevant equations



∑F = 0 = N₁ + N₂ - mg - Mg

∑τ = 0 = -1/2 N₁ + 1/2 N₂ - 1/2 Mg



3. The attempt at a solution



Those equations above are a result of a free body diagram I drew. However, if I begin to solve the equation for m, I end up with an expression in terms of N₁ (ie. 2N₁ = mg). Is there enough information in the question? By the way, I'm calculating torque from the position where the mass is placed.

Torque around an axis

1. The problem statement, all variables and given/known data

A 2.72kg ball is attached to a .65m rod that weighs 1.24kg. The rod rotates around it's axis 125 degrees. After rotating 125 degrees the ball is let go from the rod. How much torque is required to make the ball's exiting velocity 18m/s.





2. Relevant equations

t=F*d

vf=vi + at

vf^2=vi^2+2at







3. The attempt at a solution

First I solved for how much distance the ball is traveling in an arc before being released=1.43m

L = ((125)*2pi*.6558m)/360 = 1.43m.

Then I found it's needed acceleration to reach 18m/s.

18^2=0+a(1.43m), a=113.28m/s^2

Then I solved for the amount of time it will take to reach 18m/s

vf=vi+at, 18=113*t, t=.1558s

Now this is where I got stuck.

t=F*d

F=m*a, so i'm assuming (2.72kg)*113.28m/s^2=308.12N

t=308.12N*(1.43m)=440.61N*m.

Is this correct? I think this may involve angular motion which I am inexperienced with. Ps, this is for a catapult idea/robot so any help is gratefully appreciated.

Capacitors in series

I read about an example in which you had two metal plates, and in between them one third of the distance from the top plate downwards (towards the bottom plate) was made up of κ, and the rest was air. THe problem proceeded to calculate the capacitance of them in series.

I don't get how the top layer acts as a capacitor though. There's no metal plate? Can someone please help me understand what's going on?