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    Linear approximation higher order terms

    My questions are from lecture 9, MIT OCW SV Calculus, Jerison, 2009; At 27:50 he is deriving the linear approximation for the function e^(-3x)(1+x)^(-1/2)≈(1-3x)(1-1/2x)≈1-3x-1/2x+3/2x^2≈1-7/2x, for x near 0. In the last step he drops the x squared term since it is negligible(no questions so...
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    Expansion of mercury in a thermometer

    I think that I’m doing this problem correctly, but the answer seems a bit unreasonable. Can someone else check my work? A thermometer has a quartz body within which is sealed a total volume of 0.400cm^{3} of mercury. The stem contains a cylindrical hole with a bore diameter of 0.10mm. How far...
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    Damping constant and angular frequency

    While discussing ω^{'}, the angular frequency of a damped harmonic oscillator, given by: ω^{'}=\sqrt{\frac{k}{m}-\frac{b^{2}}{4m^{2}}} where k is the "springiness", m is the mass, and b is the damping constant, my book, Halliday, Resnick and Walker, says if b is small but not...
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    Frequency of a simple pendulum

    A simple pendulum of length L and mass m is suspended in a car that is traveling with constant speed v around a circle of radius R. If the pendulum undergoes small oscillations in a radial direction about its equilibrium position, what will its frequency of oscillation be? I don’t know how to...
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    Liquids; volume and mass flow rates

    Suppose that two tanks, 1 and 2, each with a large opening at the top, contain different liquids. A small hole is made in the side of each tank at the same depth h below the liquid surface, but the hole in tank 1 has half the cross-sectional area of the hole in tank 2. (a) What is the ratio...
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    Self study feedback

    I'm sure there are many like myself who are self studiers: read the book, do the problems. Right now I'm working through Halliday, Resnick and Walker, Fundamentals of Physics, and doing all the odd problems because the answers are in the back of the book. I'm also watching the MIT OCW lectures...
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    Equations and sketches on a computer

    I have been helping my daughter over the phone with first year physics and have run into a problem with communication(insert your favorite joke here). It is difficult to describe a diagram that has vector, angles, notations and also associated equations. I would like to be able to freehand...
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    Quantized energy

    Homework Statement An atom that is being monitored emits light five times without being re-excited. The energies detected for four of those emissions are 0.7 eV, 0.8 eV, 0.9 eV, and 2.0 eV. The energy information for the other emission was lost by the computer controlling the detectors, as was...
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    Energy and spring compression

    Homework Statement The spring of a gun has a spring constant, k, of 4.0lb/in. When the gun is inclined upward by 30 degrees to the horizontal, a 2.0 oz ball is shot to a height of 6.0 ft above the muzzle of the gun. (a) What was the muzzle speed of the ball? (b) By how much must the spring...
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    LaTeX Latex help

    I am trying to post a question in the physics section but the LATEX is misbehaving. It seems to be stuck on the equation for kinetic energy. Any LATEX command I put in always shows up as the kinetic energy equation. Any ideas on how to fix this? I have closed the web browser and opened it up...
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    Composite functions and domains

    Homework Statement The forward-back function is f (t) = 2t for 0\leq{t}\leq{3} , f(t)= 12-2t for 3\leq{t}\leq{6}. Graph f(f(t)) and find its four-part formula. First try t = 1.5 and 3. The Attempt at a Solution There are four possible composite functions from the two given...
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    Marginal profits

    This question is not homework(I'm not even in school). Also, I know how to get the answers but have a question regarding the reasonableness of an answer. This is taken from a calculus book in the chapter introducing the derivative. Problem: It costs C(x)=50+\frac{1}{4}x-\frac{1}{30}x^{2}...
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    Gödel's Incompleteness Theorem

    (Apologies if I am in the wrong part of the forum) What branch of mathematics does Gödel's Incompleteness Theorem deal with?(I'm guessing Logic) and does anyone know any good books at the undergraduate level that would help to lay a foundation for understanding his theorem. I am "teaching...
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    Approaching a limit

    When approaching a limit, say, x approaching 1, does x actually reach 1 or is it just infinitesimally close? In particular I'm interested in where the denominator of a function of interest contains the factor (x-1). (It's hard to show a nice example without the Latex.)
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    Extraneous roots

    Sorry for the length of the post, the problem I've included is not difficult but I wanted to have an example to help illustrate my question. solve: \sqrt{x}-\sqrt[4]{x} -2=0 . . . (x-16)(x-1)=0 The roots are 16 and 1, however when one puts them back into the original equation, 1 is...
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    Perpendicular lines

    Determine a constant real number k such that the lines AB and CD are perpendicular. A(1,2), B(4,0), C(k,2), D(1,-3). (answer given is k=-3/2) If two lines are perpendicular the product of their slopes is -1. The slope of AB is \frac{0-2}{4-1} = \frac{-2}{3} The slope of CD is...
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    Running by the pool

    The length of a rectangular swimming pool is twice its width. The pool is surrounded by a cement walk 4 ft wide. If the area of the walk is 748 square feet, determine the dimensions of the pool. A= the total area bounded by the outer edge of the surrounding walk W= the area of the walk P=...
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    Symmetry about the origin?

    Determine whether the graph of the relation is symmetric with respect to the y axis, x axis, or the origin. y=(x-3)^{3} I don't know how to produce a visual of the graph with this post but it is a graph of y=x^{3} moved 3 units to the right along the x axis. Visual examination of the graph...
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    Radical simplification

    From the book..." \sqrt[4]{(-4)^2}=\sqrt[4]{16}=2. It is incorrect to write \sqrt[4]{(-4)^2}=(-4)}^\frac{2}{4}=(-4)}^\frac{1}{2}=\sqrt{-4} ..." I understand the math involved but want to be sure of the exact reason why the first part is correct and the second is not. Is it because of the...
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