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  1. C

    Mass Observed to Increase as Velocity Increases?

    With CERN's Large Hadron Collider being in the news recently, I began to think of a question that bothered me for years about particle accelerators. It is well known in physics that mass increases as velocity does (Special Relativity if I'm not mistaken), so if these particles in the LHC are...
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    Limit Evaluation

    Homework Statement I had an exam last night and I was stuck on this problem. I went home where I had more time (and resources) and still can't figure it out, and I won't get my exam back (it was a final). \lim_{x\to0}\frac{1}{x^{2}}-\frac{1}{xsinx} Homework Equations The...
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    Which trig sub do I use?

    Homework Statement \int\frac{dx}{x(x^{2}-1)^{3/2}} Homework Equations The Attempt at a Solution I know I need to use trig sub, but which form? I can't seem to find any that fit this form.
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    Show that the Gamma function is converging

    Homework Statement The gamma function, which plays an important role in advanced applications, is defined for n\geq1 by \Gamma(n)=\int_0^{\infty} t^{n-1}e^{-t}dt (a) Show that the integral converges on n\geq1 (b) Show that \Gamma(n+1)=n\Gamma(n) (c) Show that \Gamma(n+1)=n! if n\geq1 is an...
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    Electric Potaential of two spheres

    Homework Statement Electric charge can accumulate on an airplane in flight. You may have observed needle-shaped metal extensions on the wing tips and tail of an airplane. Their purpose is to allow charge to leak off before much of it accumulates. The electric field around the needle is much...
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    Show that it satisfies the differential equation

    Homework Statement Show that y=tanh(t) satisfies the differential equation \frac{dy}{dt}=1-y^{2} with initial conditions y(0) = 0 Homework Equations The Attempt at a Solution I'm not sure how to start, but we have only dealt with one D.E., and we had moved all the...
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    Definite Integral with trig

    Homework Statement \int\stackrel{tan(8)}{tan(1)}\frac{dx}{x^{2}+1} Homework Equations The Attempt at a Solution \int\stackrel{tan(8)}{tan(1)}\frac{dx}{x^{2}+1} tan^{-1}x|\stackrel{tan(8)}{tan(1)} tan^{-1}(tan(8))-tan^{-1}(tan(1)) 8-1=7 I'm fine with this, but my instructor said...
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    Electric Field of Three Point Charges

    Homework Statement Three point charges lie along a circle of radius r at angles of 30°, 150°, and 270° as shown in the figure below. Find a symbolic expression for the resultant electric field at the center of the circle. Homework Equations \vec{E}=\frac{k_{e}q}{r^{2}} The Attempt at a...
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    Electric Field of a uniformly charged rod

    Homework Statement A uniformly charged rod of length L and total charge Q lies along the x axis as shown. Find the components of the electric field at the point P on the y-axis a distance d from the origin. There's also a part b but I should be able to figure it out easily enough...
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    L'Hopital Limit with rational exponent

    Homework Statement I need to find the limit of this using L'Hopital Rule, if it exists. lim_{x\rightarrow0^{+}}(\frac{sinx}{x})^{1/x^{2}} Homework Equations All of our examples used lim f(x)/g(x)=f'(x)/g'(x) The Attempt at a Solution We have not dealt with any limits like this that have...
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    Calculate the magnitude and direction of the net electric force

    Homework Statement Three point charges lie along a straight line as shown in the figure below, where q1 = 6.60 µC, q2 = 1.41 µC, and q3 = -1.9 µC. The separation distances are d1 = 3.00 cm and d2 = 2.00 cm. Calculate the magnitude and direction of the net electric force on each of the...
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    Inverse of a polynomial

    Homework Statement The function y=x^{2}+4x-6 has two inverses. What are they and which domains lead to these inverses? Homework Equations The Attempt at a Solution y=x^{2}+4x-6 x=y^{2}+4y-6 y(y+4)=x+6 Not really sure where to go from here.
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    Tangent line that passes through origin

    Homework Statement Find a > 0 such that the tangent line to the graph of f(x) = x^{2}e^{-x} at x = a passes through the origin. Homework Equations The Attempt at a Solution First I found the derivative to be: -e^{-x}(x-2)x , which is the slope of the function. I know the tangent...
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    Evaluate the definite integral

    Homework Statement \int^{\frac{\pi}{4}}_{0}\frac{1+cos^{2}(x)}{cos^{2}(x)} Homework Equations The Attempt at a Solution I tried u-sub, splitting it up into 2 fractions, and a trig identity (cos^2x=(1+cos2x)/2). Am i missing something? The other problems were quite easy.
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    Newton's Method General Formula

    Homework Statement The statement \sqrt[4]{a}=x means that x^{n}=a. Using this, we can approximate the radical \sqrt[n]{a} by approximating the solution to the equation x^{n}-a=0. Consider the function f(x)=x^{n}-a. We can use Newton's Method to approximate where f(x)=0 and thus...
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    Derivative Simplification

    Homework Statement y=3(x-1)^{\frac{1}{3}}-(x-1)^{2} I need to find all extrema, roots, inflection points, and concavity using first and second derivative tests. I usually do not have a problem with these, but I need to find some extrema that I know exist. The Attempt at a Solution...
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    Related Rates Ladder Problem

    Homework Statement A ladder of 25 feet long is leaning against the wall of a house. The base of the ladder is pulled away at 2 ft per second. (a) How fast is the top of the ladder moving down the wall when the base is seven feet from the wall? (b) Consider the triangle formed by the...
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    Help with a limit

    Homework Statement I'm having trouble with a homework problem: \lim_{x\to 0}\frac{\frac{1}{3+x}-\frac{1}{3}}{x} I wasn't sure how to simplify this. I tried a conjugate, but assuming I did it right, it didn't get me anywhere.
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    Practical real world uses for Colorimetry

    I need to give a short presentation about Colorimetry in my chemistry class. I plan on using the first five minutes giving an example procedure about determining the concentration using MicroLAB's spectrophotometer hardware in our lab. I would like to use that last five minutes of my...
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    Linearization of the cosine

    Homework Statement Find the linearization L(x) of f(x) = cos(x) at a = π/2 Homework Equations L(x) = f(a) + f'(a)(x-a) The Attempt at a Solution I just want to make sure I did this correctly: L(x) = cos(π/2) + -sin(π/2)(x-(π/2)) L(x) = 0-1(x-(π/2)) L(x) = -x + (π/2)...
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    Related rates yeast cells problem

    Homework Statement The number of yeast cells in a laboratory culture increases rapidly at first but levels off eventually. The population is modeled by the function below n=f(t)=\frac{a}{1+be^{-t}} where t is measured in hours. at time t=0 the population is 20 cells and is increasing...
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    Logarithmic differentiation

    Homework Statement Find y' using logarithmic differentiation if y=x^{sin(x)} Homework Equations The Attempt at a Solution lny=sin(x)ln(x) \frac{1}{y}(y')=sin(x)(\frac{1}{x})+ln(x)\cdot cos(x) (y')=[sin(x)(\frac{1}{x})+ln(x)\cdot cos(x)]y Did I do this one properly? Thanks in advance.
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    Limit definition of a derivative

    Homework Statement I had my second exam last week for my Calculus I course. I did alright, but we are supposed to correct them and bring them back for a quiz grade. However, I wasn't sure how to do this one on the test, and did not magically figure it out since then :) Find the derivative...
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    Fundamental frequency of a pipe half-filled with water

    Homework Statement A very tall pipe is partially filled as shown. (A vertical pipe is filled with water about halfway, and .85m of air is in the pipe to the open end. The bottom is closed). |**| ^ |**| | |**| .85m |__| _|_ |__| |__| |__| |__| |__| ^^ My attempt at a drawing, where...
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    Percent Yield

    Homework Statement Use the unbalanced equation NaN3 → Na + N2 to answer the following question(s). The % yield of an experiment in which two moles NaN3 was used and 21 g nitrogen was isolated is ________. Homework Equations percent yield=({actual yield}/{theoretical yield})100% The Attempt...
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    Spectator ions

    Homework Statement The spectator ions in the precipitation equation Pb(NO3)2+ 2KI --> PbI2 + 2KNO3 are ________. Homework Equations The Attempt at a Solution I've been working through a lot of practice problems for next week's final exam, and I seem to be struggling with this...
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    Equilibrium and Rate

    We just started learning this in chem, and I thought it was quite simple. I missed the particular lecture, and am now paying the price: The following chemical reaction has reached equilibrium: 2NO + O_{2} \rightarrow2NO_{2} Calculate the equilibrium constant (K_{eq}) if the concentrations...
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    Conceptual torque problem with multiple sized pulleys

    Homework Statement This is a six part conceptual question that is not difficult. However, the last part I am struggling with. What is wrong with the following student's reasoning: "The clockwise torque is greater than the counterclockwise torque because the disk is accelerating clockwise. For...
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    Converting angual velocity to linear velocity

    Homework Statement An astronaut is being tested in a centrifuge. The centrifuge has a radius of 10m and, in starting, rotates \theta=0.30t^{2}, there t is in seconds and \theta in radians. When t=5.0s, what are the magnitudes of the astronaut's (a) angular velocity, (b) linear velocity, (c)...
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    Another kinematics problem

    Homework Statement A pilot flies horizontally at 1300 km/hr at height 35m above initially level ground. At t= 0 the pilot begins to fly over ground sloping upward at an angle of 4.3 degrees. If the pilot does not change his heading, at what time will he hit the ground? Homework Equations...
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