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    Engineering Analysis of a Digital Circuit

    Homework Statement 1. (a) Remove all the internal bubbles in the circuit by applying DeMorgan's theorem so that the circuit consists of only AND gates and OR gates, and INVERTERS. INVERTERS can only be used for the inversion of inputs. Note that there is a 3-input NAND gate shared by both F1...
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    Electric Potential and a Conductor

    Homework Statement A hollow spherical conductor, carrying a net charge +Q, has inner radius r1 and outer r2=2r1 radius (see the figure ). At the center of the sphere is a point charge +Q/2. A.Write the electric field strength E in all three regions as a function of r: for r>r2,r1<r<r2...
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    First Order Linear DE System

    Homework Statement I need to solve this DE system for a lab: q_1'=2-\frac{6}{5}q_1+q_2 q_2'=3+\frac{3}{5}q_1-\frac{3}{2}q_2 Homework Equations The Attempt at a Solution I know how to use the method of elimination to solve such systems, but this is non homogeneous because of the added...
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    Jacobian Change of Variables Question

    Homework Statement Evaulate the integral making an appropriate change of variables. \int\int_R(x+y)e^{x^2-y^2}dA where R is the parallelogram enclosed by the lines x-2y=0, x-2y=4, 3x-y=1, 3x-y=8 . Homework Equations The Attempt at a Solution I'm not sure what change of variables I should...
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    Spherical Coordinates Integral

    Homework Statement Using spherical coordinates, find the volume of the solid that lies within the sphere x2+y2+z2=4, above the xy-plane and below the cone z=√(x2+y2) Homework Equations The Attempt at a Solution This is what I have so far...
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    Polar Coordinates Improper Integral Proofs

    Homework Statement (a) we define the improper integral (over the entire plane R2) I=\int\int_{R^2}e^{-(x^2+y^2)}dA=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^2+y^2)}dy dx=\lim_{a\rightarrow\infty}\int\int_{D_{a}} e^{-(x^2+y^2)} dA where Da is the disk with radius a and center the...
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    Curvature Proof

    Homework Statement show that the curvature of a plane curve is \kappa=|\frac{d\phi}{ds}| where phi is the angle between T and i; that is, phi is the inclination of the tangent line. Homework Equations The Attempt at a Solution I'm not sure how to start this one out. Any ideas?
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    Vector Application Problem

    Homework Statement A rocket burning it's onboard fuel while moving through space has a velocity v(t) and mass m(t) at time t. If the exhaust gasses escape with velocity ve relative to the rocket , it can be deduced from Newton's Second Law of Motion that...
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    Proof of r(t) and r'(t) orthogonal on a sphere

    Homework Statement if a curve has the property that the position vector r(t) is always perpendicular to the tangent vector r'(t). show that the curve lies on a sphere with center the origin. Homework Equations The Attempt at a Solution I'm not quite sure how to prove this. I...
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    Surface with specifications

    Homework Statement Find an equation for the surface consisting of all points p for which the distance from P to the x-axis is twice the distance from P to the yz-plane. Identify the surface. Homework Equations The Attempt at a Solution...
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    3 Orthogonal Vectors Problem

    Homework Statement A. Given unit vectors a, b, c in the x, y-plane such that a · b = b · c = 0, let v = a + b + c; what are the possible values of |v|? B. Repeat, except a, b, and c are unit vectors in 3-space Homework Equations The Attempt at a Solution I have solutions for both that I'm...
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    Drawing a Collection Of vectors Satisfying Cross Products

    Homework Statement Given a = <1,2,3> and b = <1,-1,-1>, sketch the collection of all position vectors c satisfying a x b = a x c Homework Equations The Attempt at a Solution I've calculated a x b = <1,4,-3> and Defining c = <x,y,z> I found a x c = <2z-3y, z-3x, y-2x>. I want to...
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    Vector Proof

    Homework Statement If c=|a|b+|b|a where a,b, and c are all non zero vectors, show that c bisects the angle between a and b Homework Equations The Attempt at a Solution I'm taking the approach to prove that the angle between b and c= the angle between c and a I have written...
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    Tricky Calculus III Problem

    Homework Statement Find the volume of the solid that lies in between both of the spheres: x2+y2+z2+4x+2y+4z+5=0 and x2+y2+z2=4 Homework Equations This is the first chapter of the calculus III material so no double or triple integrals are needed to solve this problem. The Attempt at a...
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    Improper Integral Proof

    Hi, I came across this interesting integral \int_{-\infty}^{\infty}\frac{d}{dx}(\arctan x) dx=\pi I can derive the solution analytically, but I cannot think of a geometric proof. Does anyone know geometrically why this integral is equal to pi?
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    Integration By Parts Problem

    Homework Statement Homework Equations The Attempt at a Solution I tried this problem and couldn't figure it out so I went and got the solution. However, I don't understand step 6 of the solution. I'm not sure how (n-1)\int\sin^{n-2}x(1-\sin^2x)dx=(n-1)\int\sin^{n-2}dx-(n-1)\int\sin^nx dx
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    Sketching A Graph Using Differentiation

    Homework Statement Homework Equations The Attempt at a Solution I don't understand how they get from step 4 to step 5. Wouldn't you factor a cos x out of the brackets then have (\cos x)(1-\frac{1}{6}) to the left of the brackets. Then you can multiply the (1-\frac{1}{6}) inside...
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    Finding the Intersection Of 2 Equations (difficult)

    Homework Statement Solve the following equation: e^{2x}=3x^2 Homework Equations The Attempt at a Solution I can find an approximate solution with a graphing calculator easily, but I am interested how you would find the exact solution. I can take the natural log of both sides...
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    Difference Quotient Isn't working.

    Homework Statement Find the derivative of f using the differance quotient and use the derivative of f to determine any points on the graph of f where the tangent line is horizontal. f(x)=3x^3-9x Homework Equations The Attempt at a Solution \lim_{\Delta...
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    Cancellation Limit

    Homework Statement Find the limit. \lim_{x\rightarrow0}\frac{\frac{1}{x+1}-1}{x} Homework Equations The Attempt at a Solution I have to do this analytically. Although, I know that the limit is supposed to be -1 from a graphing approach. When you substitute in 0 for x you get 0/0. How do I...
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    Polar Form of the Equation of a Conic

    Homework Statement The planets travel in an elliptical orbit with the sun as a focus. Assume that the focus is at the pole, the major axis lies on the polar axis and the length of the major axis is 2a. Show that the polar equation of orbit is given by r=\frac{(1-e^2)a}{1-e\cos\theta} here's...
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    Converting A Polar Equation to Rectangular Form; Equation of a Circle

    Homework Statement Convert the polar equation r = 2(h cos θ + k sin θ) to rectangular form and verify that it is the equation of a circle. Find the radius and the rectangular coordinates of the center of the circle. Homework Equations The Attempt at a Solution First, I...
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    Converting A Polar Equation to Rectangular Form

    Homework Statement Convert the polar equation to rectangular form. r=2sin(3θ) Homework Equations The Attempt at a Solution I can expand this out to r=2(\sin\theta\cos2\theta+\cos\theta\sin2\theta) multiply both sides by r...
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    Parametric Equations Word Problem

    Homework Statement Consider a projectile launched at a height of h feet above the ground at an angle θ with the horizontal. If the initial velocity is v0 feet per second, the path of the projectile is modled by the parametric equations x=(v[SUB]0cos θ)t and y=h + (v[SUB]0 sin θ)t-16t2. The...
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    Finding Points of Intersection by Substitution

    Homework Statement Find any points of intersection of the graphs by the method of substitution. xy+x-2y+3=0 x^2+4y^2-9=0 Homework Equations The Attempt at a Solution From the second equation I can solve for y: y=\frac{\sqrt{9-x^2}}{2} Plug it into the first equation and...
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    Graphing a Rotated Conic on a Graphing Calculator

    Homework Statement Use a graphing utility to graph the conic. Determine the angle through which the axis are rotated. x^2+xy+y^2=10 Homework Equations \cot2\theta=\frac{A-C}{B} x=x'\cos\theta-y'\sin\theta y=x'\sin\theta+y'\cos\theta The Attempt at a Solution I can find the angle of rotation...
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    Rotating a Parabola

    Homework Statement Rotate the axis to eliminate the xy-term. 3x^2-2\sqrt{3}xy+y^2+2x+2\sqrt{3}y=0 Homework Equations \cot2\theta=\frac{A-C}{B} x=x'\cos\theta-y'\sin\theta y=x'\sin\theta+y'\cos\theta The Attempt at a Solution Find the Angle of Rotation...
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    Rotating a Hyperbola

    Homework Statement Rotate the axis to eliminate the xy-term. Sketch the graph of the equation showing both sets of axis. xy-2y-4x=0 Homework Equations \cot2\theta=\frac{A-C}{B} x=x'\cos\theta-y'\sin\theta y=x'\sin\theta+y'\cos\theta The Attempt at a Solution xy-2y-4x=0 First I find the...
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    Equation of A Hyperbola

    Homework Statement Find an equation of the hyperbola with it's center at the origin. Foci:(8,0),(-8,0) Asymptotes: y=4x, y=-4x Homework Equations Equation for the asymptotes of a hyperbola with a horizontal transverse axis y=k\pm\frac{b}{a}(x-h) Equation for a hyperbola centered at (0,0) and...
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    Approximating ∏ With a Coin!

    Homework Statement Homework Equations The Attempt at a Solution I found this interesting problem in my textbook. I have heard of Buffon's needle experiment which estimates pi through probability. This problem came from a pre-calc book out of the probability section. The problem seams...
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