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    Proof of Euler's Theorem

    Homework Statement If the divisor P is a prime of the form 4q+1 then the number -1 or P-1 is certainly a residue. Homework Equations The Attempt at a Solution First the book told me to prove that 1 and -1 are the only two remainders that are their own reciprocals modP...
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    Output impedance of a Pass Transistor

    Homework Statement Calculate the output impedance of the emitter-follower circuit called a pass transistor. Assume that beta=200 **See attached diagram** Homework Equations The Attempt at a Solution Not really sure how this works, I thought it would just be 1k cause that is the...
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    Npn transistor with a diode

    Homework Statement The absolute max voltage rating for the base-emitter voltage is the lowest of all values. This low value means the B-E junction must be protected from a high reverse voltage. A diode is used for an npn transistor. Explain its function and conditions under which...
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    Charging a battery circuit

    Homework Statement Make a modification so that the battery is charged by the DC supply at a current of 10mA **See attached diagram** Homework Equations The Attempt at a Solution Well I know that I would put in a resistor in parallel with the diode. However, I don't know how...
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    Circuits with Diodes

    Homework Statement sketch the output waveforms when a 5Vp sine wave with a frequency of 100 Hz is applied to each of the following circuits **see attached diagrams** Homework Equations The Attempt at a Solution
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    Circuit with a zener diode

    Homework Statement What would happen if the load resistor was short-circuited. What would happen if the load resister was removed? -see attached diagram- Homework Equations P=Iin^2*R Pz=IzVz =(Iin-Iout)Vz=[((Vin-Vz)/R)-Iout]Vz The Attempt at a Solution I'm not sure about...
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    Current from a sine wave signal

    Homework Statement Given a sine wave signal (20V PP) connected to a 10 kilo-Ohm resistor. Calculate the current through the resistor in terms of average, rms, and peak currents. What wattage rating should the resistor have? Homework Equations Vrms=(1/Sqrt(2))(Vp) V=IR The Attempt...
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    RL to RC circuit

    Homework Statement Design a RC circuit so it has the same V vs time characteristics *See attached diagram* Homework Equations The Attempt at a Solution I really have no idea how to approach this, but I do know, RC circuit: Vc(t)=Vb(1-e^-(t/RC)) RL circuit...
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    RL to RC circuit

    Homework Statement Design an RC circuit that has the same Voltage vs. Time characteristics as the RL circuit shown *See attached Diagram* Homework Equations The Attempt at a Solution Not really sure how to approach this, but all i know is the equations: RC circuit...
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    Graph of RC circuit

    Homework Statement Sketch a graph of Vout vs time after the switch is closed *See attached diagram* Homework Equations The Attempt at a Solution I am not really sure how to do this, because I don't know if it makes a difference whether or not the resistor is before the...
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    Current in multi-battery circuits

    Homework Statement Find the current through R2 (see attached figure) Homework Equations The Attempt at a Solution Not really sure if I did this right, would like if someone would double check my work: First I tried Using Kirchhoff equations for a node and I got: I1-I2+I3=0 Next I broke...
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    Internal Resistance of a Battery problem

    Homework Statement A voltmeter whose resistance is 1000 ohms measures the voltage of a worn out 1.5V battery as .9V. What is the internal Resistance Homework Equations The Attempt at a Solution In order to do this, I Solved for the Equiv Resistance of the circuit (leaving in...
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    Current in Open Circuits

    Homework Statement Looking at the open circuit demonstrated in figure 2 (see attachment), what is the current through the ammeter? What is the voltage at point p? Homework Equations The Attempt at a Solution I am assuming if it an open circuit, I would think that the current...
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    Voltage Divider

    Homework Statement How do you measure the output of a voltage divider with real voltmeters that have effective resistances Homework Equations The Attempt at a Solution I know that for an ideal voltmeter (one with infinite resistance), the voltage divider is R2/(R1+R2), but I...
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    Diffraction Limited Object Size

    Homework Statement What is the diffraction-limited object size (at 25 cm) imposed by the numerical aperture of the eye (if the eye is a diffraction-limited optic)? Use 4mm for the iris diameter and 550 nm for the wavelength. Homework Equations Rayleigh criterion states: for angular...
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    Prisms and Images

    Homework Statement (2) For the combination of one prism and 2 lenses shown (Fig. 1), find the location and size of the final image when the object, length 1 cm, is located as shown in the figure. [Hint: Treat the prism as a mirror, but you have to take into account the image shift caused by...
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    Polarization and Fermat's principle

    Homework Statement a) A beam of light in the air strikes the surface of a smooth piece of plastic having an index of refraction of 1.55 at an angle with the normal of 60 degrees. The incident light has equal component E-field amplitudes parallel and perpendicular to the plane of incidence...
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    Hyperbolic geometry

    Homework Statement How to show that there exists a triangle whose defect is greater than 14 degrees Homework Equations The Attempt at a Solution No idea what to do here.... something about the angle of parallelism
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    Proof of Scalene Triangle

    Homework Statement Prove that a scalene triangle exists Homework Equations The Attempt at a Solution I really don't know how to approach this proof.... should I do a direct proof or maybe by contradiction? Any help would be great...
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    Simple Geometry Proof (Betweenness Axiom)

    Homework Statement Given a line l, a point A on l, and a point B not on l. Then every point of the ray AB (except A) lies on the same side of l as B. Homework Equations The Attempt at a Solution I understand why this is true, however I'm having some trouble wording my proof...
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    Differentiable functions proof

    Homework Statement Consider a function g : (a, b)-->R. Assume that g is differentiable at some point c in (a,b) and that g'(c) is not = 0. Show that there is a delta > 0 so that g(x) is unequal to g(c) for all x in V_delta(c)\{c}intersect(a,b) Homework Equations The Attempt at a...
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    Analysis derivative proof

    Homework Statement Let f : (a, b)--> R be differentiable on (a,b), and assume that f'(x) unequal 1 for all x in (a,b). Show that there is at most one point c in (a,b) satsifying f(c) = c. Homework Equations The Attempt at a Solution I think that We need to use the mean value...
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    Functions and the Intermediate Value theorem

    Homework Statement Let f : [0; 1] --> R be continuous on [0, 1], and assume that the range of f is contained in [0; 1]. Prove that there exists an x in [0, 1] satisfying f(x) = x. Homework Equations The Attempt at a Solution Well i am almost positive I need to use the...
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    Planar graph induction proof

    Homework Statement Let G be a planar graph with n vertices, q edges, and k connected components. If there are r regions in a planar representation of G, prove that: n − q + r = 1 + k Hint: Use induction on k. The base case is Euler’s formula. Homework Equations The Attempt at a...
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    Graph theory line graph proof

    Homework Statement Let G be a non-empty graph of order n whose vertices have degrees d1, . . . , dn. The line graph of G is defined as follows: the vertices of L(G) are the edges of G, and two vertices of L(G) are adjacent if they share an endpoint in G. Prove that the size of L(G) is...
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    Continuous function and limits

    Homework Statement Consider a function f : R--> R, and assume that there is a c is in (0, 1) so that |f(x) - f(y)|<= c|x -y| for all x, y in R. (a) Show that f is continuous on R. (b) Given a point y1 in R de fine a sequence by yn+1 = f(yn). Prove that yn is a Cauchy sequence (and...
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    Isolated points and continuity

    Homework Statement Let f : A --> R be a function, and let c in A be an isolated point of A. Prove that f is continuous at c Homework Equations The Attempt at a Solution I'm kind of confused by this problem.... if c is an isolated point, then the limit doesn't exist. So I can't...
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    Limit proof (using delta-epsilon definition)

    Homework Statement Let f, g : A \rightarrow R be functions and let c \in R be a limit point of A. Assume that lim(x-> c): f(x) = L and lim(x->c): g(x) = M. Use the epsilon-delta definition to prove: lim(x->c) f(x)g(x)=LM Homework Equations The Attempt at a Solution alright in...
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    Compact sets and sup/inf

    Homework Statement Show if K contained in R is compact, then supK and inf K both exist and are elements of K. Homework Equations The Attempt at a Solution Ok we proved a theorem stating that if K is compact that means it is bounded and closed. So if K is bounded that means...
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    Open and Closed sets

    Homework Statement Let E be a nonempty subset of R, and assume that E is both open and closed. Since E is nonempty there is an element a \in E. De note the set Na(E) = {x > 0|(a-x, a+x) \subsetE} (a) Explain why Na(E) is nonempty. (b) Prove that if x \in Na(E) then [a-x, a+x] \subset...
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