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...
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...
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...
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...
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...
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...
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?
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...
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...
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...
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...
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...
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...
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...
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?
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
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...