Homework Statement
Given the distance from the sun to Earth xE, the distance from the sun to venus xV and the mass of the sun m calculate the time delay due to gravity in receiving a radio signal sent from Earth to venus (and reflected back). Ignore the gravity fields of venus and earth. Venus...
Homework Statement
proof that shortest path between two points on a sphere is a great circle.
Homework Equations
Euler-Lagrange and variational calculus
The Attempt at a Solution
in sphereical coords:
N.B. \dot{\phi} = \frac{d\phi}{d\theta}
ds = \sqrt{r^{2}d\theta^{2} +r^{2}sin^{2}\theta...
Homework Statement
Hi, we're supposed to put the following integral into a power series:
\int \frac{arctan(t)}{t} dt
with 0 < t < x.
Homework Equations
n/a
The Attempt at a Solution
I just want to know whether this step is ok.
\int \frac{arctan(t)}{t} dt = \int \frac{1}{t} \int...
there is buoy bobbing gently on the open sea. we need to find
1) average density
2) natural frequency
3) amplitude when the buoy is struck by a wave 4m high (peak to trough) with period 5sec.
I've worked out parts 1 and 2:
\rho_B = \rho_w \frac{h}{H} (h = height submerged, H= total height)...
thanks, so you think I should start by working from \frac{dv}{dt} = -g -\frac{\gamma}{m}v ?
I have already done that, but the question is quite clear it asks for the diff. eq. in terms of \dot{z}(t) and then asks the student to make substitution. And it also asks for type of differential...
Homework Statement
A point mass m falls from rest through a height h. The frictional force is given by -\gamma \dot{z} and gravity by -mg.
Give the 'equation of motion' (differential equation) for the height z(t).The Attempt at a Solution
\ddot{z} = -g - \frac{\gamma}{m} \dot{z}
I thought...
Homework Statement
f: ]0, \infty[ \rightarrow \mathbb{R} is defined as
f = 0 if x is irrational and
f = \frac{1}{n}
if x = \frac{m}{n}
where m and n are co-prime
Show that f is only then continuous about x0 when x \in \frac{\mathbb{R}}{\mathbb{Q}}
The Attempt at a Solution
with x, y...
Homework Statement
the title says it all
x \rightarrow 4
for f\left(x\right) = \frac{\sqrt{1+2x}-3}{\sqrt{x}-2}
I have multiplied both top and bottom by conjugate, \sqrt{x}+2:
f\left(x\right) = \frac{\sqrt{x(1+2x)}+2\sqrt{1+2x} -3\sqrt(x)-6}{x-4}but don't know how to take this further...
Hi, thanks for the reply :smile:
Looking more at the given function i can show that it is indeed one-to-one. As to whether it is surjective I'm not really sure what my answer is telling me:
f\left(x\right) = \frac{ax+b}{cx+d} = \xi
ax+b = cx\xi+d\xi
x = \frac{d\xi - b}{a - c\xi}...
ah, i just found out about the bijective requirements for inverse functions – sorry we didn't cover this in class.
but I'm a bit confused injection and surjection seem quite the same. Isn't it enough to show that the function is one-to-one??
Homework Statement
Given the function
f(x) = \frac{ax+b}{cx+d}
where f: \mathbb{R} \backslash \left\{ \frac{-d}{c} \right\} \rightarrow \mathbb{R}
show that f is either a constant or has an inverse function.
I can see why this would be true. If a function takes all real numbers and returns...
Homework Statement
A power station has a huge tank to store water. The tank is filled by a pump which is attached horizontally to the bottom edge (the bottom of one of the walls) of the tank. In the floor of the tank is an outflow valve. The valve is subject to huge forces and after some years...
Homework Statement
Water flows at a rate of 30ml/s from an opening in the bottom of a 4m high pressure tank (a tank with a plunger type lid). Calculate the flow rate when an extra 50 kPa of pressure is applied.
Homework Equations
Bernoulli's equation and \frac{V}{t} = AvThe Attempt at a...