Starting from the point (x0, y0) = (2,3) of the function f (x) =
4x ^ 2 - 4xy + 2y ^ 2, it is to choose to proceed according to the
Steep Descent Method (x2, y2).
Use Lagrange multipliers to show that the shortest distance from
the point (x0,y0,z0)
to the plane ax+by+cz=d
Is it the perpendicular distance?
(Remind yourself of the equation of the line perpendicular to the plane.)
How to solve it
Do 5 steps, starting from x0=[1 1 1]T and
using 6S in the computation. Hint. Make sure that you solve each
equation for the variable that has the largest coefficient (why?).
Show the details
10x1 + x2 + x3 = 6
x1 + 10x2 + x3 = 6
x1 + x2 + 10x3 = 6
Starting from the first and second law of thermodynamics, derive
the fundamental equation for A in its natural variables. a) Derive
the Maxwell relation that s related to this equation. b) Show that
for df= gdx + hdy we have an exact equation if:
6. “Starting from an initial position of equilibrium with X – M
=0, an economy experiences income growth. If this growth is due to
an autonomous increase in investment, then the balance of trade
will move into a deficit, while if it is due to an exogenous
increase in exports then the balance of payments will display a
surplus, although the surplus will be less than the initial
increase in exports.”
A batted baseball is hit with a velocity of 49.7 m/s, starting
from an initial height of 4 m. Find how high the ball travels in
two cases: (a) a ball hit directly upward (b) a ball hit at an
angle of 70 degrees. also find how long the ball is in the air in
both cases
In the shot put, a heavy lead weight—the "shot"—is given an
initial velocity, starting from an initial elevation approximately
equal to the shot putter's height, say, 2.00 m. If v0 = 7.05 m/s,
find the horizontal distance traveled by the shot for the following
initial angles above the horizontal.
theta 0 =?
theta 40.0 = ?
theta 45.0 = ?
A batted baseball is hit with a velocity of 44 m/s, starting
from an initial height of 4 m. Find how high the ball travels in
two cases:
(a) a ball hit directly upward and
(b) a ball hit at an angle of 69° with respect to the
horizontal.
Also, find how long the ball stays in the air in each case.
case a:
case b:
In the shot put, a heavy lead weight—the "shot"—is given an
initial velocity, starting from an initial elevation approximately
equal to the shot putter's height, say, 2.00 m. If
v0 = 7.90 m/s, find the horizontal distance
traveled by the shot for the following initial angles above the
horizontal. (a) θ0 = 0°
(b) θ0 = 40.0°
(c) θ0 = 45.0°
In the shot put, a heavy lead weight—the "shot"—is given an
initial velocity, starting from an initial elevation approximately
equal to the shot putter's height, say, 1.96 m. If
v0 = 8.70 m/s, find the horizontal distance
traveled by the shot for the following initial angles above the
horizontal.
(a) θ0 = 0°
(b) θ0 = 40.0°
(c) θ0 = 45.0°