1.
a) Use the Chain Rule to calculate the partial derivatives.
Express the answer in terms...
1.
a) Use the Chain Rule to calculate the partial derivatives.
Express the answer in terms of the independent variables.
∂f
∂r
∂f
∂t
; f(x, y, z) = xy +
z2, x = r + s −
2t, y = 6rt, z =
s2
∂f
∂r
=
∂f
∂t
=
b) Use the Chain Rule to calculate the partial derivative.
Express the answer in terms of the independent variables.
∂F
∂y
; F(u, v) =
eu+v, u =
x5, v = 2xy
∂F
∂y
=
c)...
Derive an expression for β =
1/V(∂V/∂T)P for such a gas in
terms of b(T) ,...
Derive an expression for β =
1/V(∂V/∂T)P for such a gas in
terms of b(T) ,
db(T)/dT , P
, and Vm .
Express your answer in terms of the variables
b(T) ,
db(T)/dT , P
, and Vm .
Express the following in terms of αp, βT, Cv, Cp, S, V, T,
P.
∂G/∂P at...
Express the following in terms of αp, βT, Cv, Cp, S, V, T,
P.
∂G/∂P at constant S.
a.The derivative (∂V/∂T)P is tedious to calculate by
implicit differentiation of an equation of state such...
a.The derivative (∂V/∂T)P is tedious to calculate by
implicit differentiation of an equation of state such as
the Peng-Robinson equation. Show that calculus
permits us to find the derivative in terms of derivatives
of pressure, which are easy to find, and provide the
formula for this equation of state.
b. Using the Peng-Robinson equation, calculate the
isothermal compressibility of ethylene for saturated
vapor and liquid at the following conditions: {Tr = 0.7,
P = 0.414 MPa}; {Tr = 0.8, P...
2. Let A = {p, q, r, s}, B = {k, l, m, n}, and C...
2. Let A = {p, q, r, s}, B = {k, l, m, n}, and C = {u, v, w},
Define f : A→B by f(p) = m, f(q) = k, f(r) = l, and f(s) = n, and
define g : B→C by g(k) = v, g(l) = w, g(m) = u, and g(n) = w. Also
define h : A→C by h = g ◦ f. (a) Write out the values of h. (b) Why
is it that...
Suppose ?(?,?)=??f(x,y)=xy, ?=(−4,−4)P=(−4,−4) and
?=3?+2?v=3i+2j.
A. Find the gradient of f.
∇?=∇f= ?+i+ ?j
Note: Your answers should...
Suppose ?(?,?)=??f(x,y)=xy, ?=(−4,−4)P=(−4,−4) and
?=3?+2?v=3i+2j.
A. Find the gradient of f.
∇?=∇f= ?+i+ ?j
Note: Your answers should be expressions of x and y; e.g. "3x -
4y"
B. Find the gradient of f at the point P.
(∇?)(?)=(∇f)(P)= ?+i+ ?j
Note: Your answers should be numbers
C. Find the directional derivative of f at P in the direction of
?v.
???=Duf=
D. Find the maximum rate of change of f at P.
E. Find the (unit) direction vector in which the maximum rate...
Below are ten functions. Find the first derivative of each. All
of the derivatives can...
Below are ten functions. Find the first derivative of each. All
of the derivatives can be found by using combinations of the
constant rule, power function rule and sum-difference
rule. Do not use the product rule. It is not
needed. The degree of difficulty (more or less) increases
from (a) to (j). Be sure to show intermediate work. Five points are
awarded for correctly developed derivatives. There is no partial
credit, because an incorrect derivative is useless.
For example,...
Suppose that an economy is characterized by M = $10 trillion, V
= 2, P =...
Suppose that an economy is characterized by M = $10 trillion, V
= 2, P = base index = 1.0
Instructions: Enter your responses rounded to two decimal places
(do not include a negative sign (-) with your answers).
a. What is the real value of output (Q)?
Now assume that the Fed increases the money supply by 20 percent
and velocity remains unchanged.
b. If the price level remains constant, by how much will real
output increase?
c. If,...