find dy/dx by using implicit function theorem.
(a) x=lny
(b) x(x+y)=1
(c)ay^2+by+c=x (a, b, c are...
find dy/dx by using implicit function theorem.
(a) x=lny
(b) x(x+y)=1
(c)ay^2+by+c=x (a, b, c are constants)
when z=f(x,y), where tan(xyz)=x+y+z, find az/ax and az/ay
when z=f(x,y), where tan(xyz)=x+y+z, find az/ax and az/ay
2. (a) Let a,b ≥0. Define the function f by f(x)=(a+b+x)/3-∛abx
Determine if f has a...
2. (a) Let a,b ≥0. Define the function f by f(x)=(a+b+x)/3-∛abx
Determine if f has a global minimum on [0,∞), and if it does, find
the point at which the global minimum occurs.
(b) Show that for every a,b,c≥0 we have (a+b+c)/3≥∛abc with
equality holding if and only if a=b=c. (Hint: Use Part (a).)
Let t and s be transformations of the plane such that t(x, y) =
(x+a, y+b)...
Let t and s be transformations of the plane such that t(x, y) =
(x+a, y+b) and s(x, y) = (x+c, y+d)
where a, b, c, and d are Real numbers. Let ?(?1, ?1) and ?(?2, ?2)
be any two points in the
plane. Show that (s ◦ t)(x, y) is an isometry.
1. Consider the relations R = {(x,y),(y,z),(z,x)} and S =
{(y,x),(z,y),(x,z)} on {x, y, z}. a)...
1. Consider the relations R = {(x,y),(y,z),(z,x)} and S =
{(y,x),(z,y),(x,z)} on {x, y, z}. a) Explain why R is not an
equivalence relation. b) Explain why S is not an equivalence
relation. c) Find S ◦ R. d) Show that S ◦ R is an equivalence
relation. e) What are the equivalence classes of S ◦ R?
Suppose you have two numbers x, y ∈ Q (that is, x and y are
rational...
Suppose you have two numbers x, y ∈ Q (that is, x and y are
rational numbers).
a) Use the formal definition of a rational number to express
each of x and y as the ratio of two integers. Remember, x and y
could be different numbers!
(b) Use your results from (a) to show that xy must also be a
rational number. Carefully justify your answer, showing how it
satisfies the formal definition of a rational number.
(2) Suppose...
q.1.(a)
The following system of linear equations has an infinite number
of solutions
x+y−25 z=3
x−5 ...
q.1.(a)
The following system of linear equations has an infinite number
of solutions
x+y−25 z=3
x−5 y+165 z=0
4 x−14 y+465 z=3
Solve the system and find the solution in the form
x(vector)=ta(vector)+b(vector)→, where t is a free
parameter.
When you write your solution below, however, only write the part
a(vector=⎡⎣⎢ax ay az⎤⎦⎥ as a unit column vector with the
first coordinate positive. Write the results accurate to
the 3rd decimal place.
ax =
ay =
az =
Let S denote the set of all possible finite binary strings, i.e.
strings of finite length...
Let S denote the set of all possible finite binary strings, i.e.
strings of finite length made up of only 0s and 1s, and no other
characters. E.g., 010100100001 is a finite binary string but
100ff101 is not because it contains characters other than 0, 1.
a. Give an informal proof arguing why this set should be
countable. Even though the language of your proof can be informal,
it must clearly explain the reasons why you think the set should...
Given the function f (x, y) = ax^2
2 + 2xy + ay.y
2-ax-ay. Take
for...
Given the function f (x, y) = ax^2
2 + 2xy + ay.y
2-ax-ay. Take
for a an integer value that is either greater than 1 or less
than -1, and
then determine the critical point of this function. Then
indicate whether it is
is a local maximum, a local minimum or a saddle point.
Given the function f (x, y) = ax^2 +2 + 2xy + ay^2-2-ax-ay.
Take
for a an integer value that is either greater than 1...