Let a < c < b, and let f be defined on [a,b]. Show that f
∈ R[a,b] if and only if f ∈ R[a, c] and f ∈ R[c, b]. Moreover,
Integral a,b f = integral a,c f + integral c,b f .
Let F(x) = log5x and H(x) = x . Find the value of
F(H(25)).
Write the equation of a circle whose center is located at (-2,
-5) with a radius of 11 inches.
Given a triangle with Angle A = 45 degrees, Angle B = 60
degrees and side a = 10 inches.
Find the length of side b. (Leave answer in radical
form.)
Find the distance between a set of points whose coordinates
are A(-7, 8) and B(5, -11)....
a)Find the value or values of c that satisfy the equation
f(b)-f(a)/b-a = f′(c)in the conclusion of the Mean Value Theorem
for the function and interval. Round to the nearest thousandth.
f(x) ln(x-3), [4,7]
b)Suppose that
c(x)=3x^3-40x^2+6844x is the cost of manufacturing x items.
Find a production level that will minimize the average cost of
making x items.
c)A rectangular field is to be enclosed on four sides with a
fence. Fencing costs $6 per foot for two opposite sides,...
Let f(t) =t^2−1 and g(t) =e^t.
(a) Graph f(g(t)) and g(f(t)).
(b) Which is larger,f(g(5)) or g(f(5))? Justify your answer.
(c) Which is larger, (f(g(5)))′or g(f(5))′? Justify your
answer.
Problem 5. The operator T : H → H is an isometry if ||T f|| =
||f|| for all f ∈ H.
(a) Please, prove that if T is an isometry then (T f, T g) = (f,
g) for all f, g ∈ H.
(b) Now prove that if T is an isometry then T∗T =
I.
(c) Now prove that if T is surjective and isometry (and thus
unitary) then T T∗ = I.
(d) Give an example...
Prove
1. Let f : A→ B and g : B → C . If g 。 f is one-to-one, then f
is one-to-one.
2. Equivalence of sets is an equivalence relation (you may use
other theorems without stating them for this one).
Let
t= 20389208 mod 4 and M= t+25
a. Find integers a and b such that 0<a<M, 0<b<M
and ab= 0 (mod M)
b. Find integers a and b such that 0<a<M, 0<b<M
and ab= 1 (mod M)
Thank you in advance!