Question

2. Show that each of the following functions is O(x^2 ). Clearly state your C and k and show that |f(x)| ≤ C|x^2 | for x > k.

a. f(x) = 17x + 11

b. f(x) = x^2 + 1000

c. f(x) = floor function[x] · ceiling function[x].

Answer #1

For f(x) = x^2+6 and g(x) = x^2-5 find the following
functions.
a.) (f o g)(x)
b.) (g o f) (x)
c.) (f o g) (4)
d.) (g o f) (4)

Find each of the following functions. f(x) = 4 − 4x, g(x) =
cos(x)
(a) f ∘ g and State the domain of the function. (Enter your
answer using interval notation.)
(b) g ∘ f and State the domain of the function. (Enter your
answer using interval notation.)
(c) f ∘ f and State the domain of the function. (Enter your
answer using interval notation.)
(d) g ∘ g and State the domain of the function. (Enter your
answer using...

Find the big-O, big-Omega of the following functions (show steps
please)
a) f(n) = 5n^2 + 1
b) f(n)= (nlogn+1)*(n+1)

Consider the following functions
f(x) =x^2, g(x) = lnx, h(x) = cosx
For each of the following parts, you may use compositions,
products, and sums of thefunctions above, but no others. For
example, we can combine in the following waysh(g(x)) = cos(lnx), or
g(x)h(x) = lnxcosx, or g(x) +h(x) = lnx+ cosx
show how derivative rules apply to the function you came up
within order to produce the requested derivative.
1)A functionk(x) whose derivative is k′(x) = −tanx=
-(sinx/cosx)
2)...

Find the derivatives of each of the following functions. DO NOT
simplify your answers.
(a) f(x) = 103x (3x5+ x − 1)4
(b) g(x) = ln(x3 + x) /
x2 − 4
(c) h(x) = tan-1(xex)
(d) k(x) = sin(x)cos(x)

For each of the following pairs of functions f and g (both of
which map the naturals N to the reals R), state whether f is O(g),
Ω(g), Θ(g) or “none of the above.” Prove your answer is correct. 1.
f(x) = 2 √ log n and g(x) = √ n. 2. f(x) = cos(x) and g(x) =
tan(x), where x is in degrees. 3. f(x) = log(x!) and g(x) = x log
x.

For each of the following pairs of functions f and g (both of
which map the naturals
N to the reals R), show that f is neither O(g) nor Ω(g). Prove
your answer is correct.
1. f(x) = cos(x) and g(x) = tan(x), where x is in
degrees.

QUESTION 2 Which of the following functions is not acceptable as
a state function (wavefunction)?
A)tanx (0, ∞)
B)sin x (0, ∞)
c)cos x (0, ∞)
D)e^-x (0, ∞)

The functions f(x) = –(x + 4)^2 + 2 and g(x) = (x − 2)^2 − 2
have been rewritten using the completing-the-square method. Is the
vertex for each function a minimum or a maximum? Explain your
reasoning for each function.

Find the derivatives of the following functions
(simplify when appropriate). Please neatly show
your work.
a. f(x)= xsin^2(x)
b. f(x)=tanh(2x)+cosh(x)+4sinh(x)
c. f(x)= tan(2x)+cos(x)+4sin(x)
d. f(x)= 2√(x)-x^(2/3). Find f'(1).

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