Question

Find a function f(x) satisfying f ' '(x) = sqrt{x}, f
' (1)=0 and f(1)=0. You must explain in words what you are doing so
that someone who knows what antidifferentiation is but otherwise
does not have experience can follow your computations. Give your
final answer in what seems to you to be the simplest form.
**Use antidifferentiation, not integration.**

Answer #1

Find a function f(x) satisfying f ' '(x) = sqrt{x}, f
' (1)=0 and f(1)=0. You must explain in words what you are doing so
that someone who knows what antidifferentiation is but otherwise
does not have experience and can follow your computations. Give
your final answer in what seems to you to be the simplest form.

Find the value of C > 0 such that the function
?C sin2x, if0≤x≤π,
f(x) =
0, otherwise
is a probability density function.
Hint: Remember that sin2 x = 12 (1 − cos 2x).
2. Suppose that a continuous random variable X has probability
density function given by the above f(x), where C > 0 is the
value you computed in the previous exercise. Compute E(X).
Hint: Use integration by parts!
3. Compute E(cos(X)).
Hint: Use integration by substitution!

Let f(x,y)=1 for 0<x<1, 0<y<1 and 0 otherwise. Find
the probability density function of Z=max(X, Y)

Let f(x,y)=1 for 0<x<1, 0<y<1 and 0 otherwise. Find
the probability density function of Z=max(X, Y)

Let the function f and g be defined as f(x) = x/ x − 1 and g(x)
= 2 /x +1 . Compute the sum (f + g)(x) and the quotient (f/g)(x) in
simplest form and describe their domains.
(f + g )(x) =
Domain of (f+g)(x):
(f/g)(x) =
Domain of (f/g)(x):

Given function f(x) = c*|x| on [-3,3] and f(x) = 0 otherwise.
Find c and the corresponding CDF

Problem 1: Find a non-zero function f so that f'(x) = 3f(x).
The exponential
function f(x) = e^x has the property that it is its own
derivative.
(d/dx) e^x = e^x
f(x) = e^(3x)
f '(x) = e^(3x) *
(d/dx) 3x
= e^(3x) * 3
= 3e^(3x)
= 3 f(x)
Problem 2: Let f be your function from Problem 1. Show
that if g is another function satisfying g'(x) = 3g(x), then g(x) =
A f(x) for some constant A????

Find a function such that
f '(x)=x+1
and f(0)=6. Then f(x)=

Find the inverse function of f(x)= 1+2cos(x) and then find the
domain and the range of the function of both f and f^-1??
Find the inverse function of f(x)=sqrt(3-e^2x) and then find the
domain and the range of the function of both f and f^-1??

Let f: [0, 1] --> R be defined by f(x) := x. Show that f is
in Riemann integration interval [0, 1] and compute the integral
from 0 to 1 of the function f using both the definition of the
integral and Riemann (Darboux) sums.

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