Question

Calculus Fill in the blanks!

Suppose f(x) is a piecewise function:f(x) = x + 3, if x ≤ 2
and

f(x) = x

2

, if x > 2. Then f(2)= ——— = ——.And limx→2− f(x)=

—————————– = —————————- = ———————- =—.

Similarly, limx→2+ f(x)= —————————– = ————————- =

———————- =—————-.

Therefore, limx→2 f(x) —- f(2).

Answer #1

Calculus Fill in the blanks!
Suppose f(x) is a piecewise function:f(x) = x + 3, if x ≤ 2
and
f(x) = x
2
, if x > 2. Then f(2)= ——— = ——.And limx→2− f(x)=
—————————– = —————————- = ____________________
=___________________.
Similarly, limx→2+ f(x)= —————————– = ————————- =
———————- =—————-.
Therefore, limx→2 f(x) —- f(2).

Calculus, Taylor series Consider the function f(x) = sin(x) x .
1. Compute limx→0 f(x) using l’Hˆopital’s rule. 2. Use Taylor’s
remainder theorem to get the same result: (a) Write down P1(x), the
first-order Taylor polynomial for sin(x) centered at a = 0. (b)
Write down an upper bound on the absolute value of the remainder
R1(x) = sin(x) − P1(x), using your knowledge about the derivatives
of sin(x). (c) Express f(x) as f(x) = P1(x) x + R1(x) x...

3. For each of the piecewise-defined functions f, (i) determine
whether f is 1-1; (ii) determine whether f is onto. Prove your
answers.
(a) f : R → R by f(x) = x^2 if x ≥ 0, 2x if x < 0.
(b) f : Z → Z by f(n) = n + 1 if n is even, 2n if n is odd.

Fill in the blank with “all,” “no,” or “some” to make the
following statements true. • If your answer is “all,” explain why.
• If your answer is “no,” give an example and explain. • If your
answer is “some,” give two examples, one for which the statement is
true and the other for which the statement is false. Explain your
examples.
1. For functions g, if lim x→a+ g(x) = 2 and lim x→a− g(x) = −2,
then limx→a...

Consider the piecewise defined function
f(x) = xa− xb if 0<x<1. and
f(x) = lnxc if x≥1.
where a, b, c are positive numbers chosen in such a way that
f(x) is differentiable for all 0<x<∞. What can be said about
a, b, and c?

Let f(x) be defined below.
f(x) = { 1 if x < 1 x 2 if 1 < x ≤ 2 x + 1 if x > 2
Find limx→1+ f(x), limx→1− f(x), limx→1f(x), f(1) and
limx→2f(x)

In calculus the curvature of a curve that is defined by a
function
y = f(x)
is defined as
κ =
y''
[1 + (y')2]3/2
.
Find
y = f(x)
for which
κ = 1.

fourier expansion, piecewise function.
f(x){ pi , -1<x<0
-pi , 0<x<1

Fill in the blanks to correctly complete the sentence below.
Suppose a simple random sample of size n is drawn from a large
population with mean μ and standard deviation σ.
The sampling distribution of x has mean μx=______ and standard
deviation σx=______.

3. Fill in the blanks.
The two prokaryotic domains of life, the
________________________ and the _______________________, do
not
have nuclei.
The organisms that produced most of the oxygen in the atmosphere
of the early earth are called
______________________.
The first person to microscopically observe and describe
bacteria was _________________________________.
4.Fill in the blanks.
When __________________ carry out respiration, they transfer
_________________ from organic compounds to
terminal electron acceptors.
Phototrophs use energy from _____________________ to move
electrons and generate a ___________________...

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