Question

Consider the function f defined on R by f(x) = ?0 if x ≤ 0, f(x) = e^(−1/x^2) if x > 0.

Prove that f is indefinitely differentiable on R, and that f(n)(0) = 0 for all n ≥ 1. Conclude that f does not have a converging power series expansion En=0 to ∞[an*x^n] for x near the origin. [Note: This problem illustrates an enormous difference between the notions of real-differentiability and complex-differentiability.]

Answer #1

Consider the function f : R → R defined by f(x) = ( 5 + sin x if
x < 0, x + cos x + 4 if x ≥ 0. Show that the function f is
differentiable for all x ∈ R. Compute the derivative f' . Show that
f ' is continuous at x = 0. Show that f ' is not differentiable at
x = 0. (In this question you may assume that all polynomial and
trigonometric...

Prove that the function f : R \ {−1} → R defined by f(x) = (1−x)
/(1+x) is uniformly continuous on (0, ∞) but not uniformly
continuous on (−1, 1).

Let f : R → R be defined by f(x) = x^3 + 3x, for all x. (i)
Prove that if y > 0, then there is a solution x to the equation
f(x) = y, for some x > 0. Conclude that f(R) = R. (ii) Prove
that the function f : R → R is strictly monotone. (iii) By
(i)–(ii), denote the inverse function (f ^−1)' : R → R. Explain why
the derivative of the inverse function,...

Question 4: The function f : {0,1,2,...} → R is defined byf(0) =
7,
f(n) = 5·f(n−1)+12n2 −30n+15 ifn≥1.• Prove that for every
integer n ≥ 0,
f(n)=7·5n −3n2.
Question 5: Consider the following recursive algorithm, which takes
as input an integer
n ≥ 1 that is a power of two:
Algorithm Mystery(n):
if n = 1
then return 1
else x = Mystery(n/2);
return n + xendif
• Determine the output of algorithm Mystery(n) as a function of
n....

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?

Calculate differentiability of f(x,y,z) = x^2 + y^2 + z^2
this function is defined in R^2

Prove or give a counterexample: If f is continuous on R and
differentiable on R∖{0} with limx→0 f′(x) = L, then f is
differentiable on R.

A table of values is given for a function f(x, y) defined on R =
[1, 3] × [0, 4].
0
1
2
3
4
1.0
2
0
-3
-6
-5
1.5
3
1
-4
-5
-6
2.0
4
3
0
-5
-8
2.5
5
4
3
-1
-4
3.0
7
8
6
3
0
Estimate f(x, y) dA R using the Midpoint Rule with m = n = 2 and
estimate the double integral with m = n =...

Let f(x, y) = x^3 − 4xy^2 , x, y ∈ R. Use the definition of
differentiability to show that f(x, y) is differentiable at (2,
1).

If f is a continuous, positive function defined on the interval
(0, 1] such that limx→0+ = ∞ we have seen how to make sense of the
area of the infinite region bounded by the graph of f, the x-axis
and the vertical lines x = 0 and x = 1 with the definition of the
improper integral.
Consider the function f(x) = x sin(1/x) defined on (0, 1] and
note that f is not defined at 0.
• Would...

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