Engineering Math

Let f
(t)
=
( ) t,t2 + 1,tt+1
and let g
(t)
=
( ) t+ 1,1,t2t+1-
. Find f ⋅ g.
Let f,g be given in the previous problem. Find f × g.
Let f
(t)
=
( ) t,t2,t3
,g
(t)
=
( ) 1,t2,t2
, and h
(t)
=
(sin t,t,1)
. Find the time rate of change of the box product of the vectors f,g, and h.
Let f
(t)
=
(t,sin t)
. Show f is continuous at every point t.
Suppose
|f (x)− f (y)|
≤ K
|x − y |
where K is a constant. Show that f is everywhere continuous. Functions satisfying such an inequality are called Lipschitz functions.
Suppose
|f (x )− f (y)|
≤ K
|x− y|
^α where K is a constant and α ∈
(0,1)
. Show that f is everywhere continuous. Functions like this are called Holder continuous.
Suppose f : ℝ³ → ℝ is given by f
(x)
= 3x₁x₂ + 2x₃². Use Theorem 8.4.2 to verify that f is continuous. Hint: You should first verify that the function π_k : ℝ³ → ℝ given by π_k
(x )
= x_k is a continuous function.
Show that if f : ℝ^q → ℝ is a polynomial then it is continuous.
State and prove a theorem which involves quotients of functions encountered in the previous problem.
Let

${ 2xx22-+yy22-if (x,y) ⁄= (0,0) f (x,y) \equiv 0 if (x,y) = (0,0) .$

Find lim
(x,y)
→
(0,0)
f
(x,y)
if it exists. If it does not exist, tell why it does not exist. Hint: Consider along the line y = x and along the line y = 0.
Find the following limits if possible
1. lim_{(x,y)
  
  →(0,0)}
  2 2 xx2−+yy2
  .
2. lim_{(x,y)
  
  →(0,0)}
  x(x2−y2) (x2+y2)
  = 0.
3. lim_{(x,y)
  
  →(0,0)}
  2 42 (x(x2−+yy4))2
  . Hint: Consider along y = 0 and along x = y².
4. lim_{(x,y)
  
  →(0,0)}xsin
  ( ) --1-- x2+y2
  .
5. lim_{(x,y)
  
  →(1,2)}
  
  . Hint: It might help to write this in terms of the variables
  (s,t)
  =
  (x− 1,y− 2)
  .
Suppose lim_x→0f
(x,0)
= 0 = lim_y→0f
(0,y)
. Does it follow that

$lim f (x,y) = 0? (x,y)\to(0,0)$

Prove or give counter example.
f : D ⊆ ℝ^p → ℝ^q is Lipschitz continuous or just Lipschitz for short if there exists a constant K such that

$|f (x )- f (y)| \leq K |x - y|$

for all x,y ∈ D. Show every Lipschitz function is uniformly continuous which means that given ε > 0 there exists δ > 0 independent of x such that if
|x − y |
< δ, then
|f (x)− f (y)|
< ε.
If f is uniformly continuous, does it follow that
|f|
is also uniformly continuous? If
|f|
is uniformly continuous does it follow that f is uniformly continuous? Answer the same questions with “uniformly continuous” replaced with “continuous”. Explain why.
Let f be defined on the positive integers. Thus D
(f)
= ℕ. Show that f is automatically continuous at every point of D
(f)
. Is it also uniformly continuous? What does this mean about the concept of continuous functions being those which can be graphed without taking the pencil off the paper?

Let

( ) x2 - y42 f (x,y) = (x2-+-y4)2-if (x,y) ⁄= (0,0)

Show lim_t→0f

= 1 for any choice of

. Using Problem 11c, what does this tell you about limits existing just because the limit along any line exists.

Let f
(x,y,z)
= x²y + sin
(xyz)
. Does f achieve a maximum on the set

${ 2 2 2 } (x,y,z) : x + y + 2z \leq 8 ?$

Explain why.
Suppose x is defined to be a limit point of a set A if and only if for all r > 0, B
(x,r)
contains a point of A different than x. Show this is equivalent to the above definition of limit point.
Give an example of an infinite set of points in ℝ³ which has no limit points. Show that if D
(f)
equals this set, then f is continuous. Show that more generally, if f is any function for which D
(f)
has no limit points, then f is continuous.
Let
{xk}
_k=1ⁿ be any finite set of points in ℝ^p. Show this set has no limit points.
Suppose S is any set of points such that every pair of points is at least as far apart as 1. Show S has no limit points.
Find lim_x→0
sin(|x|x||)
and prove your answer from the definition of limit.
Suppose g is a continuous vector valued function of one variable defined on [0,∞). Prove

$lim g(|x |) = g (|x0|). x\tox0$