Engineering Math

12.2 Exercises

1. Determine which of the following functions are o
   (h)
   .
   1. h²
   2. hsin
      (h)
   3. |h|
      ^3∕2 ln
      (|h|)
   4. h²x + yh³
   5. sin
      ( ) h2
   6. sin
      (h)
   7. xhsin
      (∘ --) |h|
      + x⁵h²
   8. exp
      ( ) − 1∕ |h|2
2. Here are some scalar valued functions of several variables. Determine which of these functions are o
   (v)
   . Here v is a vector in ℝⁿ, v =
   (v1,⋅⋅⋅,vn)
   .
   1. v₁v₂
   2. v₂ sin
      (v1)
   3. v₁² + v₂
   4. v₂ sin
      (v1 + v2)
   5. v₁
      (v1 + v2 + xv3)
   6. (ev1 − 1− v ) 1
   7. (x ⋅v)
      |v|
3. Here are some vector valued functions of v ∈ ℝⁿ. Determine which ones are o
   (v)
   .
   1. (x ⋅v)
      v
   2. sin
      (v1)
      v
   3. ∘ ------- |(x ⋅v)|
      |v|
      ^2∕3
   4. ∘ |(x-⋅v)|
      |v|
      ^1∕2
   5. ( (∘ -----) ∘ ----) sin |x⋅v | − |x ⋅v|
      ⋅
      |v|
      ^−1∕4
   6. exp
      ( ) − 1∕ |v|2
   7. v^TAv where A is an n × n matrix.
4. Show that if f
   (x)
   = o
   (x )
   , then f^′
   (0)
   = 0.
5. Show that if lim_h→0f
   (x)
   = 0 then xf
   (x)
   = o
   (x)
   .
6. Show that if f^′
   (0)
   exists and f
   (0)
   = 0, then f
   p (|x|)
   = o
   (x)
   whenever p > 1.

+ class=”left” align=”middle”(v)12.3.  THE DERIVATIVE OF FUNCTIONS OF MANY VARIABLES