4.12. EXERCISES 107

Now suppose U is an open set in Rp and f : U → R is Cm+1 with x0 ∈ U . For x ∈B(x0,r)⊆U , let h(t) = f (x0 + t (x−x0)) , t ∈ (0,1) . Then

h′ (t) = ∑i

∂ f (x0 + t (x−x0))

∂xi(xi− x0i)

h′′ (t) = ∑i1,i2

∂ 2 f∂xi1∂xi2

(xi1 − x0i1)(xi2 − x0i2)

and continuing this way,

h(k) (t) = ∑i1,··· ,ik

∂ k f∂xi1∂xi2 · · ·∂xik

k

∏j=1

(xi j − x0i j

)(4.24)

Then the Taylor approximation is of the form h(1) = f (x) =

f (x0)+m

∑k=1

1k! ∑

i1,··· ,ik

∂ k f (x0)

∂xi1∂xi2 · · ·∂xik

k

∏j=1

(xi j − x0i j

)+

1(m+1)! ∑

i1,··· ,im+1

∂ m+1 f (x0 + t (x−x0))

∂xi1∂xi2 · · ·∂xim+1

m+1

∏j=1

(xi j − x0i j

)(4.25)

The last term being the remainder with t ∈ (0,1). Thus, if the (m+1)st partial derivativesare all bounded, this shows that if ∥x−x0∥ is sufficiently small, then the difference betweenf (x) and that series on the right in 4.25 other than the remainder term will also be verysmall.

4.12 Exercises1. For (x,y) ̸= (0,0) , let f (x,y) = xy4

x2+y8 . Show that this function has a limit as (x,y)→(0,0) for (x,y) on an arbitrary straight line through (0,0). Next show that this func-tion fails to have a limit at (0,0).

2. Here are some scalar valued functions of several variables. Determine which of thesefunctions are o(v). Here v is a vector in Rn, v = (v1, · · · ,vn).

(a) v1v2

(b) v2 sin(v1)

(c) v21 + v2

(d) v2 sin(v1 + v2)

(e) v1 (v1 + v2 + xv3)

(f) (ev1 −1− v1)

(g) (x ·v) |v|

3. Here is a function of two variables. f (x,y) = x2y+ x2. Find D f (x,y) directly fromthe definition. Recall this should be a linear transformation which results from mul-tiplication by a 1×2 matrix. Find this matrix.

4. Let f(x,y) =(

x2 + yy2

). Compute the derivative directly from the definition. This

should be the linear transformation which results from multiplying by a 2×2 matrix.Find this matrix.