Let f ( x ) = { e − 1 / x 2 if x ≠ 0 0 if x = 0 (a) Use the definition of derivative to compute f ′(0). (b) Show that f has derivatives of all orders that are defined on ℝ . [ Hint: First show by induction that there is a polynomial p n ( x ) and a nonnegative integer k n such that f ( n ) ( x ) = p n ( x ) f ( x ) / x k n for x ≠ 0.)
Let f ( x ) = { e − 1 / x 2 if x ≠ 0 0 if x = 0 (a) Use the definition of derivative to compute f ′(0). (b) Show that f has derivatives of all orders that are defined on ℝ . [ Hint: First show by induction that there is a polynomial p n ( x ) and a nonnegative integer k n such that f ( n ) ( x ) = p n ( x ) f ( x ) / x k n for x ≠ 0.)
Solution Summary: The author explains how to compute the value of fprime(0) by using definition of derivative.
(a) Use the definition of derivative to compute f′(0).
(b) Show that f has derivatives of all orders that are defined on
ℝ
. [Hint: First show by induction that there is a polynomial pn(x) and a nonnegative integer kn such that
f
(
n
)
(
x
)
=
p
n
(
x
)
f
(
x
)
/
x
k
n
for x ≠ 0.)
Assume that f : R → R is defined via the assignment rule f(x) = e2x. We will use the notation that f(n)(x) is the n-th derivative of f. Prove, using induction that for all n ∈ N, f(n)(x) = 2ne2x.
Note: The following facts about derivatives may be helpful: if g(x),h(x) are differentiable functions and c is a constant, then
(eg(x) )'= g'(x)eg(x) (c*h(x))'= c*(h'(x))
Apply Rolle's theorem
6
From algebraic form of function ez, prove that
|ez| = ex
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