derivation
CathyjeyLet f(x) = a^x. The goal of this problem is to explore= how the value of a affects the derivative
of f(x), without assuming= we know the rule for d/dx [a^x] that we have stated and used in earlier
work in this section.
of f(x), without assuming= we know the rule for d/dx [a^x] that we have stated and used in earlier
work in this section.
a) Use the limit definition of the der= ivative to show that
f ' (x) = lim a^x. a^h - a^x/h
 = ; h-->0
f ' (x) = lim a^x. a^h - a^x/h
 = ; h-->0
b)Explain why it is also true that
f ' (x)= a^x lim = a^h-1/h
&nb= sp; h-->0
c)Use computing technology and small values of h to estimate the valu= e of
L = lim a^h-1/h
h-->0
L = lim a^h-1/h
h-->0
when a = 2. Do likewise when a = 3.
<= br>D)Note that it would be ideal if the value of the limit L was 1, for the= n f would be a
particularly special function: its derivative would b= e simply a^x, which would mean
that its derivative is itself. By exp= erimenting with different values of a between 2 and
3, try to find a= value for a for which:
L = lim a^h-1/h=1
&nbs= p; h-->0
&nbs= p; h-->0
E) Compute ln(2) and ln(3). Wh= at does your work in (b) and (c) suggest is true about
d/dx [2^x] an= d d/dx [2^x].
d/dx [2^x] an= d d/dx [2^x].
F) How do your investigations in (= d) lead to a particularly important fact about the number
e?
e?
=
12 years ago
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