Conjecture Consider the functions f ( x ) = x 2 and g ( x ) = x 3 . (a) Graph f and f' on the same set of axes. (b) Graph g and g' on the same set of axes. (c) Identify a pattern between f and g and their respective derivatives. Use the pattern to make a conjecture about h' ( x ) if h ( x ) = x n . where " is an integer and n ≥ 2 ? (d) Find f' ( x ) if f ( x ) = x 4 Compare the result will) the conjecture in part (c). Is this a proof of your conjecture? Explain.
Conjecture Consider the functions f ( x ) = x 2 and g ( x ) = x 3 . (a) Graph f and f' on the same set of axes. (b) Graph g and g' on the same set of axes. (c) Identify a pattern between f and g and their respective derivatives. Use the pattern to make a conjecture about h' ( x ) if h ( x ) = x n . where " is an integer and n ≥ 2 ? (d) Find f' ( x ) if f ( x ) = x 4 Compare the result will) the conjecture in part (c). Is this a proof of your conjecture? Explain.
Solution Summary: The author explains that the graph of the functions is differentiable for all values of x where n is not an integer.
Conjecture Consider the functions
f
(
x
)
=
x
2
and
g
(
x
)
=
x
3
.
(a) Graph f and f' on the same set of axes.
(b) Graph g and g' on the same set of axes.
(c) Identify a pattern between f and g and their respective derivatives. Use the pattern to make a conjecture about h' (x) if
h
(
x
)
=
x
n
. where " is an integer and
n
≥
2
?
(d) Find f'(x) if
f
(
x
)
=
x
4
Compare the result will) the conjecture in part (c). Is this a proof of your conjecture? Explain.
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