. Let f(x) = 3x^2 + 5x. Using the limit definition of derivative
prove that f '(x) = 6x + 5
Then, Find the tangent line of f(x) at x = 3
Finally, Find the average rate of change between x = −1 and x =
2
Each of the following limits represents a derivative
f '(a).
Find
f(x)
and
a.
lim h→0
(5 +
h)3 − 125
h
f(x)
=
a
=
5
lim x→2
x3 − 8
x −
2
f(x)
=
a
=
2
lim h→0
52 +
h − 25
h
f(x)
=
a
=
2
Use the given function, its first derivative, and its second
derivative to answer the following:
f(x)=(1/3)x^3 - (1/2)x^2 - 6x + 5
f'(x)= x^2 - x - 6 = (x+2)(x-3)
f''(x)= 2x - 1
a) What are the intervals of increase and the intervals of
decrease
b) Identify local min and max points
c) What are the intervals where the function is concave up,
concave down and identify the inflection points
If F is a field and ?(?),?(?),h(?) ∈ ?[?]; and
h(?) ≠ 0.
a) Prove that [?(?)] = [?(?)] if and only if ?(?) ≡ ?(?)(???(
(h(?)).
b) Prove that congruence classes modulo h(?) are either disjoint
or identical.
Derive a three-point formula with the highest possible order (of
h) to approximate f '(a) and f "(a), respectively, using f(a − 2h),
f(a + h), f(a + 2h).