Given a sequence a 1 , a 2 , a 3 , … , a n , the arithmetic mean a ¯ is given by a ¯ = 1 n ∑ i = 1 n a i . Use the arithmetic mean for Exercises 103-104. Consider the sequence defined by a n = 18 , 32 , 44 , 20 , 36 , 28 , 32 , 38 . Evaluate ∑ i = 1 8 a i − a ¯ 2 .
Given a sequence a 1 , a 2 , a 3 , … , a n , the arithmetic mean a ¯ is given by a ¯ = 1 n ∑ i = 1 n a i . Use the arithmetic mean for Exercises 103-104. Consider the sequence defined by a n = 18 , 32 , 44 , 20 , 36 , 28 , 32 , 38 . Evaluate ∑ i = 1 8 a i − a ¯ 2 .
Solution Summary: The author calculates the sum of terms of the sequence with a total of n terms.
Given a sequence
a
1
,
a
2
,
a
3
,
…
,
a
n
, the arithmetic mean
a
¯
is given by
a
¯
=
1
n
∑
i
=
1
n
a
i
. Use the arithmetic mean for Exercises 103-104.
Consider the sequence defined by
a
n
=
18
,
32
,
44
,
20
,
36
,
28
,
32
,
38
. Evaluate
∑
i
=
1
8
a
i
−
a
¯
2
.
Consider the function f(x) = x²-1.
(a) Find the instantaneous rate of change of f(x) at x=1 using the definition of the derivative.
Show all your steps clearly.
(b) Sketch the graph of f(x) around x = 1. Draw the secant line passing through the points on the
graph where x 1 and x->
1+h (for a small positive value of h, illustrate conceptually). Then,
draw the tangent line to the graph at x=1. Explain how the slope of the tangent line relates to the
value you found in part (a).
(c) In a few sentences, explain what the instantaneous rate of change of f(x) at x = 1 represents in
the context of the graph of f(x). How does the rate of change of this function vary at different
points?
1. The graph of ƒ is given. Use the graph to evaluate each of the following values. If a value does not exist,
state that fact.
и
(a) f'(-5)
(b) f'(-3)
(c) f'(0)
(d) f'(5)
2. Find an equation of the tangent line to the graph of y = g(x) at x = 5 if g(5) = −3 and g'(5)
=
4.
-
3. If an equation of the tangent line to the graph of y = f(x) at the point where x 2 is y = 4x — 5, find ƒ(2)
and f'(2).
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