Finding the nth Term of a Sequence In Exercises 45-52, write an expression for the nth term of the sequence and then determine whether the sequence you have chosen converges or diverges. (There is more than one correct answer.) 1 2 − 3 , 2 3 − 4 , 3 4 − 5 , 4 5 − 6 , …
Finding the nth Term of a Sequence In Exercises 45-52, write an expression for the nth term of the sequence and then determine whether the sequence you have chosen converges or diverges. (There is more than one correct answer.) 1 2 − 3 , 2 3 − 4 , 3 4 − 5 , 4 5 − 6 , …
Solution Summary: The author analyzes the nth term of a sequence and determines whether it converges or diverges.
Finding the nth Term of a Sequence In Exercises 45-52, write an expression for the nth term of the sequence and then determine whether the sequence you have chosen converges or diverges. (There is more than one correct answer.)
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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