For Exercises 11-16, identify the statements among a-h that follow directly from the given condition about x . a. csc x is undefined. b. sec x is undefined. c. The graph of y = sec x has a relative maximum at x . d. The graph of y = csc x has a relative minimum at x . e. The graph of y = sec x has a vertical asymptote. f. The graph of y = csc x has a vertical asymptote. g. The graph of y = csc x has a relative maximum at x . h. The graph of y = sec x has a relative minimum at x . The graph of y = cos x has a relative maximum at x .
For Exercises 11-16, identify the statements among a-h that follow directly from the given condition about x . a. csc x is undefined. b. sec x is undefined. c. The graph of y = sec x has a relative maximum at x . d. The graph of y = csc x has a relative minimum at x . e. The graph of y = sec x has a vertical asymptote. f. The graph of y = csc x has a vertical asymptote. g. The graph of y = csc x has a relative maximum at x . h. The graph of y = sec x has a relative minimum at x . The graph of y = cos x has a relative maximum at x .
For Exercises 11-16, identify the statements among
a-h
that follow directly from the given condition about
x
.
a.
csc
x
is undefined.
b.
sec
x
is undefined.
c. The graph of
y
=
sec
x
has a relative maximum at
x
.
d. The graph of
y
=
csc
x
has a relative minimum at
x
.
e. The graph of
y
=
sec
x
has a vertical asymptote.
f. The graph of
y
=
csc
x
has a vertical asymptote.
g. The graph of
y
=
csc
x
has a relative maximum at
x
.
h. The graph of
y
=
sec
x
has a relative minimum at
x
.
The graph of
y
=
cos
x
has a relative maximum at
x
.
Formula Formula A function f(x) attains a local maximum at x=a , if there exists a neighborhood (a−δ,a+δ) of a such that, f(x)<f(a), ∀ x∈(a−δ,a+δ),x≠a f(x)−f(a)<0, ∀ x∈(a−δ,a+δ),x≠a In such case, f(a) attains a local maximum value f(x) at x=a .
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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