1.)If f is continuous on (, ), what can you say about its graph? (Select all that...
Question:
1.)If f is continuous on (−∞, ∞), what can you say about its graph? (Select all that apply.)
a)The graph of f has a hole.
b)The graph of f has a jump.
c)The graph of f has a vertical asymptote.
d)none of these
10.)
EXAMPLE 10 Show that there is a root of the equation
4x3 − 6x2 + 3x − 2 = 0
between 1 and 2.
SOLUTION Let
f(x) = 4x3 − 6x2 + 3x − 2 = 0.
We are looking for a solution of the given equation, that is, a number c between 1 and 2 such that
f(c) = .
Therefore we take
a = ,
b = ,
and
N =
in the Intermediate Value Theorem. We have
f(1) | = | 4 − 6 + 3 − 2 = −1 < 0 | ||
and | f(2) | = | 32 − 24 + 6 − 2 = 12 > 0. | |
Thus
f(1) < 0 < f(2);
that is N = 0 is a number between
f(1)
and
f(2).
Now f is continuous since it is a polynomial, so that the Intermediate Value Theorem says there is a number c between and such that
f(c) = .
In other words, the equation
4x3 − 6x2 + 3x − 2 = 0
has at least one root c in the open interval
.
In fact, we can locate a root more precisely by using the Intermediate Value Theorem again. Since
f(1.2) = −0.128 < 0 and f(1.3) = 0.548 > 0
a root must lie between (smaller) and (larger). A calculator gives, by trial and error,
f(1.22) = −0.007008 < 0 and f(1.23) = 0.056068 > 0.
So a root lies in the open interval
.
Expert Answer:
Answer 1 The graph of f is continuous on and therefore will not have any holes jumps or vertical asy... View the full answer
