SME
Calculus I · Problem type
The intermediate value theorem
Use this for “Show that x³ + x − 1 = 0 has a root between 0 and 1.”
Chart
Zero: x ≈ 0.6823
f(0.5) = −0.375
Shaded: 0 ≤ x ≤ 1
f at the left end f(a) = −1
f at the right end f(b) = 1
This is an example. Type your own values to start a new problem.
f(x)
=
x³
+
p
x
+
(
q
)
f(
a
)
=
f(a)
,
f(
b
)
=
f(b)
f(
m
)
=
f(m)
Formulas
f
(
a
)
=
a
3
+
p
×
a
+
q
−1
=
0
3
+
1
×
0
+
(−1)
f
(
b
)
=
b
3
+
p
×
b
+
q
1
=
1
3
+
1
×
1
+
(−1)
m
=
a
+
b
2
0.5
=
0 + 1
2
f
(
m
)
=
m
3
+
p
×
m
+
q
−0.375
=
0.5
3
+
1
×
0.5
+
(−1)
Assumptions
•
f(x) = x³ + px + q is a polynomial, so it is continuous on every interval [a, b].
•
Intermediate value theorem: if f(a) and f(b) have opposite signs, f(c) = 0 for some c between a and b.
•
One bisection step: check the midpoint m and keep the half where the sign still changes.
Step-by-step
WE KNOW
p = 1, q = −1, a = 0, b = 1
FIND
f at the left end (f(a)), f at the right end (f(b)), midpoint (m), f at the midpoint (f(m))
Step 1 · Find f at the left end (f(a))
Use f(a) = a³ + p × a + q
Put x = a into f.
f
(
a
)
=
0
3
+
1
×
0
+
(−1)
f
(
a
)
=
−1
Step 2 · Find f at the right end (f(b))
Use f(b) = b³ + p × b + q
Put
x
=
b
into
f.
f
(
b
)
=
1
3
+
1
×
1
+
(−1)
f
(
b
)
=
1
→
f
(
a
)
<
0
<
f
(
b
)
and
f
is
continuous,
so
some
root
lies
between
0
and
1
Step 3 · Find midpoint (m)
Use m = (a + b) ÷ 2
Halve the interval at its midpoint.
m
=
0 + 1
2
m
=
0.5
Step 4 · Find f at the midpoint (f(m))
Use f(m) = m³ + p × m + q
Put
x
=
m
into
f,
then
keep
the
half
where
the
sign
still
changes.
f
(
m
)
=
0.5
3
+
1
×
0.5
+
(−1)
f
(
m
)
=
−0.375
→
f
(
m
)
has
the
sign
of
f
(
a
),
so
some
root
lies
in
[0.5,
1]
Check
Put the numbers back in. Both sides should match.
−1
=
0
3
+
1
×
0
+
(−1)
✓
1
=
1
3
+
1
×
1
+
(−1)
✓
0.5
=
0 + 1
2
✓
−0.375
=
0.5
3
+
1
×
0.5
+
(−1)
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Related lessons
Limits and continuity
Main lesson
Make a piecewise function continuous
Use this for “Find k so that f(x) = 3x + k (x < 2), x² + 1 (x ≥ 2) is continuous.”
The limit of sin(kx) ÷ (mx)
Use this for “Find the limit of sin(3x) ÷ (2x) as x → 0” or “Find k so that sin(kx) ÷ (2x) → 3 as x → 0.”
Find δ for a given ε (a line)
Use this for “For lim (3x − 1) = 5 as x → 2, find δ when ε = 0.06.”
Limits at infinity of a quotient
Use this for “Find the limit of (5x² − x) ÷ (2x² + 7) as x → ∞.”
Kinds of discontinuity
Use this for “Is the discontinuity of (x² − 4) ÷ (x − 2) at x = 2 removable?”