Workshop #9
Professor D. Olles
1. Show that h(x) = x3 + 2x2 − 2x − 1 has a zero on [0, 2].
2. Sketch the graph of a function satisfying all of the following conditions.
a. f is continuous on (−∞, 2) ∪ (2, 3) ∪ (3, ∞)
b. f has a non-removable discontinuity at x = 2
c. f has a removable discontinuity at x = 3
d. limx→∞ f (x) = −1
e. limx→−∞ f (x) = 2
3. Find the values of a and b that will make f (x) is continuous everywhere.
 2
−4
x<2
 xx−2
f (x) =
ax2 − bx + 3 2 ≤ x ≤ 3

2x − a + b
x≥3
2x2 +5x−3
4. Let g(x) = 4x
2 +4x−3
a. State the interval(s) on which g is continuous.
b. State any points of discontinuity and whether they are removable
or non-removable.
5. Write the equation of a function f (x) that has a removable discontinuity
at x = −3 and a non-removable discontinuity at x = 1.
6. Suppose f is continuous on [0, 3] except at x = 0.25, and that f (0) = 1
and f (3) = 3. If N = 2, sketch a possible graph of f where f still satisfies
the conclusion of the I.V.T.
1
Solutions
1. Show that h(x) = x3 + 2x2 − 2x − 1 has a zero on [0, 2].
h(0) = −1
h(2) = 8 + 8 − 4 − 1 = 11
h(0) < 0 < h(2)
=⇒ there exists a c ∈ (0, 2) such that f (c) = 0
2. Sketch the graph of a function satisfying all of the following conditions.
a. f is continuous on (−∞, 2) ∪ (2, 3) ∪ (3, ∞)
b. f has a non-removable discontinuity at x = 2
c. f has a removable discontinuity at x = 3
d. limx→∞ f (x) = −1
e. limx→−∞ f (x) = 2
3. Find the values of a and b that will make f (x) is continuous everywhere.
 2
−4
x<2
 xx−2
f (x) =
ax2 − bx + 3 2 ≤ x ≤ 3

2x − a + b
x≥3
lim
x→2−
x2 − 4
= lim (ax2 − bx + 3)
x−2
x→2+
lim (x + 2) = 4a − 2b + 3
x→2−
4 = 4a − 2b + 3
1 = 4a − 2b
2
lim (ax − bx + 3) = lim (2x − a + b)
x→3−
x→3+
9a − 3b + 3 = 6 − a + b
10a − 4b = 3
1 = 4a − 2b
2b = 4a − 1 →
10a − 4b = 3
10a − 2(4a − 1) = 3
10a − 8a + 2 = 3
2a = 1
← a = 12
2b = 4 21 − 1
2b = 2 − 1
2b = 1
b = 21
2
2 +5x−3
4. Let g(x) = 2x
4x2 +4x−3
a. State the interval(s) on which g is continuous.
2x2 + 5x − 3
(2x − 1)(x + 3)
=
2
4x + 4x − 3
(2x − 1)(2x + 3)
3 1
x 6= − ,
2 2
[
3 1 [ 1
3
− ,
,∞
−∞, −
2
2 2
2
g(x) =
b. State any points of discontinuity and whether they are removable
or non-removable.
3
is a nonremovable discontinuity so, there is a vertical asymptote on the graph
2
1
1 7
x = is a removable discontinuity so, there is a hole on the graph at
,
2
2 8
x=−
5. Write the equation of a function f (x) that has a removable discontinuity
at x = −3 and a non-removable discontinuity at x = 1.
f (x) =
x+3
x2 + 2x − 3
6. Suppose f is continuous on [0, 3] except at x = 0.25, and that f (0) = 1
and f (3) = 3. If N = 2, sketch a possible graph of f where f still satisfies
the conclusion of the I.V.T.
3