Worksheet: 03.05 Topic: Curve Sketching Due: Nov 12 Answer the

Worksheet: 03.05 Topic: Curve Sketching Due: Nov 12 Answer the

Worksheet: 03.05
Topic: Curve Sketching
Due: Nov 12
Answer the following conceptual questions.
1. Sketch a graph of a function f (x) that is concave up on (0, 1) and is concave down on (1, 2).
2. Sketch a graph of a function f (x) that is:
(a) Increasing, concave up on (0, 1),
(b) increasing, concave down on (1, 2),
(c) decreasing, concave down on (2, 3) and
(d) increasing, concave down on (3, 4).
3. Is is possible for a function to be increasing and concave down on (0, ∞) with a horizontal asymptote of y = 1?
If so, give a sketch of such a function.
4. Is is possible for a function to be increasing and concave up on (0, ∞) with a horizontal asymptote of y = 1?
If so, give a sketch of such a function.
Given the function summary, sketch a graph of each function on the indicated interval.
5. f (x) = x2 + 2x − 3
on
[−3, 0]
Vertical Asymptotes: None
Horizontal Asymptotes: Not Applicable (f is defined on [−3, 0])
Critical numbers: x = −1
2nd Critical numbers: None
Increasing: (−1, 0]
Decreasing: [−3, −1)
Concave Up: (−∞, ∞)
Concave Down: ∅
x f (x)
-3
0
-1
-4
0
-3
1 3 1 2
x + x − x on
9
3
Vertical Asymptotes: None
6. f (x) =
[−4, 3]
Horizontal Asymptotes: Not Applicable (f is defined on [−4, 3])
Critical numbers: x = −3, 1
2nd Critical numbers: x = −1
Increasing: [−4, −3) ∪ (1, 3]
Decreasing: (−3, 1)
Concave Up: (−1, 3]
Concave Down: [−4, −1)
x
-4
-3
-1
1
3
f (x)
20/9
3
11/9
-5/9
3
1
Worksheet: 03.05
Topic: Curve Sketching
7. f (x) = x3 + 3x2 + 3
on
Due: Nov 12
[−3, 1]
Vertical Asymptotes: None
Horizontal Asymptotes: Not Applicable (f is defined on [−3, 1])
Critical numbers: x = −2, 0
2nd Critical numbers: x = −1
Increasing: [−3, −2) ∪ (0, 1]
Decreasing: (−2, 0)
Concave Up: (−1, 1]
Concave Down: [−3, −1)
x
-3
-2
-1
0
1
f (x)
3
7
5
3
7
24
on (−∞, ∞).
x2 + 12
Vertical Asymptotes: None
8. f (x) =
Horizontal Asymptotes: y = 0 (toward −∞ and +∞)
Critical numbers: x = 0
2nd Critical numbers: x = −2, 2
Increasing: (−∞, 0)
Decreasing: (0, ∞)
Concave Up: (−∞, −2) ∪ (2, ∞)
Concave Down: (−2, 2)
x
0
-2
2
f (x)
2
3/2
3/2
9. f (x) = x2 ex
on
(−∞, 1]
Vertical Asymptotes: None
Horizontal Asymptotes: y = 0 (as x → −∞)
Critical numbers: x = −2, 0
2nd Critical numbers: x = −2 −
√
2, −2 +
√
2
Increasing: (−∞, −2) ∪ (0, 1)
Decreasing: (−2, 0)
√
√
Concave Up: (−∞, −2 − 2) ∪ (−2 + 2, 1]
√
√
Concave Down: (−2 − 2, −2 + 2)
x√
−2 − 2
-2 √
−2 + 2
0
1
f (x)
0.38
0.54
0.19
0
2.72
2
Worksheet: 03.05
10. f (x) = x2 ln x
Topic: Curve Sketching
on
Due: Nov 12
(0, 5]
Vertical Asymptotes: None
Horizontal Asymptotes: Not Applicable (f is defined on (0, 5])
Critical numbers: x = 4e−2 , 0
A
2nd Critical numbers: x = 4e−6
Increasing: (4e−2 , 5]
Decreasing: (0, 4e−2 )
Concave Up: (4e−6 , 5]
Concave Down: (0, 4e−6 )
x
4e−6
4e−2
5
f (x)
Sketch a graph of each on the indicated interval.
11. f (x) = −x2 − 5x + 7
12. f (x) = x3 − x + 1
13. f (x) = e−x
2
on
on
on
(−∞, ∞)
[−2, 1]
(−∞, ∞)
x
on [−2, 2]
−1
√
15. f (x) = x 4 − x on (−∞, 4]
14. f (x) =
x2
3
Worksheet: 03.05
Topic: Curve Sketching
Selected Answers
Answer the following conceptual questions.
1. Answers will vary.
3. Yes; Answers will vary.
9. Graph: Use a graphing utility
Sketch a graph of each on the indicated interval.
Given the function summary, sketch a graph of each
function on the indicated interval.
11. Use a graphing utility
5. Graph: Use a graphing utility
13. Use a graphing utility
7. Graph: Use a graphing utility
15. Graph: See Notes
4