Exercises - Graphing functions
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In each of the following problems, sketch the graph based on listed properties.
- 1. The function f has the following properties:
D( f ) = ℝ;
f is continuous on
(−∞,0] (including
continuity from the left at 0) and on
(0,∞).
f (0) = 0,
x-intercepts are −2, 0, and 1.
Limits at endpoints:
f (−∞) = −2,
f (0-) = 0,
f (0+) = −∞,
f (∞) = ∞;
asymptotes: horizontal y = −2 at
−∞, vertical at 0, oblique
y = x − 2 at
∞.
f is increasing on
(−∞,−1]
and on (0,∞);
it is decreasing on [−1,0];
local maximum f (−1) = −1;
f ′-(0) = −3.
f is concave up on
(∞,−2]
and on [4,∞);
it is concave down on [−2,0] and on [0,4];
inflection points f (−2) = 0,
f (4) = 4;
derivatives there are f ′(−2) = 3 and
f ′(4) = 1/2.
Hint
Answer
- 2. The function f has the following properties:
D( f ) = (−4,∞);
f is continuous there.
f (0) = 0,
x-intercepts are
−2 and 0.
Limits at endpoints:
f (4+) = 2,
f (∞) = ∞;
no asymptotes.
f is increasing on
[−1,∞),
it is decreasing on (−4,−1]; local minimum
f (−1) = −1;
f ′(2) = 0, no
derivative at −2,
f ′-(−2) = −1/3,
f ′+(−2) = −3,
f ′+(−4) = −∞.
f is concave up on (−4,−2], on [−2,0], and on
[2,∞); it is concave down on
[0,2];
inflection points f (−2) = 0,
f (0) = 0;
f (2) = 2;
derivative f ′(0) = 3.
Hint
Answer
- 3. The function f has the following properties:
D( f ) = (−∞,0) ∪ (0,∞);
f is continuous there.
x-intercepts are −1 and 1, f is odd.
Limits at endpoints:
f (0+) = 1,
f (∞) = 2;
asymptotes: no vertical, horizontal y = 2 at
∞.
f is increasing on
[1,∞);
it is decreasing on (0,1]; local minimum
f (1) = 0;
f ′+(0) = −3.
f is concave up on (0,2]; it is concave down on
[2,∞);
inflection point f (2) = 1,
derivative there f ′(2) = 3.
Hint
Answer
Sketch the graphs of the following functions:
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