1. What is the vertical asymptote for the function
?
1
2. What is the horizontal asymptote for the function
?
0
3. What is the horizontal asymptote for
?
The horizontal asymptote is x =
.
The horizontal asymptote is y =
.
The horizontal asymptote is y = -
.
The horizontal asymptote is y = -8.
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4. What is the end-behavior asymptote for
5. What is the end-behavior asymptote for
?
?
6. What are the asymptotes for
y=
and x = 1
y=
and x = -1
y=
and x = -1
y=
and x = 1
?
7. What are the asymptotes for
?
y = 0 and x = 5
y = 3 and x =
y=
and x = 3
y = 2x – 5 and x = 6
8. What are the critical points of the function
?
9. What are the critical points of the function f(x) = 5 + x2?
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10. What are the critical points of the function f(x) = x2 – 4x + 1?
The critical point is x = 0.
The critical points are x = -2 and 2.
The critical points is x = 2.
The critical points is x = -2.
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11. On what intervals is
12. On what intervals is
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45
increasing?
increasing?
13. For what intervals is f(x) = x4 – 8x2 + 3 decreasing?
The graph is decreasing on the interval ( -∞ , -2).
The graph is decreasing on the interval (0, 2).
The graph is decreasing on the intervals ( -∞ , -2) and (0, 2).
The graph is decreasing on the intervals ( -2, 0) and (2, ∞ ).
14. Describe the concavity for the function f(x) = -x3 + 3x + 7.
15. Describe the concavity for the function f(x) = x3 + 3x2 – 24x + 7.
16. For what intervals is the function f(x) = x3 – 2x2 + 3x – 5 concave down?
The graph is concave down for the interval (-∞, 0).
The graph is concave down for the interval (-∞,∞ ).
The graph is concave down for the interval (-∞,
The graph is concave down for the interval (
).
, -∞ ).
17. What is the relative minimum value of the function f(x) = -x3 + 4x2 – 5x + 6?
18. What is the relative minimum value of the function f(x) = x3 + 3x2 – 9x + 5?
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19. What is the relative maximum value of the function f(x) = -2x3 + 3x2 + 36x – 15?
f(x) has a relative maximum value at x = -2, and its value is f(-2) = -59.
f(x) has a relative maximum value at x = 3, and its value is f(3) = 66.
f(x) has a relative maximum value at x = -2, and its value is f(-2) = 66.
f(x) has a relative maximum value at x = 3, and its value is f(3) = -59.
20. What is the name of the point where a graph changes concavity?
21. What are the inflection points of the graph f(x) = x3 – 6x2 + 9x – 12?
22. What are the inflection points of the graph
?
The point of inflection are at (1, 0) and (-1, 4).
The point of inflection are at (0, 2) and (-1, 4).
The point of inflection is at (0, 2).
The point of inflection is at (0, -2).
23. What are two conditions that must be met before the Extreme Value Theorem may be
applied?
24. What are the critical points of the function f(x) = x3 + 8x2 – 19x + 12 on the interval [-3.5,
3.5]?
The critical points in the given interval are x = -1 and
The critical points in the given interval are x = 1 and
The critical points in the given interval are x = 1.
The critical points in the given interval are x = 0 and -3.
25. Find all absolute and relative extreme values of the function f(x) = -x2 + 6x – 5, on the
interval [0, 5].
26. Find all absolute and relative extreme values of the function f(x) = 3x4 – 4x3 on the interval
[-1, 1].
27. Find all absolute and relative extreme values of the function f(x) = x2 – 6x + 7, on the
interval [1, 4].
28. Find all absolute and relative extreme values of the function f(x) = 4x3 – 12x2 – 36x + 25,
on the interval [-2, 2].
The absolute maximum is f(0) = 25 and the absolute minimum is f(2) = -63.
The absolute maximum is f(2) = 45 and the absolute minimum is f(-1) = -63.
The absolute maximum is f(-1) = 45 and the absolute minimum is f(2) = -63.
The absolute maximum is f(-1) = 45 and the absolute minimum is f(3) = -83.
29. Find all absolute and relative extreme values of the function f(x) = 3x4 – 4x3, on the
interval [0, 2].
The absolute maximum is f(2) = 48 and the absolute minimum is f(1) = 0.
The absolute maximum is f(2) = 16 and the absolute minimum is f(1) = -1.
The absolute maximum is f(2) = 48 and the absolute minimum is f(1) = -1.
The absolute maximum is f(2) = 16 and the absolute minimum is f(1) = 0.
30. Identify the absolute maximum and absolute minimum of the function
the interval [-2, 2].
31. What are the asymptotes of the function
, within
?
32. What are the x-intercept and y-intercept of the function
?
33. Identify the absolute maximum and absolute minimum of the function
on the interval [
,
, 4].
The absolute minimum is f(-2) = 0 and the absolute maximum is f(
)=
.
The absolute minimum is f(-2) = 0 and the absolute maximum is f( ) = - .
The absolute minimum is f(2) = 4 and the absolute maximum is f(
)=
The absolute minimum is f(2) = 4 and the absolute maximum is f(
)=- .
34. What are the x-intercept and y-intercept of the function
The x-intercept is (-2, 0). The y-intercept is (0, 0).
The x-intercept is (-2, 0). The y-intercept is (0, 4).
The x-intercept is at (-2, 0). There isn’t a y-intercept.
There isn’t an x intercept or a y-intercept.
35. What is the point of inflection for the function f(x) = x3 – 6x2 + 1?
36. What is the point of inflection for the function f(x) = x4 – 3x2 + 7?
37 . What is the point of inflection for the function f(x) = 2x3 + 12x2 – 3?
?
.
38. What is the point of inflection for the function
?
The point of inflection is (-1, -2).
The points of inflection area at (0, -3) and (3, 24).
The points of inflection are (-1, -8) and (1, -8).
The points of inflection are (-1, 8) and (1, 8).
39. What is the point of inflection for the function f(x) = x5 – 10x2?
The point of inflection is (1, -9).
The point of inflection is (-1, -9).
The point of inflection is (1, 9).
The point of inflection is (-1, -40).
40. What is the point of inflection for the function f(x) = x3 + 3x2 + 3x + 3?
The point of inflection within this interval is (-1, 2).
The point of inflection within this interval is (1, -2).
The point of inflection within this interval is (0, 3).
The point of inflection within this interval is (-3, 4).
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