MATH 140
EXAM II SAMPLE C
1. Find the horizontal asymptote of the function y =
4. Find the absolute maximum value M and the absolute minimum
4 − x2
value m of the function f (x) = 2
on the interval [−1, 2]?
x +8
3x2 + 4
.
2 − x2
a) y = 2
1
b) y =
2
c) y = 3
d) y =
a) M =
1
and m = 0
2
b) M =
1
1
and m =
3
2
c) M = 0 and m = 3
1
3
d) The function has no absolute extrema in this interval but
has a local minimum when x = 0.
e) y = −3
e) M =
π
2. Find the linearization of f (x) = tan x − cot x at a = .
4
1
and m = 0
3
a) L(x) = 1
b) L(x) = 4x − π
c) L(x) = 1 + x
d) L(x) =
x
+π
4
e) L(x) = x −
3
4
5. Find the coordinates of the point that lies on the line 2x + y = 3
and is closest to (−2, 2).
a) (2, 0)
√
b) (0, 5)
c) (−2, 2)
d)
(0, 3)
e) (−3, 3)
q
3. Find the critical number(s) of the function f (x) =
in the interval (−3, 10)?
3
(x2 − 6x + 5)
a) There are no critical numbers in this interval.
b) 1 and 5 only
c) 3 only
d) 1, 3 and 5
e) 0, 2 and 3
1
MATH 140
EXAM II SAMPLE C
6. Determine a value of c that satisfies the Mean Value Theorem for
4
the function f (x) = 5 − on the interval [1, 4].
x
a) c = 1
a) (−3, 1)
b) c = 4
c) c =
2 3
x + 2x2 − 6x + 11 find the interval for
3
which the function f is concave downward.
9. Given the function f (x) =
b) (−∞, −3)
1
2
c) (−∞, −1)
d) c = 2
d) (1, ∞)
e) There is no value c on [1, 4] that satisfies the Mean Value
Theorem.
e) (−1, ∞)
10. Find lim
x→∞
p
4x2 + 4x − 2x
a) 0
b)
1
c) 2
d) 4
e) ∞
11. Find an equation of the slant asymptote of the graph of f (x) =
2x2 + 16
.
x−3
7. Find the interval(s) on which f (x) = (x2 − 4)2/3 is increasing.
a) y = 2x −
a) (−∞, 0)
b) y =
b) (0, ∞)
8
3
34
x−3
c) (−∞, −2) and (2, ∞)
c) y = 2x + 6
d) (−2, 2)
d) y = 2x − 6
e) (−2, 0) and (2, ∞)
e) y = 2x
x
8. If f (x) = x3 − x2 − x + 1, which one of the following statements is 12. Find any local extrema for f (x) = sin x − on the interval [0, π].
2
true?
a) A local maximum for f occurs at 0.
a) An inflection point occurs when x = 1.
π
.
4
π
c) A local minimum for f occurs at .
4
π
d) A local maximum for f occurs at .
3
π
e) A local minimum for f occurs at .
3
b) A local maximum for f occurs at
1
b) An inflection point occurs when x = .
3
c) A local maximum occurs when x = 1.
d) A local minimum occurs when x =
1
.
3
e) A local maximum occurs when x =
1
.
3
2
MATH 140
EXAM II SAMPLE C
13. Given f 00 (x) = 1, f 0 (0) = 7, and f (0) = 3, find f (1).
a)
21
2
b)
10
3
c)
17
3
d)
17
2
e)
23
3
16. (9 points) An open box having a square base is to be constructed
from 48 in2 of material. What should be the dimensions of the box
to obtain a maximum volume? Show all work for full credit.
Questions 14 and 15 are true/false type. Mark a) for true, b) for
false. Each true/false question is worth 2 points.
14. If f 0 (c) = 0, then f (x) has a local extremum at x = c.
a) True
b) False
15. If f 0 (c) = 0 and f 00 (c) < 0 then f (x) must have a local maximum
at x = c.
a) True
b) False
Dimensions:
3
MATH 140
EXAM II SAMPLE C
17. (11 points) Use the function and its derivatives to find the requested 18. (11 points) The figure shows the graph of the derivative f 0 of a
information. Answer ”none” if applicable.
function f .
f (x) =
1
,
x2 − 9
f 0 (x) =
−2x
,
(x2 − 9)2
f 00 (x) =
6(x2 + 3)
(x2 − 9)3
y
a) The domain of f (x) is
.
b) The critical number(s) of f (x) is(are)
.
f 0 (x)
c) The horizontal asymptote of f (x) is
.
−2
d) The vertical asymptote(s) of f (x) is(are)
.
−1
e) f (x) is increasing on the interval(s)
.
f) f (x) is decreasing on the interval(s)
.
g) The x-value(s) of the local maximum(s) is(are)
.
h) The x-value(s) of the local minimum(s) is(are)
.
i) f (x) is concave up on the interval(s)
0
1
2
3
.
j) f (x) is concave down on the interval(s)
k) The x-value(s) of the inflection point(s) is(are)
.
NOTE: The information requested below refers to the function f . The derivative f 0 is graphed above.
On what intervals is f increasing?
On what intervals is f decreasing?
For what values of x does f have a local maximum?
For what values of x does f have a local minimum?
For what values of x is f concave up?
For what values of x is f concave down?
EXAM II- FORM A
1. E 2. B 3. D 4. A 5. D 6. D 7. E 8. B 9. C 10. B 11. C 12.
D 13. A 14. B 15. A 16. 4”x4”x2” 17. a) (−∞, −3) ∪ (−3, 3) ∪
(3, ∞) b) x = 0 c)y = 0 d) x = −3, x = 3 e)(−∞, −3), (−3, 0)
f)(0, 3) ∪ (3, ∞) g) x = 0 h) none i) (−∞, −3), (3, ∞) j) (−3, 3)
k) none 18.a) (−2, 0), (4, ∞) b) (−∞, −2), (0, 4) c) 0 d) −2, 4 e)
(−∞, −1), (1, 2), (3, 5) f) (−1, 1), (2, 3), (5, ∞)
4