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314 Final Exam

(c) transcendental
(d) indeterminate
(e) the quotient of rational numbers
4. The decimal expansion of 4/7 is
(a) 0.213535353 . . .
(b) 0.141414114 . . .
(c) 0.1357357357 . . .
(d) 0.7981818181 . . .
(e) 0.571428571428 . . .
√ √
The number 3 ’ 2
5.
(a) lies between 1 and 2
(b) is rational
(c) is a perfect square
lies between ’1 and 0
(d)
(e) lies between 0 and 1
The set {x : 3 ¤ x < 7} is
6.
(a) a closed interval
(b) an open interval
(c) a discrete set
(d) a half-open interval
(e) a half-line
The set [2, 5) © [4, 8] is
7.
{x : 4 < x < 8}
(a)
{t : 4 ¤ t < 5}
(b)
{s : 2 ¤ s ¤ 4}
(c)
{w : 2 < w < 8}
(d)
{u : 4 ¤ u ¤ 5}
(e)
The set Q © (’3, 2) is
8.
(a) in¬nite
(b) ¬nite
(c) discrete
(d) unbounded
(e) arbitrary
The set {(x, y) : x = y 2 } has graph that is
9.
(a) a line
315
Final Exam

(b) a circle
(c) a parabola
(d) a hyperbola
(e) a directrix
10. The line that passes through the point (’4, 5) and has slope 3 has
equation
x + 3y = 2
(a)
x ’ 3y = ’4
(b)
’4x + 5y = 3
(c)
3x ’ y = ’17
(d)
3x ’ 5y = 4
(e)
The line 2x + 5y = 10 has slope
11.
(a) 3
(b) 1
(c) 1/5
’1/5
(d)
’2/5
(e)
The equation 2x 2 + 2y 2 = 4 describes
12.
(a) A circle with center (0, 0) and radius 2

(b) A circle with center (0, 0) and radius 2
(c) A circle with center (2, 2) and radius 2
(d) A circle with center (4, 4) and radius 4
(e) A circle with center (2, 4) and radius 1
The equation x + x 2 + y = 0 has graph that is
13.
(a) a circle
(b) a line
(c) a parabola
(d) two crossed lines
(e) a hyperbola
14. The sine, cosine, and tangent of the angle 5π/3 (measured in radians)
are
√ √
(a) 1/2, 3/2, 3
√ √
(b) √3/2, 1/2, 1/ 3

(c) 2/2, 2/2, 1 √

(d) ’ 3/2, 1/2, ’ 3
(e) 1, 0, unde¬ned
316 Final Exam

15. The tangent, cotangent, and secant of the angle 3π/4 (measured in
radians) are

(a) ’ 3/2, ’1/2, 1
√ √
(b) 1/ 2, 1/ 2, ’1
√ √
2, ’ 2, 2
(c)

(d) 1, ’1, 3 √
(e) ’1, ’1, ’ 2

The domain and range of the function g(x) = 1 + 2x are
16.
{x : x ≥ ’1/2} and {x √0 ¤ x < ∞}
:
(a)
{x : x ≥ 1/2} and {x : 2 ¤ x ¤ 2}
(b)
{x : x ¤ ’1/2} and {y : ’2 ¤ y < ∞}
(c)
{s : 1 ¤ s ¤ 2} and {t : 2 ¤ t ¤ 4}
(d)
{x : 0 ¤ x ¤ 2} and {x : 1 ¤ x ¤ 4}
(e)
The graph of the function y = 1/|x| is
17.
(a) Entirely in the second and third quadrants
(b) Entirely in the ¬rst and fourth quadrants
(c) Entirely above the x-axis
(d) Increasing as x moves from left to right
(e) Decreasing as x moves from left to right
The graph of y = 2x/(1 + x 2 ) includes the points
18.
(a) (0, 1), (2, 4), (3, 3)
(b) (1, 1), (2, 2), (4, 4)
(’1, 1), (1, ’1), (3, 6)
(c)
(1, 1), (2, 4/5), (’2, ’4/5)
(d)
(e) (0, 0), (’4, 3), (4, 5)
Let f (x) = x 2 + x and g(x) = x 3 ’ x. Then
19.
f —¦ g(x) = (x 2 + x)x and g —¦ f (x) = (x 2 ’ x)2x
(a)
f —¦ g(x) = (x 2 + x)3 + x, g —¦ f (x) = (x 3 ’ x)2 + x
(b)
f —¦ g(x) = (x 3 ’x)2 +(x 3 ’x) and g —¦ f (x) = (x 2 +x)3 ’(x 2 +x)
(c)
f —¦ g(x) = (x 2 + x) · (x 3 ’ x) and g —¦ f (x) = (x 2 + x)/(x 3 ’ x)
(d)
f —¦ g(x) = (x 2 + x) + (x 3 ’ x) and g —¦ f (x) = (x 3 ’ x)x +x
2
(e)

Let f (x) = 3 x + 1. Then
20.
f ’1 (x) = x 3 ’ 1
(a) √
f ’1 (x) = 3 x ’ 1
(b)
317
Final Exam

f ’1 (x) = x 3 ’ x
(c)
f ’1 (x) = x/(x + 1)
(d)
f ’1 (x) = x 3 ’ 1
(e)
a 3 · b’2
21. The expression ln 4 ’3 simpli¬es to
c /d
3 ln a ’ 2 ln b ’ 4 ln c + 3 ln d
(a)
3 ln a + 2 ln b + 4 ln c ’ 3 ln d
(b)
4 ln a ’ 3 ln b + 2 ln c ’ 4 ln d
(c)
3 ln a ’ 4 ln b + 3 ln c ’ 2 ln d
(d)
4 ln a ’ 2 ln b + 2 ln c + 2 ln d
(e)
2 ’ln b3
The expression eln a
22. simpli¬es to
2a · 3b
(a)
2a
(b)
3b
a 2 · b3
(c)
a2
(d)
b3
6a 2 b3
(e)
x2 if x < 1
The function f (x) =
23. has limits
if x ≥ 1
x
2 at c = 1 and ’1 at c = 0
(a)
1 at c = 1 and 4 at c = ’2
(b)
0 at c = 0 and 3 at c = 5
(c)
’3 at c = ’3 and 2 at c = 1
(d)
1 at c = 0 and 2 at c = 2
(e)
x
The function f (x) = 2
24. has limits
x ’1
(a) 3 at c = 1 and 2 at c = ’1
(b) ∞ at c = 1 and 0 at c = ’1
(c) 0 at c = 0 and nonexistent at c = ±1
(d) 2 at c = ’2 and ’2 at c = 2
(e) ’∞ at c = 1 and +∞ at c = ’1
x3 if x < 2
The function f (x) = √
25. is continuous at
if x ≥ 2
x
x = 2 and x = 3
(a)
318 Final Exam

= 2 and x = ’2
(b) x
= ’2 and x = 4
(c) x
= 0 and x = 2
(d) x
= 2 and x = 2.1
(e) x
The limit expression that represents the derivative of f (x) = x 2 + x at
26.
c = 3 is
[(3 + h)2 + (3 + h)] ’ [32 + 3]
(a) lim
h
h’0
[(3 + 2h)2 + (3 + h)] ’ [32 + 3]
(b) lim
h
h’0
[(3 + h)2 + (3 + h)] ’ [32 + 3]
(c) lim
h2
h’0
[(3 + h)2 + (3 + 2h)] ’ [32 + 3]
(d) lim
h
h’0



Y
[(3 + h)2 + (3 + h)] ’ [32 + 4]
(e) lim
FL
h
h’0
x’3
If f (x) = 2
27. then
AM

x +x
1
(a) f (x) =
2x + 1
TE



x2 ’ x
(b) f (x) =
x’3
(c) f (x) = (x ’ 3) · (x 2 + x)
’x 2 + 6x + 3
(d) f (x) =
(x 2 + x)2
x 2 + 6x ’ 3
(e) f (x) =
x2 + x
If g(x) = x · sin x 2 then
28.
(x) = sin x 2
(a) f
(x) = 2x 2 sin x 2
(b) f
(x) = x 3 sin x 2
(c) f
(x) = x cos x 2
(d) f
(x) = sin x 2 + 2x 2 cos x 2
(e) f
If h(x) = ln[x cos x] then
29.
319
Final Exam

1
h (x) =
(a)
x cos x
x sin x
h (x) =
(b)
x cos x
cos x ’ x sin x
h (x) =
(c)
x cos x
h (x) = x · sin x · ln x
(d)
x cos x
h (x) =
(e)
sin x
If g(x) = [x 3 + 4x]53 then
30.
g (x) = 53 · [x 3 + 4x]52
(a)
g (x) = 53 · [x 3 + 4x]52 · (3x 2 + 4)
(b)
g (x) = (3x 2 + 4) · 53x 3
(c)
g (x) = x 3 · 4x
(d)
x 3 + 4x
(e) g (x) = 2
2x + 1
31. Suppose that a steel ball is dropped from the top of a tall building. It takes
the ball 7 seconds to hit the ground. How tall is the building?
(a) 824 feet
(b) 720 feet
(c) 550 feet
(d) 652 feet
(e) 784 feet
The position in feet of a moving vehicle is given by 8t 2 ’ 6t + 142. What
32.
is the acceleration of the vehicle at time t = 5 seconds?
12 ft/sec2
(a)
8 ft/sec2
(b)
’10 ft/sec2
(c)
20 ft/sec2
(d)
16 ft/sec2
(e)
Let f (x) = x 3 ’ 5x 2 + 3x ’ 6. Then the graph of f is
33.
concave up on (’3, ∞) and concave down on (’∞, ’3)
(a)
concave up on (5, ∞) and concave down on (’∞, 5)
(b)
concave up on (5/3, ∞) and concave down on (’∞, 5/3)
(c)
concave up on (3/5, ∞) and concave down on (’∞, 3/5)
(d)
concave up on (’∞, 5/3) and concave down on (5/3, ∞)
(e)
320 Final Exam

7
Let g(x) = x 3 + x 2 ’ 10x + 2. Then the graph of f is
34.
2
(a) increasing on (’∞, ’10/3) and decreasing on (’10/3, ∞)
(b) increasing on (’∞, 1) and (10, ∞) and decreasing on (1, 10)
(c) increasing on (’∞, ’10/3) and (1, ∞) and decreasing on
(’10/3, 1)
(d) increasing on (’10/3, ∞) and decreasing on (’∞, ’10/3)
(e) increasing on (’∞, ’10) and (1, ∞) and decreasing on
(’10, 1)
Find all local maxima and minima of the function h(x) = ’(4/3)x 3 +5x 2 ’
35.
4x + 8.
= 1/2, local maximum at x = 2
(a) local minimum at x
= 1/2, local maximum at x = 1
(b) local minimum at x
= ’1, local maximum at x = 2
(c) local minimum at x
= 1, local maximum at x = 3
(d) local minimum at x
= 1/2, local maximum at x = 1/4
(e) local minimum at x
Find all local and global maxima and minima of the function h(x) = x +
36.
2 sin x on the interval [0, 2π].
(a) local minimum at 4π/3, local maximum at 2π/3, global minimum
at 0, global maximum at 2π
(b) local minimum at 2π/3, local maximum at 4π/3, global minimum
at 0, global maximum at 2π

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