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Calculus One Sample Multiple Choice Questions.
Choose the answer that best fits the given problem and shade the corresponding letter in the space provided on
your scantron from. Each problem is worth 4 points.
1. Let f ( x) =
x
( x − 4)
2
. Which of the statements in (a) through (d) is false?
(a) lim f ( x) does not exist
(b) lim f ( x) = +∞
(c) x = 4 is a vertical asymptote of the curve y = f (x)
(d) f (x) has a jump discontinuity at x = 4
x →4
x →4
(e) none, each of the statements above is true
x2 − 4
2. Evaluate lim−
.
x →2
x−2
(b) − 4
(a) + ∞
(d) − ∞
(e) none of these
(d) 1
(e) none of these
2−t − 2
.
t
3. Evaluate lim
t →0
(a) Does not exist
4. Assume that
(c) 4
3
(c) −
(b) +∞
1
2 2
8 − x 4 ≤ f ( x ) ≤ 3 − cos x for all x in the interval − 1 ≤ x ≤ 1 . Find lim f ( x ) .
(a) 1
x →0
(b) 2
(c) 3
(d) 0
(e) none of these
5. Let f ( x) = x and let n be any integer. Which of the statements in (a) through (d) is false?
(a) lim− x = n − 1
x→n
(b) f (x) is continuous at x = π
(d) f ( x) has a jump discontinuity at x = n
(e) none, each of the statements above is true
(c) n = n
Calculus One Sample Multiple Choice Questions.
The next two problems refer to the function y = f ( x ) whose graph is pictured below.
y
x
6. The discontinuity at x = 3 is called
(a) a removable discontinuity
(b) an infinite discontinuity
(d) a jump discontinuity
(e) none of these
(c) a reprehensible discontinuity
7. lim f ( x ) =
x→3
(b) +∞
(a) DNE
(c) 2
(d) 1
(e) none of these
8. Suppose that lim g ( x) = 2. Which of the statements in (a) through (d) must be true?
x →7
(a) g (x) is both continuous from the left at x = 7 and continuous from the right at x = 7
1
1
=
(c) g (7) = 2
(d) (a), (b), and (c) must be true
(b) lim
x →7 g ( x )
2
(e) none of the statements above has to be true.
x −1
. Which of the statements in (a) through (d) is false?
x ( x + 2)
(a) f has vertical asymptotes at x = 0 and at x = −2
(b) lim f ( x) = −∞
9. Let f ( x) =
2
x →0
(d) f has infinite discontinuities at x = 0 and at x = −2
(e) none, each of the statements above is true
(c) lim f ( x) = +∞
x →−2
Calculus One Sample Multiple Choice Questions.
10. For this problem let f and g be the functions whose graphs are shown below. Find u ′(3)
if u ( x ) = g ( f ( x ) ) .
f
(a) −
2
3
g
(b) 0
(c)
2
3
(d) −
1
9
(e) none of these
11. A table of values for f, g, f ′ , and g ′ is given. Below. Find H ′(1) if H ( x ) = g ( f ( x )) .
f (x )
3
1
7
x
1
2
3
(a) 36
g (x )
2
8
2
(b) 32
f ′( x )
2
5
7
g ′( x )
6
7
8
(c) 16
(d) 8
12. Find an equation of the normal line to the curve y =
1
5
(a) y = − x +
2
2
(b) y = −2 x + 4
(c) y =
(e) none of these
3x + 1
at the point (1, 2).
x2 +1
1
3
x+
2
2
(d) y = 2 x
(e) none of these
13. Find and simplify y ′′ if x 4 + y 4 = 1 .
(a)
3x 2 y 4 + 3 x 6
y7
(b) −
2x 2
y5
6x 4
y7
(e) none of these
(c) − 7130 cos 7 x
(d) 7130 cos 7 x
(e) none of these
(c) ± x
(d) 1
(e) none of these
(c) −
3x 2
y7
(d) −
14. Find D130 (sin 7 x)
(a) − 7130 sin 7 x
(b) 7130 sin 7 x
15. Find f ′ ( x ) if f ( x ) = x .
(a) x
(b) − x
Calculus One Sample Multiple Choice Questions.
16. If g is a differentiable function, find an expression for the derivative of y =
(a)
g ( x ) − x g ′( x )
x
(b)
2
x g ′( x ) − g ( x )
x
(c)
2
1
g ′( x )
(d)
g ( x ) − x g ′( x )
x
.
g (x )
(e)
[g (x )]
2
x g ′(x ) − g (x )
[g (x )]2
17. A particle moves according to the law of motion s = t 3 − 12t + 3, t ≥ 0 , where t is measured in seconds and
s in feet. Find the total distance traveled by the particle in the first 3 seconds.
(a) 21 ft
(b) 22 ft
(c) 23 ft
(d) 24 ft
(e) none of these
18. The following limits represents the derivative of some function at some number a. Find the function f ( x )
and the number a .
1
1
−
3.
lim 3 + h
h→ 0
h
(a) f ( x ) = x , a = 3
(d) f ( x ) = ( x + 3)
−1 2
(b) f ( x ) = x −1 2 , a = 3
, a=3
(c) f ( x ) =
x, a = 3
(e) none of these
For the next five problems consider the following situation: A particles moves according to the law s = f ( t ) ,
where t is measured in seconds and s is in meters. Suppose the following data are given.
t
0
1
2
3
4
5
6
7
s
0
25
32
27
16
5
0
7
ds / dt
36
15
0
-9
-12
-9
0
15
d 2 s / dt 2
-24
-18
-12
-6
0
6
12
18
19. At what time(s) is the particle at rest?
(a) 0
(b) 2 and 6
(c) 0 and 6
(d) 4
(e) 0, 2, 4, and 6
Calculus One Sample Multiple Choice Questions.
20. At t = 0 the particle is:
(a)
(b)
(c)
(d)
(e)
moving in the negative direction and slowing down.
moving in the positive direction and speeding up.
moving in the positive direction and slowing down.
moving in the negative direction and speeding up.
none of these
21 At t = 3 the particle is:
(a)
(b)
(c)
(d)
(e)
moving in the negative direction and slowing down.
moving in the positive direction and speeding up.
moving in the positive direction and slowing down.
moving in the negative direction and speeding up.
none of these
22. At t = 5 the particle is:
(a)
(b)
(c)
(d)
(e)
moving in the negative direction and slowing down.
moving in the positive direction and speeding up.
moving in the positive direction and slowing down.
moving in the negative direction and speeding up.
none of these
23. What is the speed of the particle when the acceleration is 0?
(a) − 12
(c) ±12
(b) 12
(d) 0
(e) none of these
For the next three problems let f be a continuous function whose derivative is sketched below.
y'
2
1
x
1
2
3
4
5
6
-1
-2
24. On what intervals is f increasing?
(a)
( 0, 4 )
(b)
( 0, 2 ) , ( 2, 4 )
(c) (1, 2 ) , ( 3,5 )
(d) (1, 2 )
(e) none of these
Calculus One Sample Multiple Choice Questions.
25. At what value(s) of x does f have a local maximum?
(a) x = 2
(b) x = 4
(c) x = 5
(d) (a) and (b) are correct
(e) (a) and (c) are correct
26. State the x-coordinates of all inflection points.
(a) x = 1
(b) x = 2.5
27. Evaluate: lim
x → −∞
(a)
x+2
2x 2 + 3
(c) x = 5.5
(d) (a), (b), and (c) are correct
(e) x = 4
.
2
(b)
3
1
(c) −
2
1
2
(d) 0
(e) none of these
28. Which of the following is true at x = c about the twice differentiable function pictured below.
c
(a) f > 0, f ′ > 0, and f ′′ > 0
(b) f < 0, f ′ < 0, and f ′′ < 0
(d) f < 0, f ′ > 0, and f ′′ > 0
(e) f < 0, f ′ < 0, and f ′′ > 0
(c) f < 0, f ′ > 0, and f ′′ < 0
29. Let f be a differentiable function whose domain is all real numbers. The sign graph for f ′ near x = 1 is
shown below. What can be said about the function at x = 1?
++++++ − − − − − −
sign of f ′
1
(a) x = 1 is a critical number of f
(d) (a) and (b) are correct
(b) f has a local maximum at x = 1 (c) f has an absolute maximum at x = 1
(e) (a), (b), and (c) are correct
30. Find the absolute maximum and absolute minimum values of f on the given interval.
f ( x) = sin x + cos 2 x,
(a) maximum value = 1, minimum value = 0
(c) maximum value =
(e) none of these
5
4
, minimum value = 0
[− π 2 , π 2]
(b) maximum value =
5
4
, minimum value = -1
(d) maximum value = 1, minimum value = -1