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Review of Calculus BC I
Math
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1. Let
f
(
x
) =
x
)
for
x
> 0
, and
f
(
x
) = 0
for
x
≤ 0
. Which of the following is true?
f
is continuous
f
is continuous at all points
x
except
0
f
is continuous only at the points
x
> 0
.
f
is continuous only at the points where
x
≤ 0
.
2. Compute
.
2
2/5
1/2
3/5
3. Let
g
(
x
) = 1/
x
(with domain

x
 > 0
). Which of the following statements is true of
g
?
g
is not differentiable
g
is continuous at all but one point of its domain
g
fails to be continuous at infinitely many
x
values.
g
is continuous on its entire domain
4. The function
f
(
x
) = sin(
x
)
is invertible when defined on which of the following domains?
[
Π
, 0]
[
pi
/2,
Π
/2]
[0,
Π
]
[0, 2
Π
]
5. What is the value of
x
sin(1/
x
)
?
1
0
pi
undefined
6. Let
f
(
x
) =
ax
+
b
. The inverse function of
f
(
x
)
has slope
1/
a

a
a
/
b
b
/
a
7. Let
f
(
x
) =
x
^{2}
for
x
≠1
and let
f
(1) = 2
. Then
f
(
x
)
equals
2
3/2
0
1
8. The degree of the product of two degree
3
polynomials is equal to
9
27
6
3
9. Which of the following points is equal to the image of some real number in the domain of the function
f
(
x
) = 2 sin(2
x
)
?
2
3
2
Π
Π
10. The inverse function of
f
(
x
) = (1/3)
e
^{2x}
is
( 1/2)log(3
x
)
(1/2)log(3
x
)
(1/2)log(
x
/3)
(1/3)log(
x
/2)
11. Find the derivative of
sin(
e
^{x}
)
.
x
cos(
e
^{x}
)
e
^{x}
cos(
e
^{x}
)

e
^{x}
cos(
e
^{x}
)
( 1/
e
^{x}
)cos(
e
^{x}
)
12. Compute the derivative of
.
13. Find the derivative of
cos(sin(
x
)))
.
cos(sin(
x
))sin(
x
)
sin(sin(
x
))sin(
x
)
sin(sin(
x
))cos(
x
)
sin(sin(
x
))cos(
x
)
14. Compute the derivative of
25
x
^{2}
+ 4
.
50
x
50
x
^{2}
50
x
^{2}
+ 4
25
x
15. What is the derivative of
log
x
?
(log
x
)/
x
log 1
1/
x
1/
x
16. What is the derivative of

x

at
x
= 0
?
1
1
0
undefined
17. Find the derivative of
e
^{2x+1}
.
(2
x
+ 1)
e
^{2x+1}
2
e
^{2x+1}
2
e
^{2x}
(2
x
+ 1)
e
^{2x}
18. Compute the derivative of
1/(
x
^{2}
+ 1)
.
(2
x
)/(
x
^{2}
+1)
^{2}
(2
x
)/(
x
^{2}
+ 1)
( 2
x
)/(
x
^{2}
+1)
^{2}
1/(2
x
)
19. What is the derivative of
at
x
= 1/2
?
1/
1/
1/
1/2
20. Find the third derivative of
f
(
x
) = 2 sin(3
x
)
.
54 cos(3
x
)
18 cos(3
x
)
18 sin(3
x
)
54 cos(3
x
)
21. What is the derivative of
tan(
x
)
at
x
=
Π
/2
?
undefined
1
0
1
22. Compute the derivative of
f
(
x
) = (
x
^{2}
+1)log
x
.
(
x
^{2}
+1)/
x
 2
x
log
x
(
x
^{2}
+1)
e
^{x}
+2
x
log
x
(
x
^{2}
+1)
e
^{x}
2
x
log
x
(
x
^{2}
+1)/
x
+ 2
x
log
x
23. Find
f'
(
x
)
, where
f
(
x
) =
e
^{x}
+
x
.
1
undefined
0
1
24. Compute the derivative of
x
^{3}
+
x
+ 1/
x
+ 1/
x
^{3}
.
3
x
^{2}
+ 1
3
x
^{2}
+1  1/
x
^{2}
3/
x
^{4}
3
x
^{2}
+1 + 1/
x
^{2}
+3/
x
^{4}
3
x
^{2}
+1 + 1 + 1/(3
x
^{2}
)
25. Find the values of
x
at all the critical points
f
(
x
) =
x
^{3}
/3 
x
^{2}
+
x
.
1
,
2
0
,
2
1
0
,
1
26. Find the derivative of
1/tan(
x
)
.
x
tan(
x
)
1/(tan
^{2}
(
x
))
1/(sin
^{2}
(
x
))
1/(tan
^{2}
(
x
))
27. What is the minimum value of
e
^{x}

x
?
0
1
does not exist
1
28. Find the maximum value of
sin(
x
) + cos(
x
)
.
2
1
29. Find the minimum value of
x
^{2}
 6
x
+ 11
on the interval
[2, 5]
.
2
3
4
2
30. Find all values of
x
at which the derivative of
f
(
x
) = log
x
 
x
is zero.
1
1
,
1
0
1
31. Use implicit differentiation to find the slope of the tangent line to the graph of the equation
x
^{2}
y
^{4}
= 2
x
^{2}
+
y
at the point
(1,  1)
.
2/5
2/5
5/2
5/2
32. Which of the following is an antiderivative of
1/
x
^{2}
?
(
x
 1)/
x
(
x
^{2}

x
)/
x
x
/(
x
+ 1)
(
x
+ 1)/
x
33. What is the
y
intercept of the tangent line to the graph of
x
^{2}
at
x
= 3
?
9
3
3
9
34. Find the second derivative of
x
log
x
.
1 + 1/
x
1 + 1/
x
1/
x
1/
x
^{2}
35. What is the
x
intercept of the tangent line to the graph of
e
^{x}
at
x
= 0
?
2
1
0
1/2
36. Compute the maximum vertical distance between the graphs of
cos(
x
)
and
sin(
x
)
.
1/
1

37. If an ostrich's velocity at time
t
is given by
log(
t
^{2}
)
, what is the ostrich's acceleration at time
t
= 4
?
2
3
1
1/2
38. What are the values of
x
at inflection points of
x
^{3}
3
x
^{2}
+
x
+ 1
?
1
1
,
1
0
,
1
1
,
0
39. Compute
L
_{2}
(sin(
x
), 
Π
, 0)
.

Π
/2

Π
Π
/2
0
40. Find
L
_{4}
(sin(
x
), 
Π
, 0)
.
Π
( 1  1/
)
(
Π
/2)(1  1/
)
(
Π
/2)( 1  1/
)
(
Π
/2)( 1 +
)
41. Compute
4
x
^{3}
dx
.
19
16
65
81
42. Find
cos(
x
)
x
.
1
0
Π

Π
43. Compute
(1/
x
)
dx
.
(log 2)/2
log 1
log 2
log 3
44. Find the upper Riemann sum
U
_{48}
(5, 1, 2)
.
10
12
3
5
45. Compute
(1  cos
^{2}
(
x
))
dx
+
( sin
^{2}
(
x
))
dx
.
2
Π
0
1

46. Which of the following is an antiderivative of
4(
x
+ 1)
^{3}
?
(
x
+ 1)
^{4}
+ 3
4(
x
 1)
^{3}
5(
x
+ 1)
^{4}
+ 1
5(
x
+ 1)
^{4}
47. Find
t
^{2}
 1
dt
.
t
^{2}
 1
x
^{2}
 1
2
x
2
t
48. Compute
sin(2
x
)
dx
.
Π

Π
1
0
49. What is
( 1/
x
^{2}
)
dx
?
1
undefined

Π
2
50. The term 'indefinite integral' is equivalent to which of the following terms?
definite integral
implicit derivative
antiderivative
indefinite derivative
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