CALCULUS BC – Worksheet #1
dy
In each of problems 1–27 a function is given. In each problem, choose the alternative that is the derivative, dx , of the
function.
1. y = (4x + 1)2(1 – x)3
A) (4x + 1)2(1 – x)2(5 – 20x)
B) (4x + 1)(1 – x)2(4x + 11)
2
•C) 5(4x + 1)(1 – x) (1 – 4x)
D) (4x + 1)(1 – x)2(11 – 20x) E) –24(4x + 1)(1 – x)2
2–x
7
6x – 5
9
7
7 – 6x
2. y = 3x + 1
•A) –
B)
C) –
D)
E)
(3x + 1)2
(3x + 1)2
(3x + 1)2
(3x + 1)2
(3x + 1)2
1
1
(3 – 2x)3/2
1
2
3. y = 3 – 2x
A)
•B) –
C) –
D) – 3 – 2x
E) 3 (3 – 2x) 3/2
3
2 3 – 2x
3 – 2x
4. y =
2
(5x + 1)3
A) –
30
(5x + 1)2
5. y = 3x2/3 – 4x1/2 – 2 A) 2x1/3 – 2x–1/2
6.
7.
8.
9.
10.
11.
•E) 2x–1/3 – 2x–1/2
1
4x – 1
1
1
4
1
y=2 x –
A) x +
B) x–1/2 + x–3/2 C)
•D)
+
E)
+
2 x
x x
4x x
x
4x x
x
x x
x
+
1
1
x
+
1
y = x2 + 2x – 1
•A) y
B) 4y(x + 1) C)
D) – 2
E) none of these
(x + 2x – 1)3/2
2 x2 + 2x – 1
x
1 – 2x2
1
1
1 – 2x2
y=
A)
B)
C)
D)
•E) none of these
(1 – x2)3/2
1 – x2
(1 – x2)1/2
1 – x2
1 – x2
y = cos x2
A) 2x sin x2 B) –sin x2
C) –2 sin x cos x •D) –2x sin x2 E) sin 2x
2
2
y = sin 3x + cos 3x
A) –6 sin 6x •B) 0 C) 12 sin 3x cos 3x D) 6(sin 3x + cos 3x) E) 1
x
e
ex
1
1
ex – 2
y = ln x
A) x – x
B) x
•C)
D)
0
E)
e –1
e –1
e –1
1 – ex
ex – 1
A)
13. y = ln(sec x + tan x)
14. y = cos2 x
ex – e–x
15. y = x
e + e–x
17.
18.
19.
20.
21.
C)
1
x
12. y = tan–1 2
16.
–6
10
30
D) – 3 (5x + 1) –4/3 E)
(5x + 1)4
(5x + 1)4
9
2
2
B) 3x–1/3 – 2x–1/2 C) 5 x5/3 – 8x3/2 D) 1/3 – 1/2 – 2
x
x
•B) – 30(5x + 1)–4
4
4 + x2
B)
1
2 4 – x2
•A) sec x
C)
1
B) sec x
2
4 – x2
D)
1
2 + x2
sec2 x
C) tan x + tan x
•E)
2
2
x +4
1
D) sec x + tan x
1
E) – sec x + tan x
A) – sin2 x
•C) – sin 2x D) 2 cos x E) –2 sinx
4
1
A) 0
•D) x
E) 2x
(e + e–x)2
e + e–2x
x
1
2x2 + 1
2x2 + 1
y = ln x x2 + 1
A) 1 + 2
•B)
C)
D)
E) none of these
x +1
x(x2 + 1)
x x2 + 1
x x2 + 1
1
x
1
1
1
y = ln x + x2 + 1
A) x + 2
B)
C) 1 •D) x2 + 1
E) x +
x +1
2
x +1
2 x2 + 1
1
1
1
2
1
1
1
1
1
y = x2 sin x (x ≠ 0)
A) 2x sin x – x2 cos x
•B) – x cos x
C) 2x cos x
D) 2x sin x – cos x
E) – cos x
1
1
cos 2x
y = 2 sin 2x
•A) – csc 2x cot 2x B) 4 cos 2x
C) –4 csc 2x cot 2x D)
E) – csc2 2x
2 sin 2x
2
ln x
2(ln x) y
2y
y = xln x ( x > 0)
A) x
B) 2 x
•C)
D) x
E) (ln x) xln x–1
x
1
x
x
1–x
–1
y = xtan–1 x – ln x2 + 1
A) 0 B)
– 2
•C) tan–1 x D)
2 + tan x – x E) 1 + x2
2
x
+
1
1
+
x
1–x
(
B) 2 sin x cos x
2
B) 1 C) x
(e + e–x)2
)
(
)
22. y = e–x cos 2x
•A) –e–x(cos 2x + 2 sin 2x) B) e–x(sin 2x – cos 2x)
D) –e–x(cos 2x + sin 2x) E) –e–x sin 2x
C) 2e–x sin 2x
sec x tan x
tan x
sec2 x tan x
B)
C) 2 sec x tan2 x
•D)
x
x
x
2
3 ln x
24. y = x ln3 x A)
B) 3 ln2 x C) 3x ln2 x + ln3 x D) 3(ln x + 1) • E) none of these
x
23. y = sec2 x
A)
E) 2 sec2 x tan x
1 + x2
25. y =
1 – x2
A) –
26. y = ln (x 2)
2
A) x
27. y = sin–1 x –
1 – x2
CALCULUS BC – Worksheet #1
4x
–4x3
2x
•B)
C)
D)
(1 – x2)2
(1 – x2)2
1 – x2
4x
(1 – x2)2
B)
A)
1
2x
1
C) 2x
1
1
•D) x
1
x
E)
2
B)
4
1 – x2
E)
1+x
x2
E)
2 1 – x2
1 – x2
1 – x2
1 – x2
In each of Problems 28–31 a pair of equations is given which represents a curve parametrically.
dy
In each, choose the alternative that is the derivative dx .
sin t
1–cos t
sin t
1–x
1–cos t
28. x = t – sin t and y = 1 – cos t
•A) 1–cos t B) sin t
C) cos t–1 D) y
E) t–sin t
29. x = cos3 θ and y = sin3
A) tan3
B) – cot
C) cot
•D) – tan
E) – tan2
–t
e
30. x = 1 – e–t and y = t + e–t
A)
B) e–t –1 C) et+1 D) et–e–2t •E) et–1
1–e–t
1
1
31. x = 1 – t and y = 1 – ln(1 – t) (t < 1) A) 1 – t
•C)
1
B) t – 1 • C) x
D)
D)
(1 – t)2
t
1
1+x
E) 1 + lnx
dy
In each of the Problems 32-35, y is a differentiable function of x. In each, choose the alternative that is the derivative dx .
3x2
x – 3y2
3x2 – 1
1 – 3y2
32. x3 – xy + y3 = 1
A)
33. x + cos (x + y) = 0
•A) csc (x + y) – 1
34. sin x – cos y – 2 = 0
B)
A) – cot x
•C)
y – 3x2
3y2 – x
B) csc (x + y)
B) –cot y
D)
3x2 + 3y2 – y
x
x
C) sin (x + y)
cos x
C) sin y
D)
E)
3x2 + 3y2
x
1
1 – x2
•D) – csc y cos x
E)
E)
1 – sin x
sin y
2 – cos x
sin y
3x + y
A) x – 5y
y – 3x
3x + 4y
•B) 5y – x
C) 3x + 5y D)
E) none of these
x
d2y
1
36. If x = t2 – 1 and y = t4 – 2t3, then when t = 1,
is A) 1 B) –1 C) 0 D) 3 •E) 2
dx2
35. 3x2 – 2xy + 5y2 = 1
37. If ƒ(x) = x4 – 4x3 + 4x2 – 1, then the set of values of x for which the derivative equals zero is
A) {1, 2} B) {0, –1, –2} C) {–1, +2} D) {0} •E) {0, 1, 2}
3
1
38. If ƒ(x) = 16 x , then ƒ ‘ ‘ ‘ (4) is equal to
•A) 16
B) – 4 C) – 2
D) 0 E) 6
2
24
6
1
39. If ƒ(x) = ln x, then ƒiv (x) is
A) 3
B) 5
C) 4
D) – 4
•E) none of these
x
x
x
x
d2y
1
1
40. If a point moves on the curve x2 + y2 = 25, then, at (0, 5),
is A) 0 B) 5
C) – 5 •D) – 5
2
dx
E) nonexistent
d2y
is
dt2
A) ac2(sin t + cos t) • B) – c2y C) – ay D) – y E) a2c2 sin ct – b2c2 cos ct
42. If ƒ(x) = x4 – 4x2, then ƒ(iv) (2) equals A) 48 B) 0 •C) 24 D) 144 E) 16
x
43. If ƒ(x) =
, then the set of x’s for which ƒ ‘ (x) exists is A) all reals B) all reals except x = 1 and x = –1
(x – 1)2
41. If y = a sin ct + b cos ct, where a, b, and c are constants, then
C) all reals except x = –1
44. If y = (x – 1)2 ex, then
d2y
=
dx2
1
D) all reals except x = 3 and x = –1
A) ex(x – 1)2
df
45. If ƒ(x) = e–x ln x, then, when x = 1, dx is
46. If y =
• E) all reals except x = 1
B) ex(x2 – 2x – 1) • C) ex(x2 + 2x – 1)
A) 0
B) nonexistent
x2 + 1 , then the derivative of y2 with respect to x2 is
•A) 1
2
C) e
B)
1
•D) e
x2 + 1
2x
C)
D) 2ex(x – 1)
E) none
E) e
x
2
2(x + 1)
2
D) x
E)
x2
x2 + 1
CALCULUS BC – Worksheet #1
1
47. If ƒ(x) = 2
and g(x) = x , then the derivative of ƒ(g(x)) is
x +1
A)
– x
2
(x + 1)2
48. If x =
eθ cos
•B) – (x + 1)–2
and y = e sin
C)
– 2x
2
(x + 1)2
, then, when
49. If x = cos t and y = cos 2t, then
1
1
E)
2
2 x (x + 1)
(x + 1)
π dy
= 2 , dx is A) 1 B) 0 C) eπ/2
D)
d2y
is A) 4 cos t
dx2
• B) 4
4y
C) x
D) –4
D) nonexistent
•E) –1
E) –4 cot t
1
50. If y = x2 + x, then the derivative of y with respect to 1 – x is
•A) (2x + 1)(x – 1)2
B)
2x + 1
(1 – x)2
C) 2x + 1
D)
3–x
(1 – x)3
E) none of these
(1+ h) 6 − 1
is A) 1 B) -1 •C) 6 D) ∞ E) nonexistent
h
3
8+ h −2
1
lim
52. h →0
A) 0 •B) 12 C) 1 D) 192 E) ∞
is
h
1
lim ln(e + h) −1
53. h →0
is
A) 0 • B) e C) 1 D) e E) nonexistent
h
cos
x
−1
lim
54. x →0
is
A) –1 •B) 0 C) 1 D) ∞ E) none of these
x
51.
lim
h →0
55. The function ƒ(x)=x2/3 on [–8,8] does not satisfy the conditions of the mean–value theorem because
A) ƒ(0) is not defined B) ƒ(x) is not continuous on [–8,8] C) ƒ ’ (–1) does not exist
D) ƒ(x) is not defined for x<0 •E) ƒ ’ (0) does not exist
56. If ƒ(a) =ƒ(b)=0 and ƒ(x) is continuous on [a,b], then
A) ƒ(x) must be identically zero
•B) ƒ ’ (x) may be different from zero for all x on [a,b]
C) there exists at least one number c, a < c < b, such that ƒ ’ (c) = 0
D) ƒ ’ (x) must exist for every x on (a,b)
E) none of the preceding is true
57. If c is the number defined by Rolle’s Theorem, then for ƒ(x)=2x3–6x on the interval
0 ≤ x ≤ 3 , c is
•A) 1 B) –1 C) 2 D) 0 E) 3
1
1
1
58. If h is the inverse function of ƒ and if ƒ(x) = x , then h’ (3) =
A) –9 •B) –9 C) 9 D) 3 E) 9
59. Suppose y = ƒ(x) = 2x3–3x. If h is the inverse function of ƒ, then h’(y) =
1
1
1
1
–6x2–3
A)
B)
– 3 •C)
D)
E) none of these
2
2
2
6y –3
6x
6x –3
(2x2–3)2
dy
60. Suppose y=ƒ(x) and x=ƒ–1(y) are mutually inverse functions. If ƒ(1)=4 and dx = – 3 at
dx
x = 1, then dy at y = 4 equals
1
•A) – 3
1
B) – 4
1
C) 3
D) 3
E) 4
dx
61. Let y=ƒ(x) and x=h(y) be mutually inverse functions. If ƒ ’ (2) = 5, then what is the value of dy at
y = 2?
A) –5
1
B) –5
1
C) 5
D) 5
62. If ƒ(x) = xsinx for x > 0, then ƒ ’ (x) =
A) (sinx)xsinx–1 B) xsinx(cosx)(lnx)
C)
•E) It cannot be determined from the information given.
sinx
sinx⎡sinx + (cosx)(lnx)⎤
⎣ x
⎦
x + (cosx)(lnx) •D) x
E) xcosx + sinx
CALCULUS BC – Worksheet #1
63. Suppose
lim
x →0
g(x) − g(0)
= 1. It follows that
x
A) g is not defined at x = 0
B) g is not continuous at x = 0
C) The limit of g(x) as x approaches 0 equals 1
•D) g ’ (0) = 1
E) g ’ (1) = 0
dy
sec(xy)
sec(xy)–y
1+sec(xy)
64. If sin(xy) = x, then dx =
A) sec(xy) B)
•C)
D) –
E) sec(xy) – 1
x
x
x
1
lim
x
65. x →0 + x =
A) 0 B) e •C) 1 D) e E) none of these