Name:
Quiz 2
Read all directions carefully. You must show all work to receive credit. No notes, book, calculators,
mp3 players or phones are allowed during this quiz. Write clearly and make sure to indicate your final
answer.
1. (4pts) Give the general antiderivative of f (x) = 2 sin x cos x +
f (x) =
2 sin x cos x +
3x7 − 2
x
=
2 sin x cos x +
3x7 2
−
x
x
=
2 sin x cos x + 3x6 −
=
2 sin x cos x + 3x6 − 2 ⋅
or
3x7 − 2
.
x
2
x
1
x
So,
F (x) =
=
or
sin2 (x) +
3 6+1
x − 2 ln ∣x∣ + C
6+1
3
sin2 (x) + x7 − 2 ln ∣x∣ + C
7
3
= − cos2 (x) + x7 − 2 ln ∣x∣ + C
7
2. (2pts) Which definite integral is the largest? Do NOT compute any of the integrals!!!!
27
3
∫3 x dx
27
27
3
∫3 (x + 1)dx
2
∫3 x dx
27
∫3
(x3 + 1)dx
because
x2 ≤ x3 ≤ x3 + 1 on [3, 27]
MTH 252
1
Quiz 2
Name:
5
3. (5pts) Using the limit definition and Riemann sums, evaluate ∫1 (4x − 3)dx
4
4i
∆x = and xi = 1 +
n
n
n
4i
4
5
lim ∑ [4 (1 + ) − 3]
∫1 (4x − 3)dx = n→∞
n
n
i=1
n
=
lim ∑ [(4 +
n→∞
i=1
n
=
lim ∑ [1 +
n→∞
i=1
4
16i
) − 3]
n
n
16i 4
]
n n
=
n
4 64i
lim ∑ [ + 2 ]
n→∞
n
i=1 n
=
lim ∑
n→∞
=
lim
64 n
1 n
∑4 + 2 ∑i
n i=1
n i=1
lim
1
64 n (n + 1)
⋅ 4n + 2 ⋅
n
n
2
n
=
n
4
64i
+ ∑ 2
i=1 n
i=1 n
n→∞
n→∞
32
1
64
n (n + 1)
= lim
⋅ 4
n +
⋅
2
n→∞ 2
n
n
=
=
=
MTH 252
lim 4 +
32 (n + 1)
n
lim 4 +
32n 32
+
n
n
n→∞
n→∞
lim 4 + 32 +
n→∞
=
4 + 32
=
36
32
n
2
Quiz 2