Stony Brook University  MAT 320 Midterm II
Solutions
1. (25 points) The functions
f
and
g
are continuous on the interval [
a, b
]
and differentiable on (
a, b
). If
f
(
a
) =
g
(
a
) and
f
(
b
) =
g
(
b
), prove that
there is a point
c
, with
a < c < b
, where
f
0
(
c
) =
g
0
(
c
).
•
Apply Rolle’s Theorem to
f

g
.
2. (25 points) The function sin
x
is infinitely differentiable on
R
, and its
derivatives cycle through cos
x,

sin
x,

cos
x,
sin
x
. Let
P
k
(
x
) be the
k
th Taylor polynomial, based at 0, for sin
x
. Prove that
lim
k
→∞
P
k
(
x
) = sin
x
for every
x
∈
R
.
•
Choose
x
. By Taylor’s Theorem, there exists
c
between 0 and
x
such that
sin
x

P
k
(
x
) =
f
(
k
+1)
(
c
)
x
k
+1
(
k
+ 1)!
.
Since

f
(
k
+1)
(
c
)
 ≤
1, we have

sin
x

P
k
(
x
)
 ≤ 
x

k
+1
/
(
k
+1)!. The
proof then follows from lim
n
→∞
a
n
/n
! = 0, true for any
a
∈
R
.
3. (25 points) The function
f
is continuous and twice differentiable on
the interval [
a, b
], with both derivatives positive there:
f
0
(
x
)
>
0 and
f
00
(
x
)
>
0 for every
a
≤
x
≤
b
. Suppose that
f
(
a
)
<
0 and
f
(
b
)
>
0.
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 Fall '10
 Mark
 Intermediate Value Theorem, Mean Value Theorem, Continuous function, Xn

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